I will paraphrase from MathSciNet review number MR0202821 (34 #2680), written by B. Chang:
Let $G$ be a group of order $p^sh$, with $(p,h)=1$ and let $d_k$ be the number of subgroups of $G$ of order $p^{s−k},\,\, 0\leq k\leq s.$ The main results are that:
1\. $d_k=1 \mod p$
2\. If a Sylow p-subgroup S is elementary abelian or cyclic, then $d_k$ is congruent mod $p^{k+1}$ to the number of subgroups of S of order $p^{s−k}$.
3\. If S is not cyclic then $d_1=1+p \mod p^2$.
**Note** (July 26th 2018, NH) one can find the publication here!