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.