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

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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

Similar facts

Proof

Case p = 2

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