In $A_7$,
1) Are all subgroups of order 168 are conjugate? ($A_7$ contains a simple group of order 168).
2)Does it contain an abelian group of order 12? What is the largest order of abelian group?
3)What is the Sylow-2 subgroup of $A_7$?
[I am trying to get some other way to prove that there is only one simple group order 168.
In a simple group $G$ of order $168$, the number of Sylow-3 subgroups is $7$, or $28$ . If it is $7$ , then $G$ is contained in $A_7$, and we get an abelian subgroup of order $12$, in $A_7$ . Certainly, $A_7$ does not contain an element of order $12$, hence cyclic subgroup of order $12$. therefore, it may happen that $A_7$ may contain $\mathbb{Z}_2 \times \mathbb{Z}_6$. I want to know whether this is possible? Also, to know about intersection of Sylow-2 subgroups with each other, I want to know their structure.]