Subgroups of simple groups are as diverse as possible

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