Quotients of Solvable Groups are Solvable I just proved that subgroups of solvable groups are solvable. So given that $G$ is solvable there is $1=G_0 \unrhd G_1 \unrhd \cdots \unrhd G_s=G$ where $G_{i+1}/G_i$ is abelian and for $N$ a normal subgroup we know it is solvable so there is $1=N_0 \unrhd N_1 \unrhd \cdots \unrhd N_{r}=N$ where $N_{i+1}/N_i$ is abelian. 
I'm trying to construct the chain for the entire group $G/N$ using the Lattice Isomorphism Theorem but am stuck. Is it possible to do this?

(Corrected) Note that since the chain starts at $1<N$, there exists a maximun $i$ such that $G_i<N$. Multiplying and quotienting out that part of the chain gives a chain of quotients $1\lhd NG_i/N\lhd \cdots \lhd G/N$ and $(NG_{i+1}/N)/(NG_i/N)\simeq NG_{i+1}/NG_i\hookrightarrow G_{i+1}/G_i$ is abelian. The argument is essentially the same as the following: suppose that $\eta:G\to H$ is onto and $G$ is solvable. Then $H$ is solvable.
There's another part to the story: if $G/N$ and $N$ are solvable, show that $G$ is solvable too. 
