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).
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