We define the upper density $\rho (G)$ of a finite group $G$ as the ratio of the number of finite groups of order $<n$ which contain a subgroup isomorphic to $G$ to the number of groups of order $<n$, in the limit as $n \rightarrow \infty$ (if such a limit exists, but we can always take limsup/liminf).

For example, $\rho (G) =1$ for $G$ is the trivial group.

I would like to investigate the value of $\rho (C_2) $. In general, this seems like a hard problem, so lets simplify things and talk about abelian groups only.

So we define $ \rho (G) = lim_{n \rightarrow \infty} \frac{ \textrm{number abelian groups order} \ < n \ \textrm{with subgroup iso.} \ G}{ \textrm{number of abelian groups order} < n}$

I tried some rough bounds but I still end up with horrible summations that I can't seem to get much out of.

Some basic facts that will be useful:

Number of non-isomorphic abelian groups of order $p_1 ^{\alpha_1} \cdots p_k ^{\alpha_k}$ is $p( \alpha_1) \cdots p( \alpha_k)$ where $p(r)$ is the number of partitions of $r$.

We want to count primes somehow, so maybe a result Bertrand's Postulate would be useful (although will probably only give weak bounds) https://en.wikipedia.org/wiki/Bertrand%27s_postulate

- The number of abelian groups of order $n$ is a multiplicative function.

Any ideas would be appreciated.