I think either outstrip or overshadow might work if you are wanting to keep this sentence structure, though as Oldcat said you may want to add words in to specify that the elite derives from either group. You might consider the word "constituent" (being a part of the whole, i.e. each group is a constituent of the ruling elite, though this may be confusing in a political sense since constituent can also mean a voting member).
I think overshadows works particularly well for your use, especially since it keeps the tense you want.
An unappointed ruling elite that overshadows both of the member groups.
An unappointed ruling elite that overshadows both of its constituent groups.
Or:
An unappointed ruling elite that outstripped both of the member groups.
You could also say "outsrips" but that seems to me a strange use of tense for that word in this context.
Otherwise an entire rephrasing like thecrease suggested might work well.
