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 (?), .
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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.