If I take a finite collection of n x n invertible matrices and generate a group G under matrix multiplication, is it the case that there always exists a maximal normal solvable group R from which I can decompose G into G = S * R for some other group S? Furthermore, given numerical matrices and assuming that type of decomposition exists, is it feasible to determine if S is a subgroup of U(n) the n x n unitary matrices?
Basically, I would like to get something similar to the Levi decomposition for a Lie Algebra but for matrix groups under multiplication. I was wondering if anyone could point me towards some theorems that might be helpful.
For connected linear algebraic groups over a field $k$ one has indeed a Levi decomposition, provided $k$ has characteristic zero. A Levi factor then is a reductive complement to the unipotent radical of the group. In prime characteristic there may be no Levi factor. For a discussion see, for example, the article Levi decompositions of a linear algebraic group by G. M. Mcninch, section $3$.