 Exactly two composition series for a simple group times a prime cyclic group

The only nontrivial proper normal subgroups of this direct product of groups are its two factors. Indeed a normal subgroup meeting $S\times 1$ nontrivially contains it by the simple group property. Otherwise each of its commutators with $S\times 1$ lies in that trivial intersection. Its first coordinates therefore lie in $Z(S)=1$, so it is contained in $1\times C_p$. Both factors are simple and each quotient by one factor is isomorphic to the other. Hence precisely two composition series occur, one through each factor. This is compatible with the Jordan–Hölder theorem, since the two series have the same simple factors in opposite orders. For $S=A_5,p=2$, the group has order $120$.