PRIME FACTORS AND THEIR EXPONENTS IN THE SPORADIC FINITE SIMPLE GROUPS   JUNE 30TH 2010
     
  2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
M11        4 2 1   1                              
M12        6 3 1   1                              
M22        7 2 1 1 1                              
M23        7 2 1 1 1       1                      
M24        10 3 1 1 1       1                      
J1           3 1 1 1 1     1                        
J2 = HJ    7 3 2 1                                
J3           7 5 1       1 1                        
J4           21 3 1 1 3       1 1 1 1   1            
HS          9 2 3 1 1                              
McL         7 6 3 1 1                              
He          10 3 2 3     1                          
Ru          14 3 3 1   1       1                    
Suz         13 7 2 1 1 1                            
O'N         9 4 1 3 1     1     1                  
Co3 = .3 10 7 3 1 1       1                      
Co2 = .2 18 6 3 1 1       1                      
Co1 = .1 21 9 4 2 1 1     1                      
F5 = HN   14 6 6 1 1     1                        
Ly           8 7 6 1 1           1 1             1  
F3 = Th   15 10 3 2   1   1     1                  
Fi22        17 9 2 1 1 1                            
Fi23        18 13 2 1 1 1 1   1                      
Fi24'        21 16 2 3 1 1 1   1 1                    
F2 = B     41 13 6 2 1 1 1 1 1   1       1          
F1 = M    46 20 9 6 2 3 1 1 1 1 1   1   1   1     1
  2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71

  

General Wikipedia lemmas on sporadic groups:
http://en.wikipedia.org/wiki/Sporadic_group
and http://en.wikipedia.org/wiki/Category:Sporadic_groups

The pariahs among the sporadic finite simple groups are the six groups not involved in the Monster group.

The pariah prime numbers are those which do not appear as prime factors in the orders of the sporadic finite simple groups. They are 53, 61 and all primes from 73 onwards. The ratio of pariah primes against the happy family of primes is infinity against eighteen, so one creates an oxymoron by this definition.

The GCD of the orders of all sporadic finite simple groups is 120.

Finite simple groups in Wikipedia:  list  -  classification