Is a cyclic group considered a more specialized form of a "plain-ole" group?

Question

(edited question) I'm trying to identify (for context) the hierarchy of groups, rings and fields and then where cyclic groups "fit". For example, I've seen groups represented at the top of the hierarchy, then rings, then fields. I'm wondering if a cyclic group is considered to be in the category--groups or its own separate category?

(original question) I'm trying to get a high level view of groups, rings and fields and then within fields cyclic groups and I'm wondering if a cyclic group is considered to be in the family of just plain-ole groups?

Answer 1

Every cyclic group is a group but not every group is a cyclic group.

Answer 2

Yes, a cyclic group is a group.

Answer 3

Cyclic groups are groups, but they're nice because they're easy to think about (they can be visualized as, say, a cycle), are abelian, and have plenty of other nice properties that always hold no matter the order.