The Ordo Hydra is an extremist faction of the secret Illuminati society.
Both the mainstream Illuminati and its more extremist faction are devoted to saving humanity, but where the Illuminati's intent is to sacrifice to the Emperor his immortal sons, the Sensei, the Ordo Hydra's plan is to link all of mankind through the Hydra warp entity, offering salvation through total enslavement. Mankind would be ultimately in the control of the masters of the Ordo Hydra. The combined power of mankind would be enough to totally destroy all that threatens its survival.
Many of the Ordo Hydra's members are Inquisitors, in effect making the ordo overlap into the Inquisition and making it, at least seemingly, a faction of both the Inquisition and the Illuminati.