It is widely known that for every finite group $G$, $G$ is nilpotent If and only if every maximal subgroup of $G$ is a normal subgroup.
But I don't know if there is an infinite non-nilpotent group whose every maximal subgroup is a normal subgroup.
Thank you for your help.
Best Answer
Yes, there is. There is an example of a non-nilpotent group with the normalizer condition, i.e., $H<N_G(H)$ for all proper subgroups $H$. It was constructed by Heineken and Mohamed in 1968. The paper is
A group with trivial centre satisfying the normalizer condition, J. Algebra 10, 368-376 (1968).
Edit: This group is seriously wonky. From their paper, it has: