This does not work. They are a combination of more than one quadra.
Here's what I reckon:
Ravenclaw: Alpha NTs, ILI
Slytherin: Beta STs (maybe with some EIEs)
Gryffindor: Gamma SFs, LIE
Hufflepuff: Deltas (possibly the only house which generally contains all Deltas, but I could see ESEs here)
I don't know where IEIs could go.