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. 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?

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?