Finite groups with some nonnormal subgroups of non-prime-power order

Abstract

We prove that a group G with exactly three classes of nonnormal proper subgroups of the same non-prime-power order is nonsolvable if and only if G ≃ A₅, and a group G with exactly four classes of nonnormal proper subgroups of the same non-prime-power order is nonsolvable if and only if G ≃ PSL(2,7) or PSL(2,8). Moreover, we prove that any group G with at most nine classes of nonnormal nontrivial subgroups of the same order is always solvable except for G ≃ A₅, PSL₂(7) or SL₂(5).