Central factor not implies direct factor

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 $$G$$.