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 ∼ = A 5 , and a group G with exactly four classes of nonnormal proper subgroups of the same nonprime-power order is nonsolvable if and only if G ∼ = P SL(2, 7) or P SL(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 ∼ = A 5 , P SL 2 (7) or SL 2 (5).