I would like to know results on the structure of a finite group $G$ which possesses a maximal subgroup $H$, with $H$ solvable. More precisely, about
supplements of $H$, that is, decompositions $G=HK$ where the intersection of $H$ and $K$ is not necessarily the trivial subgroup. I know the question is a bit vague, but I would appreciate references to results of the form "there exists a supplement $K$ which satisfies ... "



Groups with a maximal subgroup which is solvable
The following references answer the question in case of almost simple groups:


1. [The maximal factorizations of the finite simple groups and their automorphism groups](https://bookstore.ams.org/memo-86-432/) by M. W. Liebeck, C. E. Praeger, and J. Saxl.


The book describes all possible factorizations by maximal subgroups of non-abelian finite simple groups. 


2. [Factorizations of almost simple groups with a solvable factor, and
Cayley graphs of solvable groups](https://arxiv.org/pdf/1408.0350.pdf) by Cai Heng Li and Binzhou Xia.


This unpublished paper classifies all almost simple groups factorizing on a solvable subgroup.


3. [On finite factorizable groups](https://doi.org/10.1016/0021-8693(84)90046-2) by Zvi Arad and Elsa Fisman.


The paper classifies all non-abelian finite simple groups $G=AB$ such that $\gcd(|A|,|B|)=1$. Assume $|B|$ is odd. Then $G$ is isomorphic to one of the following groups:


* $A\_n$ ($n\geq5$ is a prime) and $A\cong A\_{n-1}$;
* $M\_{11}$ and either $A$ is solvable or $A\cong M\_{10}$;
* $M\_{23}$ and either $B$ is Frobenius of order $11\cdot23$ or $B$ is cyclic of order $23$ and $A\cong M\_{22}$;
* $PSL(2,q)$ where either $q\in\{11,29,59\}$ and $A\cong A\_5$ or $3<q\not\equiv1\pmod4$ and $A$ is solvable;
* $PSL(r,q)$ with $r$ an odd prime such that $\gcd(r,q-1)=1$ and either $G\cong PSL(5,2)$ and $|B|=5\cdot31$ or $A$ is a maximal parabolic subgroup such that $PSL(r-1,q)$ is involved in $A$. In particular, $B$ is either cyclic or Frobenius.


