2) A prime ordered group is always cyclic. Lagrange's theorem suffices to show it is simple because the order of a subgroup must divide the order of the group, so the only subgroups are the trivial one and the group itself. \
Sylow's theorem guarantees the existence of a sylow p subgroup for each prime dividing the order of the group. It is just in this case the only prime dividing the group's order is 59. So it is the only subgroup.
4) has a cyclic subgroup of order 30 (the group of rotations) any subgroup of index 2 is normal. So it has a normal subgroup
1 and 3 are easily done if you actually understand what Sylow's theorem is.