A group $G$ is called perfect iff $G’ = G$.
A finite group $G$ is called immaculate iff its order is equal to the sum of orders of its proper normal subgroups.
Does there exist a finite group $G$, that is both perfect and immaculate at the same time?
Motivation behind this question:
If I am not mistaken, the notion of «immaculate groups» first appeared in this MO question by @TomLeinster. In the first paragraph of the question they said, that considered calling such groups «perfect» (because it is a group theoretic analogue of perfect numbers), however the term "perfect groups" was already taken. However, it would be interesting to know whether some of the immaculate groups are indeed perfect…