Equivariant homotopy commutativity for $G=C_{pqr}$

We investigate the combinatorial data arising from the classification of equivariant homotopy commutativity for cyclic groups of order $G=C_{p_1 \cdots p_n}$ for $p_i$ distinct primes. In particular, we will prove a structural result which allows us to enumerate the number of $N_\infty$-operads for $C_{pqr}$, verifying a computational result.