Question: The [Whitney graph isomorphism theorem](https://en.wikipedia.org/wiki/Line_graph#Whitney_isomorphism_theorem) gives an example of an extraordinary exception: a very general statement holds except for **one** very specific case.

Another example is the [classification theorem for finite simple groups](http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Statement_of_the_classification_theorem): a very general statement holds except for **very few (26)** sporadic cases. 

I am looking for more of **this kind** of theorems-with-not-so-many-sporadic-exceptions 

> 
> (***added:***) where the exceptions don't come in a row and/or
>  in the beginning - but are scattered truly [sporadically](https://en.wiktionary.org/wiki/sporadically).
> 
> 
> 

(A late thanks to Asaf!)

Answer: 
Here's a neat result that holds for all finite groups *except* if the Monster shows up as a quotient:

> 
> Let $G$ be a finite group and $p,q$ primes dividing $|G|$. If $G$ contains no element of order $pq$, then either
> 
> 
> 1. the Sylow $p$-subgroups or the Sylow $q$-subgroups are abelian, or
> 2. $G/O\_{\{p,q\}'}(G)$ is the Monster and $\{p,q\}=\{5,13\}$ or $\{7,13\}$.
> 
> 
> 

 [[source]](http://www.mathematik.uni-kl.de/%7Emalle/download/elemords.pdf)
The only spheres which admit almost complex structures are $S^2$ and $S^6$.