Central factor is not finite-join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) not satisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about central factor|Get more facts about finite-join-closed subgroup propertyGet more facts about join-closed subgroup property|

Statement

Statement with symbols

It is possible to have a finite group $G$ with subgroups $H, K$ such that both $H$ and $K$ are central factors of $G$ but the product $HK$ (which in this case is also the join of subgroups $\langle H, K \rangle$) is not a central factor.

Definitions used

Central factor

Further information: Central factor

A subgroup $H$ of a group $G$ is termed a central factor if $HC_G(H) = G$, or equivalently, every inner automorphism of $G$ restricts to an inner automorphism of $H$.

Proof

Example of the extraspecial group

For any prime $p$, either of the two isomorphism classes of extraspecial groups of order $p^5$ gives a counterexample.