Groups with a Bounded Number of Conjugacy Classes of Non-normal Subgroups

Let ν(G) denote the number of conjugacy classes of non-normal subgroups of a groupG. We prove that ifGis a finite group and ν(G) ≠ 0, then there is a cyclic subgroupCof prime power order contained in the centre ofGsuch that the order ofG/Cis a product of at most ν(G) + 1 primes. We also obtain a bound in the opposite direction, thus obtaining a criterion for a group to have a bounded number of conjugacy classes of non-normal subgroups. These results extend to infinite groups, with the subgroupCbeing the infinite Prüferp-group, but only whenGhas finitely many non-normal subgroups. This is to be expected because of the existence of monsters of the type constructed by S. V. Ivanov and A. Yu. Olʹshanskii. The structure of groups with infinitely many non-normal subgroups falling into finitely many conjugacy classes is also studied.