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

A central factor of a group need not be a direct factor.

Related facts

• Normal not implies central factor
• Normal not implies direct factor
• Characteristic not implies direct factor

Proof

Let $G$ be cyclic group of order four and $H$ be the unique subgroup of order two, comprising the squares (or the elements whose order divides two). Then:

• $H$ is a normal subgroup of $G$.
• $H$ is not a direct factor of $G$: In fact, it is the only proper nontrivial subgroup of $G$.