All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order. Equivalently, all its composition factors are abelian.
All its chief factors (i.e., the successive quotient groups for any chief series for the group) are elementary abelian groups.
The trivial group is a finite solvable group.
Any prime power is solvability-forcing, because prime power order implies nilpotent and nilpotent implies solvable.
Any product of two prime powers, i.e., any number of the form , with primes, is solvability-forcing. See order has only two prime factors implies solvable (this result is also termed Burnside's -theorem).
Any odd number is solvability-forcing. See odd-order implies solvable. This result, also called the odd-order theorem or the Feit-Thompson theorem, is highly nontrivial.
Any square-free number i.e., any number that is a product of pairwise distinct primes. See square-free implies solvability-forcing.
Any finite simple non-abelian group is a finite group that is not solvable. See classification of finite simple groups for a list of finite simple non-abelian groups.
Further, any group that contains a finite simple non-abelian group as a subgroup, has a finite simple non-abelian group as a quotient group, or admits a finite simple non-abelian group as a subquotient must be non-solvable.
symmetric group:S5, special linear group:SL(2,5), direct product of A5 and Z2: All of these are groups of order 120 which have alternating group:A5 as one of their composition factors, and are hence neither simple nor solvable.
projective special linear group:PSL(3,2): This is a simple non-abelian group of order 168.
Below is a list of group properties such that a finite group has the property if and only if it is a finite solvable group.