About the maximal condition on subgroups and normality Let $G$ be a group which satisfies the maximal condition on subgroups. Now, let $H$ be a subgroup containing a non trivial conjugate (i.e. there exists an element $x\in G,\,\notin H $ such that $H^x\leq H$). Is it true that, in this case, $H$ is normal in $G$? How can we prove that? Is solvability of any help?

Note

A group $G$ is said to satisfy the maximal condition on subgroups iff it is impossibile to find an infinite strictly ascendant chain of subgroups.
Observe that any finite nilpotent group $\;G\;$ both fulfills the maximal condition for subgroups and also for any proper subgroup $\;H\;$ we have that $\;H\lneqq N_G(H)\;$ which means there exists $\;x\in G\setminus H\;\;s.t.\;\;H^x\le H\;$ ...and this doesn't make the subgroup $\;H\;$ normal.
I may be missing something, but the maximal condition simply tells you that if $H^x \leq H$ then $H^x = H.$ In other words, $H$ has normalizer strictly containing $H.$ It does not seem to imply that $H$ is normal.