Normal 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., normal subgroup) need not satisfy the second subgroup property (i.e., direct factor)
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Statement

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

Related facts

• Characteristic not implies direct factor
• Normal not implies characteristic
• Central factor 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 $G$.