Landau's Theorem on conjugacy classes for normal subgroups

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes for any positive integer k. We show that, for any positive integers n and s, there exists only a finite number of finite groups G, up to isomorphism, having a normal subgroup N of index n which contains exactly s non-central G-conjugacy classes. We provide upper bounds for the orders of G and N, which are used by using GAP to classify all finite groups with normal subgroups having a small index and few G-classes. We also study the corresponding problems when we only take into account the set of G-classes of prime-power order elements contained in a normal subgroup.