No common composition factor with quotient group not implies complemented

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup having no common composition factor with its quotient group) need not satisfy the second subgroup property (i.e., complemented normal subgroup)
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Statement

It is possible to have a finite group $G$ having a normal subgroup $N$ such that $N$ and $G/N$ have no common composition factors, but $N$ is not a permutably complemented subgroup of $G$.

Related facts

Similar facts

• No nontrivial homomorphism to quotient group not implies complemented

Opposite facts

• Normal Hall implies permutably complemented

Proof

Further information: special linear group:SL(2,5)

Let $G=SL(2,5)$, the special linear group of $2\times 2$ matrices over the field of five elements. Let $N=Z(G)$. Then, $N$ is a subgroup of order two and $G/N$ is isomorphic to the alternating group of degree five, which is simple. Thus, $N$ and $G/N$ have no common composition factors. However, $N$ has no complement in $G$.