The Machine Orthodoxy
Led by Elesh Norn, the white-aligned faction believes that Phyrexia must adhere to the teachings of the Father of Machines. They intend to indoctrinate everyone about the truth, even if it means tearing them limb from limb and reattaching the parts into a creation that is worthy of Phyrexia. They are dogmatic, hierarchal, and orthodox in their methods.
The Apostles of Karn: The Sect of the Creator's DestinyEdit
The Apostles of Karn want a leader that rules all of Phyrexia, and have chosen Karn to take on the role of Father of Machines. Karn really wasn't up to it, so they acted as his jailers, simultaneously grooming him to take on his role as Father of Machines. Considering Karn had his perfection and blessed oil purged from him after this sect was revealed, it is probably safe to assume it doesn't have as much influence as it once had, assuming it still exists at all.
The Flesh Singularity: The Sect of Total UnityEdit
Skin is the prison of the blessed and the stronghold of the heretic.—Argent Etchings, plate 64, passage 17
Infact this sect might even fear it to some extent.
The Porcelain Legion: The Sect of the Ideal FormEdit
Where the Flesh Singularity wants to turn all life into a single Phyrexian organism, The Porcelain Legion wants to turn all life into a Phyrexian super organism. The Porcelain Legion views 'the Ideal Form' as the optimal form of a Phyrexian for that Phyrexian's job.