Let $\pi$ be a set of primes. Our definition for Hall $\pi$-subgroups states: A Hall $\pi$-subgroup of $G$ is a subgroup $H$ where $|H|$ is product of elements of $\pi$ and $|G:H|$ is product of $P\setminus\pi$ where $P$ is the set of all primes. So basically $|H|$ and $|G:H|$ are coprime.
My script says, that a solvable group G always has a Hall $\pi$-subgroup for any $\pi$. I think I'm missing something here, because I don't understand how this works for example for $S_4$ and $\pi = \{5\}$. How is there a subgroup of $S_4$ that is a product of elements of $\pi$ to begin with?