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 ... "

$\begingroup$ This seems a very difficult question. For example, nonsolvable groups with ALL maximal subgroups solvable were extremely difficult to classify ( done by J.G. Thompson in the 1960s). $\endgroup$ – Geoff Robinson Nov 9 '18 at 20:35

$\begingroup$ I see. Well, partial results, even if they are small, are appreciated. Thanks! $\endgroup$ – Goa'uld Nov 9 '18 at 22:28
The following references answer the question in case of almost simple groups:
 The maximal factorizations of the finite simple groups and their automorphism groups by M. W. Liebeck, C. E. Praeger, and J. Saxl.
The book describes all possible factorizations by maximal subgroups of nonabelian finite simple groups.
 Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups by Cai Heng Li and Binzhou Xia.
This unpublished paper classifies all almost simple groups factorizing on a solvable subgroup.
 On finite factorizable groups by Zvi Arad and Elsa Fisman.
The paper classifies all nonabelian 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 $A\cong A_{n1}$;
$M_{11}$ 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 $\gcd(r,q1)=1$ and either $G\cong PSL(5,2)$ and $B=5\cdot31$ or $A$ is a maximal parabolic subgroup such that $PSL(r1,q)$ is involved in $A$. In particular, $B$ is either cyclic or Frobenius.