Collection of groups satisfying a weak normal replacement condition

Statement
Suppose $$\mathcal{S}$$ is a collection of finite $$p$$-groups, i.e., groups of prime power order where the underlying prime is $$p$$. We say that $$\mathcal{S}$$ satisfies a weak normal replacement condition if whenever a finite $$p$$-group $$P$$ contains a subgroup isomorphic to an element of $$\mathcal{S}$$, $$P$$ also contains a normal subgroup isomorphic to an element of $$\mathcal{S}$$.

Stronger properties

 * Weaker than::Collection of groups satisfying a universal congruence condition
 * Weaker than::Collection of groups satisfying a strong normal replacement condition

Examples/facts


Threshold values
This lists threshold values of $$k$$: the largest value of $$k$$ for which the collection of $$p$$-groups of order $$p^k$$ satisfying the stated condition satisfies a weak normal replacement condition. The nature of all these is such that the weak normal replacement condition is satisfied for all smaller $$k$$ but for no larger $$k$$. The between $$a$$ and $$b$$ below means that the minimum known value is $$a$$ and the maximum known value is $$b$$.

