Group Structure from Subgroup and Cyclic Subgroup Counts

For a finite group \(G\), let \(\sub(G)\) be the number of subgroups of \(G\), let \(\cyc(G)\) be the number of cyclic subgroups, and let \(π(G)\) be the number of distinct prime divisors of \(|G|\). We study the normalized counts \(λ(G)=\sub(G)/2^{π(G)}\) and \(η(G)=\cyc(G)/2^{π(G)}\). We prove tha