If a proper subgroup of maximal order is solvable and normal, then the group is solvable. Generalization.
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
104080
Group with order $pq$ has subgroups of order $p$ and $q$
111375
Normality of subgroups and Sylow groups under homomorphisms...
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
19352
Normal subgroup of index m contains every element of order n where m and n are relatively prime
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
51359
Prove that every subgroup of an abelian group is a normal subgroup.
103423
Normal subgroup where G has order prime
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
40513
A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
138406
If N and every subgroup of N is normal in G then G/N is abelian .
20386
Any subgroup of index $p$ in a $p$-group is normal.
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple