They never really explained if he was an unofficial member or if he is just a friend that can occasionally order everyone in the school around? I would love to see people's opinion on this.
It could always be OgaTouhoushinki lol.
Personally, I see Oga as an unofficial member of the TKKH, kind of in the way similar to having a "right-hand man". Based on his feats and incredible abilities, he should be allowed to join as the 5th member of the TKKH that represented Ishiyama High, but how would they name that? TOKKH?
I wonder if the Rokkisei see Oga as a member of TKKH or if they even know about the group. Oga even has his own gang Furiuchi, Alaindeon, Hilda, Oga, Beel, Lamia and maybe Aoi.
Oga never got the official role as leader even though Toujou and the others regard him as the strongest. Since the school was destroyed, the guys at St. Ishiyama only know TKKH.