So we know about the Senzus, but what clans are under the umbrella of Goku?
The reason I haven't given a list is because there are hundreds of them, and even then they fragment more every generation. The Clan system is
strained to breaking by the fact that the plan to unite the race failed and the population controls being removed wasn't followed by the steps needed to make that a feasible move. So each Lord has to ride herd on the disputes of hundreds of people and remember
all of them, at the risk of mortally offending somebody from a point of misremembered trivia.
That said, there are a few big ones in particular who serve as the important players in Clan politics.
House Mato are traditionalists and isolationists. They are an old and prestigious House that dates back to before the Clan system was even formalized, and their support was a large part of what allowed Lady Garla to formalize that system. They are, obviously, named for Son Mato, and have an odd and amusing tendency to do as House Vegeta does and name their heirs after their House name, to predictable results. They are loyalists without peer to Lords who tend to reinforce the distance from the human world and preserve the Masquerade, but act as lightning rods for dissent against reformers. They are fervent members of the Ancestor Cult, to the point that they mandate their children's education in it.
House Balor, your Sensei's House, is Mato's diametric opposition. They accept the Masquerade with poor grace at the best of times, and the most combative of them even refuse to wear their Masques at all, to the result of being largely exiled to isolation. As a result, they are outsiders, but vocal ones who tend to collect support when public opinion is swinging against the Masquerade. They are also
violently hostile to the Ancestor Cult and opposed its formalization with open conflict. The resulting sectarian violence threatened the Masquerade itself until Lady Garla intervened. Not every member of House Balor feels this way, but they make damn sure that their heir always does. The most even-keeled of them are at best merely disdainful of the Cult and Masquerade; from what you know of Sensei, that's where he falls.
House Stauber is a branch House from House Goku itself, and is fiercely loyalist at all times. That said, their politics don't always agree; like House Mato, they're isolationists, and when Lords try to break the Masquerade open Stauber often disagrees. That said, they've only openly opposed their Lord once, during the reign of Lord Oni, and even then only once Carrick Vegeta died fighting Oni. Their entrance into the conflict, however, proved decisive, and they were the ones to bring Oni down, exhausted as he was from his duel with Carrick. They are esteemed beyond measure by the majority of the Clan, and their name is a watchword for integrity, reliability, and honor. Additionally, unlike House Mato, Stauber tends to oppose the Ancestor Cult, viewing them as unacceptably independent from the Lords and all too likely to break the Masquerade if some arcane religious imperative demands it (Head Stauber's words, not mine).
House Peat is
legendarily pro-Ancestor Cultist; since the Cult's first reformation, they have had three family members become the High Priest, and they take the sitting High Priest's words as quite literal gospel. And they are also the
reason why House Stauber is so hostile to the Cult; Peat is highly opposed to the Masquerade, favoring open interaction with the humans of Garenhuld and an aggressive approach to returning to the stars. Members of the House have been part of three separate Super Saiyan rebellions, and their continued
existence causes the Lords Goku a good deal of embarrassment and lost respect when dealing with the Lords Vegeta, who historically tend not to allow second chances; while not every Lord Vegeta takes Yammar's "root, branch, and seed," approach, they
do historically execute every adult and adopt the children out among separate Houses with new names. The fact that House Peat has not been punished for their failure (or refusal) to impose internal discipline on their rebellious members makes Vegetans in general look on "Gokun law" as an oxymoron, and the House is in general a pariah in the modern world. But their position as important members of the Ancestor Cult, and especially the fact that the current High Priest
is a Peat, mean that they still have some power left to them.
Those are four of the big ones. You'll notice that they represent the extremes from, "isolationist to anti-Masquerade," to, "pro- to anti-Cult." Those are the major axes of Saiyan political life, and while sub-factions of course exist, the four above Clans are held to be the Gokun epitome of the axes.
@PoptartProdigy, will we get a malus to gaining the Instant Kamehameha due to it being offensive or will we lack one since we'd just be using two skills at once?
I presume you mean
this?
You've already used it.