Question: How to determine the number of all the Sylow subgroups of $A_8$.
I'm trying to use some Sylow Theorems (especially the $3^{rd}$ Sylow theoreom).
My approach: (I'm not sure, but so far I've got)
The $2-$Sylow subgroups are self normalizing, so there are $315$ of them.
There are $280$ ,$3-$Sylow subgroups.
Not sure about $5$ and $7$ Sylow subgroups yet, also not sure about what I did with $2$ and $3$.
Should I also calculate all the $p$ Sylow subgroups as well?
Also, I know that $GL(2,4) \cong A_8$, for every class of $GL(2,4)$ I've calculated the size of the class and the order of the elements, and found all the representatives for all the conjugacy classes in that group, I believe it also could help me somehow.