ON NON-NORMAL NON-ABELIAN SUBGROUPS OF FINITE GROUPS

In this paper we prove that a finite group G having at most three conjugacy classes of non-normal non-abelian proper subgroups is always solvable except for G congruent to A(5), which extends Theorem 3.3 in [Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sinica (English Series) 27 (2011) 891-896.]. Moreover, we show that a finite group G with at most three same order classes of non-normal non-abelian proper subgroups is always solvable except for G congruent to A(5).