Why is Lagrange's theorem so important? Sure, |H| divides |G|. but why is this important? I dont seem to get why it is.
Well, I think Lagrange can do squat since he's been dead, but this theorem of his is VERY important in finite group theory since, to begin with,it gives a very nice and easy-to-check test to rule out possible subgroups of groups. For example, a group of order 10 can only have proper subgroups of order 2 and 5 (besides the trivial one), and that already rules out 7 possible orders.
You have to wait until you advance a little more in your group theory studies to begin to realize how basic and important this result is.
Tonio