The Independent Democratic Conference is a group of six renegade Democrats who effectively let the GOP control the New York State Senate, despite its nominal Democratic majority.
I do not expect there to be an Independent Republican Conference in the U.S. Senate. It will be a 52-48 Republican majority (barring something truly shocking in Louisiana's runoff) -- a two-seat Democratic gain (pickups in Illinois and New Hampshire).
But what is plausible -- maybe -- is that a cohort of Senate Republicans might be willing to break from the past eight years policy of absolute, resolute, kneejerk party line voting and join with Democrats to insure there will be some actual oversight of the Trump administration.
Who are the likely candidates to take up that mantle?
The leader almost certainly would have to be Sen. Ben Sasse (R-NE). He was one of the earliest, most consistent, and most outspoken critics of Trump from within the GOP (here's his column on Trump's victory, tealeaf it yourself). That's one -- not because it's guaranteed, but because if he doesn't take the lead I can't imagine any caucus forming. Who else?
The supposedly moderate Susan Collins (ME) is an obvious possibility, but she's never exactly been renowned for her backbone. It'd be a major change for her to start bucking her party on a regular basis. But if ever there was a time for her to grow an actual spine, it'd be now.
Lindsey Graham (SC) could be a possibility. He's likewise been pretty critical of Trump, and has some personal grudges against Trump's wing of the party. His colleague Tim Scott (SC), as the only Black Republican in the Senate, is a complete wild card on this -- I wouldn't normally slot him in unless Trump goes so avowedly White supremacist that he can't not say something.
John McCain (AZ) ... well, who knows what he's thinking these days. I don't have a lot of faith. Jeff Flake might actually be a more realistic shot from this rapidly purpling state.
Marco Rubio (FL) and Ted Cruz (TX)? Don't make me laugh. Both have raced to snuggle up to Trump after getting blown apart by him in the primaries.
Chuck Grassley (IA), Orrin Hatch (UT), and maybe Pat Roberts (KS) might be old enough to do the whole "elder statesmen" thing. None of them will suffer any repercussions if they don't, though.
Dean Heller (NV) might look at Joe Heck's defeat and feel the need to avoid a similar fate. Or he might think that Heck was undone by his late wince away from Trump.
The Democratic Party is in a routed state right now. It will recover, but it will take time. In the meantime, it'll be up to congressional Republicans to decide if they want to put brakes on Trump or let him run wild. Democrats are, for the short-term at least, out of the equation: the last eight years have shown that a unified Republican majority can completely, utterly, entirely shut out the Democratic minority if they want to.
The ball is in your court, Sasse.