On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups - Jiangtao Shi; Cui Zhang (The Scientific World Journal)
Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)|≤2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).