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finite group; number of subgroups of possible orders
Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\{1, q+1\}$ for some prime $q$. In particular, $G$ is supersolvable under this condition.
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