There are Five Great Houses, with their attendant territories as per the (rather stylised)
map of the Empire:
Sumeragi, Fenghuang, Hoou, Eisenwald, and
Peyrac-Beausoli. The Imperial House is not
officially labeled as a Great House, but then it doesn't need to be, it's the
Imperial House. House Akasha is wealthy and has a certain degree of power of influence, but nothing on the scale of the Great Houses: it helps that Yuidad, Yuimom and Yui are considered
members of the Imperial Family, so they get to tap into the Imperial House's influence and prestige for now.
IfOnce Ahri has children, that'll push Yui out of the immediate line of succession; she'll receive Letters Patent as Duchess* of Akasha (or whatever title Her Majesty Empress Meigyoku bestows upon her), and then GG no re, glhf, House Akasha be independant now as a cadet branch of the Imperial House (as opposed to "our heir-daughter is the Heir Presumptive to the Imperial throne").
That said, there is a non-zero chance that Nenek will stipulate that Yui gets her title and all her riches when she dies, because, y'know, favorite (more like *only*
) grandchild, so it's not like she'll be out in the cold, even if she hasn't been saving her salary quite like Johnny
. Yui lives paycheck to paycheck off her Vice Admiral's salary, with some of it put aside for savings/investment, but she does have access to House Akasha's finances, and while she's generally quite responsible with her money, you don't do things like
"buy Michelin Star quality ingredients for your ship's mess" and
"buy replacement super comfortable chairs for CIC and bridge" on an Admiral's salary
But generally she just makes sure to stay out of the way of Daddy's accountants and Daddy's financial advisors and Daddy's lawyer who know this stuff better than her.
*She will inevitably have to talk Ahri out of trying to make her a Grand Duchess.
Note the map of the Empire here, btw: