The below written lines are taken from book title " Visual Group Theory" by Natham Carter
The most general way to deconstruct a large group into two factors is called taking quotients. It reveals not only direct products but also semi direct products. It gives us deep insight into groups structure.
Question : Why quotients in groups needed? I have heard about division in numbers but in groups how it make sense? 8/4 = 2, but $C_4 / C_2$ ? How researcher in mathematics comes up with the idea of quotient?
I have seen this question Why the term and the concept of quotient group?
my opinion and self experience is that the quotient groups are famous and very important for group theorists since the have important behave and they built by normal subgroups which are very very important in group theory.somehow quotient groups play the role of fraction in numbers(that is if G\N is quotient group, we called G without considering N inside) but the whole story is about algebraic theorist like normal groups and normal subgroups so the quotient groups are important to them.and be careful u never can compare quotient with divisions but in some groups. for example u can study $\frac{z}{n{z}}$ which z is integer numbers.