Subgroups of Symmetric and Dihedral groups

[Avatrin]

I am having problem working with the objects in the title. Working with permutations, rotations and reflections is fine, but I have problem with the following:

Showing a subgroup is or is not normal (usually worse in the case of symmetric groups)

Finding a subgroup of order n.

Showing that there is no subgroup of order n.

I cannot remember encountering many exercises that helped me learn to work with subgroups of symmetric and dihedral groups. Are there any strategies I can follow, and, even better, any sets of exercises anybody here recommends?

 

[Avatrin]

My exams are over, and luckily, I didn't need much about this topic. Also, I guess my questions above were too broad.