It was clear that they were a traitorous faction within the Shogunate. In Gintama' episode 16, it's stated by Sougo that it's basically known that the Hitotsubashi were behind some bad stuff, but if they were called out for it, they could cause a lot of trouble for Shigeshige. But if they're only a part of the Shogunate, how could they have enough power to overthrow it, and how would that not be noticed?
In the same episode, Sougo says the Bakufu/Shogunate doesn't want any trouble with the Hitotsubashi. So if they're so dangerous, why weren't they taken out already? Just what kind of powerful organization are they?
I'm just having trouble understanding the premise of the Hitotsubashi.