Characterization of finite groups with a unique non-nilpotent proper subgroup

Abstract

‎We characterize finite non-nilpotent groups $G$ with a unique non-nilpotent proper subgroup‎. ‎We show that $|G|$ has at most three prime divisors‎. ‎When $G$ is supersolvable we find the presentation of $G$ and when $G$ is non-supersolvable we show that either $G$ is a direct product of an Schmidt group and a cyclic group or a semi direct product of a $p$-group by a cyclic group of prime power order‎.