The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,B)$ such that:

- $G$ is a finite simple group,
- $A$ and $B$ are maximal subgroups of $G$,
- $AB=G$.

**Question**: What is the classification with the following *additional* assumption?

- No extra intermediate: if $(A \cap B) < H < G$ then $H \in \{A,B\}$.

*Remark*: by GAP computation, below is the classification for $|G|<2\cdot 10^6$:

- $(A_6, \ A_5, \ A_5)$,
- $(A_8, \ A_7, \ 2^3:A_1(7))$,
- $(M_{12}, \ M_{11}, \ M_{11})$,
- $(C_2(2^2), \ A_1(2^4):2, \ A_1(2^4):2)$,
- $(C_3(2), \ A_8:2, \ ^2A_2(3^2):2)$.

This post was inspired by an exchange with Pablo Spiga.