Classification of finite simple groups
The Classification of finite simple groups is a megatheorem which states that every finite simple group belongs to one of eighteen infinite families of simple groups, or to one of 26 sporadic simple groups.
Contents
The eighteen families
Here are the families, up to isomorphism. Note that these families are oneparameter, twoparameter or threeparameter families. Each parameter varies either over prime numbers or over natural numbers. Many of the families have a few small exceptions that turn out not to be simple groups.
Also, some of these families have intersections, i.e., there are some groups that occur in multiple families. The intersection of any two families is finite: there are only finitely many groups that are simultaneously in two distinct families.
No.  Family name  Nature of parameters  Notation for group  Chevalley notation (if applicable)  Order  Exceptions(not simple)  Links to proofs 

1  cyclic groups of prime order  prime number  or      No proper nontrivial subgroup implies cyclic of prime order, prime order implies no proper nontrivial subgroup  
2  alternating group  natural number    A5 is simple, alternating groups are simple  
3  projective special linear group  natural number (degree), prime power (field size)  Projective special linear group is simple  
4  Chevalley group of type B  odd natural number (degree), prime power  , , . Although is not simple, is.  
5  projective symplectic group  even natural number (degree), prime power (field size)  ,  Projective symplectic group is simple  
6  Chevalley group of type D  even natural number (degree), prime power (field size)  , , (so simple for  
7  Suzuki group  Parameter , effectively  , so  
8  Ree group  Parameter , effectively  , so 
10 more families need to be entered in the table above.
Collisions between families
Here are some of the infinite collisions:
Collision  Precedence convention (if any) 

. In other words, for all .  We denote the group as or . 
for all . In other words, for all .  We denote the group as . 
for all . In other words, for all . Note that the case is already covered in the preceding collision.  We denote the group as . 
Here is the list of finite and isolated collisions by family pairs:
First family  Second family  All the collision cases  Proof 

projective special linear group  projective special linear group  alternating group:A5: and also projective special linear group:PSL(3,2): and also . 

alternating group  projective special linear group  alternating group:A5: alternating group , also projective special linear group and . alternating group:A6: alternating group , also projective special linear group alternating group:A8: alternating group , also projective special linear group . 
projective special linear group equals alternating group in only finitely many cases 
The table needs to be completed.
The twentysix sporadic simple groups
Group name  Symbol  Order  Prime factorization of order  Number of conjugacy classes 

Mathieu group:M11  7920  10  
Mathieu group:M12  95040  15  
Mathieu group:M22  443520  12  
Mathieu group:M23  10200960  17  
Mathieu group:M24  244823040  26  
Janko group:J1  175560  15  
Janko group:J2 (also called the HallJanko group)  or  604800  21  
Janko group:J3  50232960  21  
Janko group:J4  86775571046077562880  62  
Conway group:Co1  4157776806543360000  101  
Conway group:Co2  42305421312000  60  
Conway group:Co3  495766656000  42  
Fischer group:Fi22  64561751654400  65  
Fischer group:Fi23  4089470473293004800  98  
derived subgroup of Fischer group:Fi24  1255205709190661721292800  183  
HigmanSims group  44352000  24  
McLaughlin group  898128000  24  
Held group  4030387200  33  
Rudvalis group  145926144000  36  
Suzuki sporadic group  448345497600  43  
O'Nan group  460815505920  30  
HaradaNorton group  273030912000000  54  
Lyons group  51765179004000000  53  
Thompson group  90745943887872000  48  
baby monster group  4154781481226426191177580544000000  [SHOW MORE]  184  
monster group  808017424794512875886459904961710757005754368000000000  [SHOW MORE]  194 
List of simple nonabelian groups of small order
Further information: List of simple nonabelian groups of small order
The simple abelian groups are precisely the groups of prime order, and there is one such group for each prime number.
The first few simple nonabelian groups are listed below:
References
Expository article references
 A brief history of the classification of the finite simple groups by Ronald Mark Solomon, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Volume 38,Number 3, Page 315  352(Year 2001): An expository paper by Ronald Mark Solomon describing the 110year history of the classification of finite simple groups.^{Weblink (PDF)}^{More info}
 Paper:AschbacherCFSG^{More info}