Reference request on determining whether or not a group of finite order is simple or not

Question

I want to know the best (most efficient and simple is what I'd classify as best) techniques to tell whether or not a group of finite order is simple. These are standard exercises but I don't know of any book that, say, gives me many different ways to classify the groups of finite order up to $1000$ or something, for example, and this or material similar to it would be really helpful to me.

Answer 1

Wikipedia provides a list of finite simple groups of small order. As you will see, there are only 5 non-cyclic simple groups of order < 1000. You can also use GAP small groups library.

EDIT. You can also look at Orders of non abelian simple groups.