Subgroups of simple groups are as diverse as possible

For a finite group G$G$, let σ(G)$\sigma (G)$ be the number of subgroups of G$G$ and σι(G)$\sigma _\iota (G)$ the number of isomorphism types of subgroups of G$G$. Let L=Lr(pe)$L=L_r(p^e)$ denote a simple group of Lie type, rank r$r$, over a field of order pe$p^e$ and characteristic p$p$. If r≠1$r\ne 1$, L≇2B2(21+2m)$L\not\cong {}^2 B_2(2^{1+2m})$, there are constants c,d$c,d$, dependent on the Lie type, such that as re$re$ grows p(c−o(1))r4e2⩽σι(Lr(pe))⩽σ(Lr(pe))⩽p(d+o(1))r4e2.\begin{align*}\hskip6.5pc p^{(c-o(1))r^4e^2} & \leqslant \sigma _{\iota }(L_r(p^e)) \leqslant \sigma (L_r(p^e)) \leqslant p^{(d+o(1))r^4e^2}.\hskip-6.5pc \end{align*}For type A$A$, c=d=1/64$c=d=1/64$. For other classical groups 1/64⩽c⩽d⩽1/4$1/64\leqslant c\leqslant d\leqslant 1/4$. For exceptional and twisted groups, 1/2100⩽c⩽d⩽1/4$1/2^{100}\leqslant c\leqslant d\leqslant 1/4$. Furthermore, 2(1/36−o(1))k2)⩽σι(Altk)⩽σ(Altk)⩽24(1/6+o(1))k2.\begin{align*}\hskip6.5pc 2^{(1/36-o(1))k^2)} & \leqslant \sigma _{\iota }(\operatorname{Alt }_k) \leqslant \sigma (\operatorname{Alt }_k)\leqslant 24^{(1/6+o(1))k^2}.\hskip-6.5pc \end{align*}For abelian and sporadic simple groups G$G$, σι(G),σ(G)∈O(1)$\sigma _{\iota }(G),\sigma (G)\in O(1)$. In general, these bounds are best possible among 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.