Abstract
Considering the quantitative properties of some particular subgroups of a finite group, we prove that (1) a non-solvable group $G$ has exactly 5 non-subnormal non-supersolvable proper subgroups if and only if $G\cong A_5$ or $SL_2(5)$. (2) a non-solvable group $G$ has exactly 5 non-subnormal non-2-nilpotent proper subgroups if and only if $G\cong A_5$ or $SL_2(5)$. (3) a non-solvable group $G$ has exactly 16 non-subnormal non-2-closed proper subgroups (or two same order classes of non-subnormal non-2-closed proper subgroups) if and only if $G\cong A_5$ or $SL_2(5)$. Our results improve some known related research.
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