Collection of groups satisfying a universal non-divisibility condition

Definition
Suppose $$\mathcal{S}$$ is a collection of finite $$p$$-groups, i.e., groups of prime power order for a prime number $$p$$. We say that $$\mathcal{S}$$ satisfies a universal non-divisibility condition if the following equivalent conditions hold:


 * 1) For any finite $$p$$-group $$P$$ containing a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of subgroups of $$P$$ isomorphic to elements of $$\mathcal{S}$$ is not divisible by $$p$$.
 * 2) For any finite $$p$$-group $$P$$ containing a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of subgroups of $$P$$ isomorphic to elements of $$\mathcal{S}$$ is not divisible by $$p$$.
 * 3) For any finite $$p$$-group $$Q$$ and any normal subgroup $$P$$ of $$Q$$ containing a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of normal subgroups of $$Q$$ isomorphic to elements of $$\mathcal{S}$$ and contained in $$P$$ is not divisible by $$p$$.
 * 4) For any finite $$p$$-group $$P$$ that contains a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of p-core-automorphism-invariant subgroups of $$P$$ is not divisible by $$p$$.
 * 5) For any finite group $$G$$ containing a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of subgroups of $$G$$ isomorphic to an element of $$\mathcal{S}$$ is not divisible by $$p$$.

Stronger properties

 * Collection of groups satisfying a universal congruence condition

Weaker properties

 * Collection of groups satisfying a strong normal replacement condition
 * Collection of groups satisfying a weak normal replacement condition