Abstract:

In this paper， the finite non-nilpotent group with every non-abelian subgroup being subnormal  is investigated.It’s shown that such groups are solvable, and then give a necessary and sufficient condition and some structures for this kind of groups by taking into account the Sylow subgroups. As an application, the supersolvable group with the above property is discussed.

Key words: subnormal subgroup； A-group; non-nilpotent group; super-solvable group