I know that there are plenty of infinite groups with no maximal subgroups - classic example is the additive group of rational numbers, $\mathbb{Q}$.

Moreover, I know of the result for finite groups that states that a subgroup $H$ of a (finite) group $G$ is a maximal normal subgroup if and only if $G/H$ is simple.

But does there exist a necessary and sufficient condition for any group, finite or infinite, to have a maximal normal subgroup?

I need to answer the question "is it true that any group has a maximal normal subgroup?", and while I know that $\mathbb{Q}$ has no maximal subgroups, it does have normal ones (namely, $\mathbb{Z}$). So, I would like to ask, in addition, are there any examples of infinite groups that do have maximal subgroups, but none of them are normal?

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    $\begingroup$ Because $\Bbb Q$ is abelian, every subgroup is normal. Also, $A_5\times \Bbb Z$ is infinite, and has plenty of proper, non-normal subgroups that are not contained in any proper, normal subgroup. This there are maximal subgroups that are not normal. $\endgroup$ – Arthur Mar 17 '17 at 21:15
  • $\begingroup$ @Arthur that is true. $\endgroup$ – ALannister Mar 17 '17 at 21:17
  • $\begingroup$ The statement that $N$ is a maximal normal subgroup of $G$ if and only if $G/N$ is simple is true for all groups, not just for finite groups. Also note that all finitely generated groups have maximal subgroups and maximal npormal subgroups. $\endgroup$ – Derek Holt Mar 17 '17 at 22:30

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