Congruence condition on number of subloops of given prime power order in nilpotent loop of prime power order

Statement
Suppose $$L$$ is a fact about::finite nilpotent loop (i.e., a fact about::nilpotent loop whose underlying set is finite) of order a prime power $$p^k$$. Suppose $$0 \le r \le k$$. Then, the number of subloops of $$L$$ of order $$p^r$$ is congruent to 1 mod $$p$$.

Related facts

 * Congruence condition on number of subgroups of given prime power order
 * Sylow subloops exist for Sylow primes in finite Moufang loop