For a symmetric group $S_n$, we have known clearly the proper nontrivial normal subgroups of it: if n=3 or $\geq 5$, $S_n$ has only one normal subgroup; if n=4, $S_n$ has two normal subgroups; otherwise, $S_n$ has no normal subgroups.
As a generalization of $S_n$ is the signed symmetric group:https://groupprops.subwiki.org/wiki/Signed_symmetric_group. Could someone tell me what are the normal subgroups of signed symmetric groups? (I think they should be very similar like $S_n$) Thank you.
In brief, in ATLAS notation, for $n \ge 5$ there are $9$ normal subgroups of the signed permutation group $2^n\!:\!S_n$.
These have structures $1$, $2$, $2^{n-1}$, $2^n$, $2^{n-1}\!\!:\!A_{n}$, $2^n\!:\!A_n$, $2^{n-1}\!\!:\!S_n$ (two subgroups) and $2^n\!:\!S_n$.
Note that the normal subgroup $2^{n-1}$ consists of diagonal matrices of determinant $1$ in the standard linear representation. Also $A_n$ acts irreducibly on $2^{n-1}$ when $n$ is odd, but when $n$ is even, the subgroup $2$ is contained in the subgroup $2^{n-1}$, with irreducible action of $A_n$ on the quotient of order $2^{n-2}$.
There are 11 normal subgroups when $n=4$, 9 when $n=3$, and 6 when $n=2$.