Why are subgroups important when regarding groups? What information do they provide us? H of G
closed as too broad by Matt Samuel, Henning Makholm, Adam Hughes, Shailesh, Moishe Kohan Dec 12 '16 at 5:03
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
I think the Sylow theorems give a good example of how studying subgroups of a given finite group can yield a better understanding of that group. For instance, some standard exercises in a group theory class include:
Show that no group of order $56$ is simple.
Show that every group of order $15$ is abelian.
And so forth. (Credit: I got those examples from Prof. Kimberly Elce's website)
There are more sophisticated examples of results along these lines, but this should already be enough to show that one use of subgroups is to aid in classifying groups.
I would also argue that studying subgroups is of independent interest. More generally, if I want to understand a class of mathematical structures (groups, rings, topological spaces, linear orderings, ...), then I also want to understand how objects in that class relate to each other: in particular, how can I build new ones from old ones? This question often splits into three sub-questions:
How can I transform one object into another?
How can I combine lots of objects into one big object?
How can I take one object, and "shrink" it to get another object?
In the context of groups, these correspond to (among other things) group homomorphisms, direct products, and subgroups. (And indeed, the HSP theorem in universal algebra shows that these operations are sufficeint to build a well-behaved class of structures from some "starting examples"!)