Thanks, is the claim you're making that any (or many?) formal group coming from a ring spectrum that admits an H-infinity complex orientation has only "special" subgroups? Because the reason I actually posted this question is that I had convinced myself that something like that is true!

If you haven't seen Dmitri Pavlov's "Structured Brown Representability via Concordance" then I can say with high confidence you'd have fun poking around through it.

Could you include a reference or give a little more context for this "regularized euler class" ?

@DavidRoberts No, it wasn't recorded

I think one way to characterize those bundles (although it might be more tautological than what you're looking for) is that they are the tensor product (over R) of a complex vector bundle on Y with the line bundle on Y determined by the C_2 principal bundle X.