Jonathan Bernstein makes an easy call:
Fox is part of the Republican Party. American political parties are made up of both formal organizations (such as the RNC) and informal networks. Fox News Channel, then, is properly understood as part of the expanded Republican Party, just like Hill staff of GOP Members of Congress, or pollsters who only work for Republicans, or activists who volunteer for Republican campaigns, or think tanks that generate legislation for Republicans to support. ... Fox is simply part of the communications arm of the party.
The tricky part is that FNC isn't only a component of the Republican Party. It's also a business, so it may have profit motivations beyond its partisan goals (both on the organizational and individual level, of course).