Let $\mathcal{G}$ be the set of finite groups and for $G \in \mathcal{G}$, let $S(G)$ be the set of subgroups of $G$. I am interested in the ratio $R(G)=|S(G)|/|G|$. It is easy to show that by picking $G$ appropriately, $R(G)$ can be made arbitrarily large or arbitrarily close to zero. I am interested in some deeper properties of the set $R=(R(G) : G \in \mathcal{G})$, such as:

(1) For which $x \in \mathbb{R}$ do there exist sequences of finite groups $G_1, G_2, ...$ such that the sequence $R(G_1), R(G_2), ... $ converges to $x$?

(2) Does $R$ contain a (non-empty) interval $(a,b) \in \mathbb{Q}$?

(3) Which integers belong to $R$?

(4) How do these properties change when $\mathcal{G}$ is replaced by the set of finite abelian groups? Finite simple groups?