Let's for any cardinal $\alpha$ define $S_\alpha$ as the group of all permutations of a set of cardinality $\alpha$.
Is there some sort of classification of normal subgroups of $S_\alpha$?
For $\alpha < \aleph_0$ it is well known: they are $E$, $A_\alpha$ and itself (and also $C_2 \times C_2$ in the special case when $\alpha = 4$)
However, what is for $\alpha \geq \aleph_0$?
Here we have the following subgroups, that are obviously normal:
1)For any cardinal $\beta$, such that $\aleph_0 \leq \beta \leq \alpha$, we have the subgroup $S_\alpha^{\leq \beta}$ of all permutations with the cardinality of their support $\beta$ or less.
2)For any cardinal $\beta$, such that $\aleph_0 \leq \beta \leq \alpha$, we have the subgroup $S_\alpha^{< \beta}$ of all permutations with the cardinality of their support strictly less than $\beta$.
3)The subgroup $A_\alpha$ of all permutations, that are equal to a product of even number of transpositions
4)The trivial subgroup
My question is: is there anything else? And if there is, then what?