Subgroups of simple groups are as diverse as possible

For a finite group $G$, let $σ(G)$ be the number of subgroups of $G$ and $σ_ι(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$ and characteristic $p$. If $r\neq 1$, $L\not\cong {^2 B_2}(2^{1+2m})$, then there are constants $c,d$, dependent on the Lie type, such that as $re$ grows $p^{(c-o(1))r^4e^2}\leqσ_ι(L_r(p^e))\leqσ(L_r(p^e)) \leq p^{(d+o(1))r^4e^2}.$ For type $A$, $c=d=1/64$. For other classical groups $1/64\leq c\leq d\leq 1/4$. For exceptional and twisted groups $1/2^{100}\leq c\leq d\leq 1/4$. Furthermore, $2^{(1/36-o(1))k^2)}\leqσ_ι(\mathrm{Alt}_k)\leq σ(\mathrm{Alt}_k)\leq 24^{(1/6+o(1))k^2}.$ For abelian and sporadic simple groups $G$, $σ_ι(G),σ(G)\in O(1)$. In general these bounds are best possible amongst groups of the same orders. Thus with the exception of finite simple groups with bounded ranks and field degrees, the subgroups of finite simple groups are as diverse as possible.

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