That question now looms over American politics after a week in which the party was thrust out of power in Washington DC and ripped in two by Donald Trump’s anti-democratic demands.
With the double victory for Democratic senators in Georgia, the Republicans lost their control of the Senate, just as they have done with the White House and the House of Representatives.
With the unprecedented scenes on Tuesday, a day of bloodshed in the US Capitol and a president vowing to overturn an election, the Trumpists and traditionalists finally split.
You need only peer back a month or two into the past to find a time when Donald Trump and his Republican congressmen were still largely joined at the hip.
The marriage had always been one of convenience. Almost every sitting Republican senator and congressmen opposed Mr Trump winning their party’s presidential nomination in 2016.
But installed in the White House after a shock election victory and with a vice-like grip on the Republican base, Mr Trump’s political power over his congressmen was immense.
There were moments on policy that they were willing to stand up to him en masse. When he sided with Vladimir Putin over US intelligence chiefs on Russian election meddling, for example, or his proposed troop withdrawal from Syria.
But those senators who chose to repeatedly challenge him publicly, such as Jeff Flake of Arizona or Bob Corker of Tennessee, were forced to retire for fear of a primary challenger.
Even Mitt Romney, now the standard-bearer of moderate Republicanism and a fierce critic of Mr Trump, accepted the president’s endorsement when he first ran to be Utah’s senator in 2018. But this week the gulf cracked back open.
With the binding agent of Oval Office occupancy almost gone, Mr Trump’s undemocratic demand for Congress to reject the election result proved a step too far.
No more so is this embodied than the disintegration of the Trump-Mike Pence relationship. The US vice president has been obsequiously loyal to Mr Trump throughout their four years in the White House.
Unlike many of the Trump inner circle, who snipe at him behind his back to reporters, Mr Pence is renowned for keeping schtum and invariably defends his boss during controversies.
So it was all the more remarkable that the president, spurned by the courts and Republican state legislatures in his bid to falsely overturn Joe Biden’s election win, flipped on Mr Pence.
The president had become convinced that Mr Pence, in his role overseeing proceedings in the Senate as the election results were confirmed, could intervene to hand him victory.
Time and time again in tweets last week and then when speaking before the cheering crowd on that fateful Wednesday Mr Trump implored and threatened his vice president to act.
“Mike Pence, I hope you’re going to stand up for the good of our Constitution and for the good of our country,” Mr Trump said. “If you’re not, I’m going to be very disappointed in you.”
Mr Pence resisted, saying rejecting the election results would be unconstitutional. Mr Trump publicly damned him. Then the pro-Trump mob swarmed the US Capitol and Mr Pence went into hiding.
The vice president, who reportedly did not even receive a call from Mr Trump to check on his safety as he hid in place, was left apoplectic, according to one long-time friend.
The Republican senator James Inhofe said Mr Pence expressed a sense of betrayal by Mr Trump “after all the things I’ve done for him”.
In a twist of fate, it is Mr Pence who – if he wishes to use it – has the political leverage to force Mr Trump from office early via the 25th Amendment, in theory at least.
If he concluded that was the best path and convinced a majority of the cabinet likewise then Mr Pence could remove Mr Trump. However the signs are he has decided against the move.
Mr Pence is not alone on the Right in breaking with Mr Trump. Rupert Murdoch’s Wall Street Journal has called on him to resign before the January 20 inauguration to avoid impeachment.
Republicans as varied as John Kelly, the retired general who was Mr Trump’s chief of staff, and Larry Hogan, the Maryland governor and frequent critic, want him to be removed.
But any moderates convinced this week will begin the end of Mr Trump’s insurgent takeover of their party would best pause and assess some stark realities also on display.
Even after Congress reconvened following the violence to certify the election results, eight Republican senators and more than 130 Republican House members voted against doing so.
Ted Cruz and Josh Hawley, the senators for Texas and Missouri who have adopted Trumpian rhetoric as they seek the 2024 Republican nomination, led the charge in the Senate, to condemnation from some fellow senators.
The fact that Mr Trump raised his vote total by 11 million between the 2016 to 2020 elections is a reminder of the sway he held just a few months ago over the party faithful. Much will depend on whether that grip remains.
A shock poll from YouGov showed that 45 per cent of Republicans supported the storming of the Capitol, despite the outcry.
Mr Trump appears intent on continuing a political career of some kind. Little-noted in his Thursday night conciliation video was a line near the end explicitly talking to supporters: “Our incredible journey is only just beginning.”
Will that be within the Republican Party? Perhaps. Mr Trump is widely speculated to seek the party’s 2024 presidential nomination and has teased as much in rallies this week.
But so too could he declare political civil war. Addressing his supporters before their march on the Capitol, he decried Republicans who did not support his false claims of election victory.
“They’re weak Republicans. They’re pathetic Republicans,” he said. He vowed to “primary the Hell” out of those not with him, meaning challenging them when they face re-election.
Less talked about but perhaps just as significant in the battle for the soul of the Republican Party is the other side.
Who will be stand up to be the anti-Trump, the fresh face of moderate Republicanism who can become a rallying point for traditionalists and perhaps even win the 2024 nomination?
There are names on that wing getting attention. Tim Scott, the party’s only black senator who has an inspiring personal backstory and gave a widely praised speech at the party’s summer convention, is one.
Ben Sasse, the Nebraska senator who was scathing about Mr Trump in a leaked phone call before the election and called his actions this week “wicked”, is another.
Then there are old stalwarts who may not seek the presidential nomination, like Mr Romney, the defeated 2012 nominee, but have a position of prominence to shape the party’s future.
Ultimately, it is the voters that will decide. After half a decade marching to Mr Trump’s tune, will the Republican base stick with him after Wednesday?
Or will they gravitate towards a softer form of conservatism, based around its more traditional principles of free markets, free speech and free elections?
The dying days of the Trump presidency have revealed the depth of division at the heart of the Grand Old Party. What comes next, however, has rarely been so opaque.