Normal not implies central factor

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) need not satisfy the second subgroup property (i.e., central factor)
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Statement

A normal subgroup of a group need not be a central factor.

Related facts

• Normal not implies direct factor
• Central factor not implies direct factor
• Abelian normal not implies central

Proof

Further information: dihedral group:D8, Klein four-subgroups of dihedral group:D8, cyclic maximal subgroup of dihedral group:D8

Suppose $G$ is the dihedral group of order eight, $H$ is the cyclic subgroup of order four in $G$, and $K_1$ and $K_2$ are the two Klein four-subgroups. Then, the three subgroups $H$, $K_1$, and $K_2$ are all normal in $G$. However, none of them are central factors. In fact, they are all self-centralizing subgroups of $G$.