The main difference between Claire and Frank is that Frank occasionally does things to help people without expecting anything in return (getting Meechum re-hired is a perfect example). Even though Frank is generally heartless and goes so far as murdering his opponents on several occasions, he does, at the very least, possess a modicum of basic empathy.
Claire, on the other hand, is an irredeemable solophist. She has either coldly calculated each move she’s made or, on some occasions, acted completely out of petty self-interest and lack of foresight. Even her sudden and unexpected defense of Corrigan, the jailed gay rights activist in Russia, was simply a power play that almost certainly got all of the man’s fellow Russian gay rights activists tortured if not killed outright shortly afterward… and why? Because Corrigan hung himself, thereby taking away Claire’s power over him. Claire can’t abide anyone, other women included, holding any sort of real or imagined power over her under any circumstances. She is willing to utterly destroy people regardless of the scale of their “power” over her or their intentions even when they’re on her side. That is a chilling trait for anyone to possess, regardless of gender, sex, or any other personal attribute.
If you think any sort of a world in which Claire Underwood would be hailed as a hero is a good thing, you are indeed a sociopath. Frank should never be hailed as a hero either, but at least he has a glimmer of humanity in him. The same glimmer that Claire utterly lacks and has, on many occasions, knowingly used against Frank himself. That is why she gets more hate than Frank; it has nothing to do with the fact that she is a woman, and I’m truly sorry (and more than a little disturbed) that you can’t see it.