Why must the Sylow 2-subgroup of an order 56 group have non-trivial intersection

Question

I can see that the Sylow 7-subgroups of a group of order 56 must have trivial intersection due to Lagranges theorem, but why does this nor apply to the Sylow 2-subgroups?

Answer 1

The only possible subgroups of a group of order $7$ are the trivial group and the whole group. The intersection of two subgroups is of a larger group itself a subgroup. Two subgroups of order $7$ therefore either intersect in a subgroup of order $7$ i.e. they are identical, or in a subgroup of order $1$.

Subgroups of order $8$ can intersect in non-trivial subgroups of order $2$ or order $4$.