Non-soluble groups with few conjugacy classes of non-normal subgroups

Abstract

For a finite group G, let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. We determine all finite non-soluble groups G with ν(G) < 14. As a consequence, all finite groups G with ν(G) ≤ 6 or \({\nu(G)\in \{ 8,11,12\}}\) are soluble.