In order to give a further study of the influence of the number of non-TI-subgroups on the solvability of fi-nite groups,we first investigate non-abelian non-TI-subgroups and prove that if G is a non-solvable group having at most 21 non-abelian non-TI-subgroups,G is isomorphic to the alternating group A5.Moreover,we investigate non-subnormal non-abelian non-TI-subgroups and we also have that G is isomorphic to A5 if G is a non-solvable group having at most 21 non-subnormal non-abelian non-TI-subgroups.
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