Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders
$$\sigma(G) = \sum\_{H \le G} |H|.$$
Note that if $C\_n$ is cyclic of order $n$ then $\sigma(C\_n) = \sigma(n)$, with $\sigma$ the usual [divisor function](https://en.wikipedia.org/wiki/Divisor_function). Consider the functions $$\sigma\_{-}(n)= \min\_{|G|=n} \sigma(G), \ \ \ \ \ \ \sigma\_{+}(n)= \max\_{|G|=n} \sigma(G). $$

This post is about a characterization of the extremizers, i.e. the finite groups $G$ such that $\sigma(G) = \sigma\_{\pm}(|G|)$.

$$\begin{array}{c|c} n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15 \newline \hline 
\sigma\_{-}(n)&1&3&4&7&6&12&8&15&13&18&12&28&14&24&24\newline \hline
\sigma(n)&1&3&4&7&6&12&8&15&13&18&12&28&14&24&24 \newline \hline
\sigma\_{+}(n)&1&3&4&11&6&16&8&51&22&26&12&60&14&36&24 \end{array}$$

We can observe in above table that $\sigma\_{-}(n) = \sigma(n)$, and it holds for all $n < 256=2^8$ (by checking on GAP).

**Question 1**: What are the finite groups $G$ such that $\sigma(G) = \sigma\_{-}(|G|)$? Exactly the cyclic groups?

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Next, consider the prime factorization of $n$ $$n=\prod\_{i=1}^r p\_i^{n\_i},$$ then, candidates which come in mind for $\sigma(G) = \sigma\_{+}(|G|)$ are the product of prime order cyclic groups: $$G = \prod\_{i=1}^r C\_{p\_i}^{n\_i}.$$
It works often but not always, as $\sigma(S\_3) = \sigma\_{+}(|S\_3|) = \sigma\_{+}(6) = 16$ whereas $\sigma(C\_2 \times C\_3) = 12$; but $S\_3 = C\_3⋊C\_2$, moreover, for $n \le 60$, all the models I found are semi-direct product of prime order cyclic groups.

**Question 2**: What are the finite groups $G$ such that $\sigma(G) = \sigma\_{+}(|G|)$?

Are there semi-direct product of prime order cyclic groups? Or at least supersolvable?