Solvability-forcing number

This article defines a property that can be evaluated for natural numbers

Definition

A natural number is said to be solvability-forcing if it satisfies the following equivalent conditions:

Every group of that order is solvable
It has no non-prime divisor which is simple-feasible. In other words, no divisor of it occurs as the order of a simple non-Abelian group

Odd number: For full proof, refer: Odd-order implies solvable (also known as the Feit-Thompson theorem or odd-order theorem)
A number whose order has at most two distinct prime factors. For full proof, refer: Order has only two prime factors implies solvable (also known as Burnside's p^aq^b theorem
A number whose order is the product of three distinct primes. For full proof, refer: Order is product of three distinct primes implies normal Sylow subgroup
Square-free number: A number whose order is a product of distinct primes. For full proof, refer: Square-free implies solvability-forcing
Nilpotency-forcing number
Abelianness-forcing number
Cyclicity-forcing number