Does there exist some sort of classification of finite "characteristically simple" groups?
62000
$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$
130937
Embeddings of finite groups into $\mathrm{GL}_n(\mathbb{Z})$
31252
"Efficient version" of Cayley's Theorem in Group Theory
111168
Automorphisms of a finite non-abelian group
80381
Groups with "few" subgroups
96614
Finite number of subgroups $\Rightarrow$  finite group
115574
Finitely presented group with fewer relations than generators.
40846
When can a (finite) group be written as the quotient of some other group by its center?
96878
A finite group with the property that all of its proper subgroups are abelian
63451
Is there some sort of classification of all minimal non-cyclic groups?
9232
Group with proper subgroups infinite cyclic
63162
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
42582
Infinite group with only two conjugacy classes
8746
Groups of order $pq$ without using Sylow theorems
66806
A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$
85222
Direct sum of finitely-many non-abelian groups
12955
Classification of $n\times n$ real matrices up to congruence