 Solution

The face action gives $|G|=60$, as calculated above. Every normal subgroup contains the identity and is a union of whole conjugacy classes. Thus its possible order is $1$ plus a subset sum of $15,20,12,12$. The distinct resulting numbers areBy Lagrange's theorem for finite groups, a subgroup order divides $60$. Only $1$ and $60$ in this list do so. Consequently the normal subgroups are $\{1\}$ and $G$, and the rotational symmetry group of a regular icosahedron is a simple group.