Omega subgroups are homomorph-containing

Statement
Suppose $$G$$ is a group of prime power order (i.e., a finite $$p$$-group for some prime number $$p$$). Then, the omega subgroups of $$G$$, defined as:

$$\Omega_j(G) := \langle x \mid x^{p^j} = e \rangle$$

are homomorph-containing subgroups of $$G$$.

Related facts

 * Omega subgroups not are subisomorph-containing
 * Omega subgroups not are subhomomorph-containing