Congruence condition fails for number of central factors in group of prime power order

Statement

It is possible to have the following situation: $p$ is a prime number, $P$ is a finite p-group of order $p^k$, and there exists $r$ with $0 \le r \le k$ such that the number of subgroups that are central factors (i.e., the product of the subgroup with its centralizer is the whole group) of $P$ of order $p^r$ is a nonzero number that is not congruent to 1 mod $p$.

Related facts

Opposite facts

• Congruence condition on number of subgroups of given prime power order tells us that the opposite is true if we are looking at all subgroups or at all normal subgroups.

Similar facts

• Congruence condition fails for number of normal subgroups of given prime power order: Note that this states that the number of normal subgroups of a given prime power order may be a nonzero number not congruent to 1 modulo the prime, but to construct a counterexample, we need to move to an ambient finite group that is not itself a p-group.

Proof

Case $p = 2$

Further information: central product of D8 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2