Let $G$ be a non-abelian finite simple group, let $p$ be a prime dividing the order of $G$, and let $P < G$ be a Sylow $p$-subgroup.

Is there a list of all the cases where $N_G(P)$ is maximal? If not, would it be feasible/difficult to list them all, using results from the literature on maximal subgroups of finite simple groups? Is there already a list somewhere in the literature?

If $G$ is sporadic group, I looked at the ATLAS and I think here are the cases where a Sylow $p$-normalizer is maximal in $G$ (hopefully I did not miss any cases):

- M11, $p = 3$
- M23, $p = 23$
- J2, $p = 5$
- Co1, $p = 7$
- Co2, $p = 5$
- McL, $p = 5$
- He, $p = 5, 7$
- Th, $p = 5, 7, 31$
- Fi24', $p = 29$
- B, $p = 47$
- M, $p = 41$
- J1, $p = 2, 3, 5, 7, 11, 19$ (every prime divisor of order)
- O'N, $p = 3$
- J3, $p = 3$
- Ru, $p = 5$
- J4, $p = 11, 29, 37, 43$
- Ly, $p = 37, 67$
- T, $p = 5$

I think the case where $G$ is an alternating group follows from the answer of Derek Holt here, using the classification of $2$-transitive groups.

- $\operatorname{Alt}_5$, $p = 2,3,5$
- $\operatorname{Alt}_6$, $p = 3$
- $\operatorname{Alt}_p$, for $p = 13, 19$ or $p \geq 29$

So that leaves the simple groups of Lie type, of which there are many. The sporadic and alternating case, and computations in some small cases suggest that perhaps it is relatively rare for $N_G(P)$ to be maximal.

isirreducible, then it will (I think) lie inside a field-extension subgroup... $\endgroup$