Well, think of a simple example, like the integers Z. Z, 2Z, 3Z, etc. are all subgroups of Z, and in those cases, they are isomorphic to Z sub n, the cyclic group under addition modulo n. Also, there exists other types of infinite groups, such as the infinite alternating group.
Mathematicians treat infinity differently though - there are different levels of infinity. Set size is denoted by the cardinality of a set, and two sets have the same cardinality if there exists a bijection between them. The partial order that is assigned to cardinality is the less than or equal to symbol, in which a set A has smaller cardinality than a set B if there exists an injection from A to B. When talking about infinite sets though, you can have more interesting things happen. For example, take the set of natural numbers, and take the subset of all even natural numbers. Surely there exists a bijection between the two (notably mapping every natural number n to 2n), so they have the same cardinality, even though the set of all even natural numbers is surely a subset of the natural numbers. So, the cardinality of the natural numbers is the smallest cardinality that a set can have.
Also noteworthy is that the real numbers have a strictly larger cardinality than the natural numbers, since the interval (0, 1) has a bijection into the real numbers, and the cardinality of (0, 1) is not equal to the cardinality of the natural numbers since there does not exist a bijection between the two (the famous Cantor diagonalization argument is used to prove this).
Also, I figured out that question I just posed, it's quite easy.