Maximal subgroups with arithmetically large index

Question

It is well known that a maximal subgroup of a finite solvable group has prime power index. On the other hand, there exist nonabelian finite simple groups with maximal subgroups whose index is divisible by every prime divisor of the order of the group. One example is $\operatorname{Alt}(4) \times \operatorname{Sym}(3)$ in the Mathieu group $M_{12}$. I can see with a computer that this is very rare.

Question: List the couples $(G,H)$ where $G$ is a finite simple group and $H<G$ is a maximal subgroup such that every prime dividing $|G|$ already divides $|G:H|$.

One motivation comes from https://www.gtseminar.xyz/documents/2025-4-09-Luca.pdf. In the last slide I ask for the stronger condition of $|G:H|$ being a multiple of the exponent of $G$.

Edit: In two comments below, Sean Eberhard has found two infinite families. Geoff Robinson pointed out that the question is equivalent to listing the couples $(G,H)$ where $G$ is a finite simple group and $H$ is a maximal subgroup not containing any Sylow subgroup.

Answer 1

If $G = A_n$ then I would rather try to classify the non-examples. Suppose $H < A_n$ is maximal. If $H$ is intransitive or imprimitive then everything is explicit and we have a formula for the index, so just go away and check that and report back. Now assume $H$ is primitive and suppose $H$ contains a Sylow $p$-subgroup of $A_n$, where $p \le n$. If $p = 2$ then this implies $H$ is of exponential order, so there must be finitely many examples (most likely just $\mathrm{GL}_3(2) \le A_7$ and $\mathrm{AGL}_3(2) \le A_8$). Now assume $p > 2$. Then $H$ contains a $p$-cycle. By Jordan's theorem, it follows that $p \in \{n-2, n-1, n\}$. The examples can be listed: see Jones [1] Theorem 1.2.

Next, we can look for low-rank examples by looking for small-order maximal subgroups. See King [2].

1. From [2, Corollary 2.2(f,h)], $G = \mathrm{PSL}_2(p)$ has $H = S_4$ as a maximal subgroup whenever $p \equiv \pm 1 \pmod 8$, and $|G:H| = p(p-1)(p+1)/48$, so this is an example whenever $288 \mid (p^2-1)$. Similarly, $H = A_5$ is a maximal subgroup whenever $p \equiv \pm 1 \pmod{10}$, of index $|G:H| = p(p-1)(p+1)/120$ so we have another example whenever $3600 \mid (p^2-1)$. (As Derek Holt rightly points out, $H = A_4$ does not give an example.)

2. From [2, Theorem 2.5(j,k)] there are various examples with $G = \mathrm{PSL}_3(p)$, I think. For example $H = \mathrm{PSL}_2(7)$ is a subgroup (apparently maximal) whenever $-7$ is a square mod $p$ and $|G:H| = p^3(p^3-1)(p^2-1)/(3,p-1)/168$, so this is an example whenever $-7$ is a square mod $p$ and $(p^3-1)(p^2-1)/(3,p-1)$ is divisible by $7056$, e.g., for all $p \equiv 1 \pmod {7056}$.

3. From [2, Theorem 2.8(j)], $G = \mathrm{PSp}_4(p)$ and (for example) $H \cong A_6$ is an example with $|G:H| = p^4(p^2-1)(p^4-1) / 360$, and this works whenever $(p^2-1)(p^4-1)$ is divisible by $10800$.

4. "etc etc"

[1] Jones, Gareth A., Primitive permutation groups containing a cycle., Bull. Aust. Math. Soc. 89, No. 1, 159-165 (2014). ZBL1297.20003.

[2] King, Oliver H., The subgroup structure of finite classical groups in terms of geometric configurations., Webb, Bridget S. (ed.), Surveys in combinatorics 2005. Papers from the 20th British combinatorial conference, University of Durham, Durham, UK, July 10–15, 2005. Cambridge: Cambridge University Press (ISBN 0-521-61523-2/pbk). London Mathematical Society Lecture Note Series 327, 29-56 (2005). ZBL1107.20035.