Nonsolvable proper Hall subgroup of symmetric group is contained in symmetric group on subset of size one less

Statement
Suppose $$n$$ is a natural number and $$S_n$$ is the symmetric group on the set $$\{ 1,2,3, \dots, n \}$$. Suppose $$H$$ is a proper fact about::Hall subgroup of $$S_n$$ such that $$H$$ is not solvable. Then, there exists $$i \in \{ 1,2,3,\dots,n\}$$ such that $$H$$ is contained in the stabilizer of $$i$$. In other words, $$H$$ is contained in the subgroup of $$S_n$$ given by the symmetric group on $$\{ 1,2,3, \dots, n \} \setminus \{ i \}$$.