Here is the very small collection of things I know:

• the coefficient of $E_n$ in the molecular decomposition is the number of normal subgroups, which is $3$ for $n\geq 5$.
• the coefficient of $s_n$ in the character is the number of conjugacy classes of subgroups https://oeis.org/A000638, because each molecular species in the molecular decomposition contains precisely one copy of the trivial representation.