I am reviewing some Algebra as well. I THINK that this is problem

E is a subgroup of but it is not a normal subgroup

If A is a subgroup of G. Then A is a normal subgroup if for all

Note that this is a Set equality.

For you so E isn't normal

Then the defintion of a Quoteint Group is

If H is a normal subgroup of G, the group G/H that consists of the cosets of H in G is called the quotient groups.

Conclusion: I think that E must be Normal not just a subgroup to construct a quotient group.

I will keep looking.