I was asked to prove: no group can have a minimal normal subgroup isomorphic to a $\mathrm{Syl}_2(A_7)$.
I think I should find some property that $\mathrm{Syl}_2(A_7)$ has but not a minimal normal subgroup. So then I thought that it a $\mathrm{Syl}_2(A_7)$ always has characteristic subgroup, i.e. exist $G\ char\ \mathrm{Syl}_2(A_7)$, then the image of G under isomorphism will contradict minimality of "group can have a minimal normal subgroup isomorphic to a $\mathrm{Syl}_2(A_7)$". But I get stuck then about whether it is true that $\mathrm{Syl}_2(A_7)$ always has characteristic subgroup.
Maybe this is not the correct way to prove it. Can anyone help me?