There is a theorem that states every subgroup of a solvable group is solvable. Does this imply that if a group is not solvable then it cannot be a subgroup of any solvable group? For instance, $S_5$ is not a subgroup of any solvable group. That seems like a strong claim, so I just want to make sure I understand this correctly.
If this is true, does this have other implications/applications?
Yes that follows from the (fairly) well-known result that solvability is hereditary.
This is an example of the contrapositive. (So your question is an easy one.)
That's if a subgroup of a group is not solvable, then we would have that the parent group can't be solvable.
As far as consequences, note that since $A_5\le S_n $ for $n\gt4$, we get that such $S_n $ are not solvable. For $A_5$ is not solvable.
This is the key step in the Abel-Ruffini theorem, that there's no closed form for solving quintic polynomials and those of higher degree.
Note that as a special case we get the statement that a finite abelian group can't have a non-solvable subgroup.
