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

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. Keywords