There are at most two finite simple groups of any order

Statement
Let $$n$$ be a natural number. Then, there are at most two finite simple groups of order $$n$$.

The smallest value of $$n$$ for which there are two non-isomorphic simple groups of order $$n$$ is $$20160 = 2^6 \cdot 3^2 \cdot 5 \cdot 7$$. The two groups in this case are the alternating group of degree eight (which is also isomorphic to the projective special linear group $$PSL(4,2)$$) and projective special linear group:PSL(3,4).

Infinite families
Note that there are examples other than the infinite families given below.

Below, $$q = p^r$$ is a prime power denoting the size of the field over which we are considering stuff.

First few examples
These include both the infinite families and the examples not arising from infinite families.

Proof
All known proofs of this fact employ the classification of finite simple groups, along with some explicit computations.