Groups with finitely many isomorphism classes of non-pronormal subgroups

A subgroup H of a group G is said to be pronormal if H and H^g are conjugate in 〈H,H^g〉 for every element g of G. The behaviour of pronormal subgroups in finite or infinite groups has been often investigated and, in particular, the structure of (generalized) soluble groups in which all subgroups are pronormal is known. Here it is proved that any (generalized) soluble group in which non-pronormal subgroups fall into finitely many isomorphism classes either is minimax or a group in which all subgroups are pronormal.