1) Is S7 X {0} a normal subgroup of S7 X Z7?

2) Are there any subgroups G such that S7 X {0} < G < S7 X Z7?

3) If yes to 2), are any of the subgroups G normal?

The original question was if the group S7 X {0} is a maximal normal subgroup of the product group S7 X Z7, but this is as far as I got in solving it