Both of the people above are correct.

Only elezen can be members of the High Houses of Ishgard, because that requires you to be descended from the original knights twelve. That in itself becomes a very important plot point later on in Heavensward. Except for 1 single person, there is no mixing between elezen and hyur so it is impossible to be a hyur member of a high house.

Hyur, and just about anyone in Ishgard, can become a minor noble after killing a legit dragon. Once you do that, you and your descendants are pretty much set.