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Sporadic finite simple groups catalogue

A catalogue of 26 objects with explicit orders, simplicity conditions, and category views.

Core idea

26 separately identified objects and parameterized families. Individual finite groups and constrained families retain distinct entries. Orders are exact integers or formulas; simplicity is stated under each entry's parameter restrictions. An isomorphism is a recorded relationship, not a reason to collapse the entries.

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Sporadic group

ObjectOrderSimplicityKind of entry
241⋅313⋅56⋅72⋅11⋅13⋅17⋅19⋅23⋅31⋅472^{41}\cdot3^{13}\cdot5^{6}\cdot7^{2}\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47simple under the stated constraintsspecified object
217⋅39⋅52⋅7⋅11⋅132^{17}\cdot3^{9}\cdot5^{2}\cdot7\cdot11\cdot13simple under the stated constraintsspecified object
218⋅313⋅52⋅7⋅11⋅13⋅17⋅232^{18}\cdot3^{13}\cdot5^{2}\cdot7\cdot11\cdot13\cdot17\cdot23simple under the stated constraintsspecified object
221⋅316⋅52⋅73⋅11⋅13⋅17⋅23⋅292^{21}\cdot3^{16}\cdot5^{2}\cdot7^{3}\cdot11\cdot13\cdot17\cdot23\cdot29simple under the stated constraintsspecified object
210⋅33⋅52⋅73⋅172^{10}\cdot3^{3}\cdot5^{2}\cdot7^{3}\cdot17simple under the stated constraintsspecified object
214⋅36⋅56⋅7⋅11⋅192^{14}\cdot3^{6}\cdot5^{6}\cdot7\cdot11\cdot19simple under the stated constraintsspecified object
24⋅32⋅5⋅112^{4}\cdot3^{2}\cdot5\cdot11simple under the stated constraintsspecified object
26⋅33⋅5⋅112^{6}\cdot3^{3}\cdot5\cdot11simple under the stated constraintsspecified object
27⋅32⋅5⋅7⋅112^{7}\cdot3^{2}\cdot5\cdot7\cdot11simple under the stated constraintsspecified object
27⋅32⋅5⋅7⋅11⋅232^{7}\cdot3^{2}\cdot5\cdot7\cdot11\cdot23simple under the stated constraintsspecified object
210⋅33⋅5⋅7⋅11⋅232^{10}\cdot3^{3}\cdot5\cdot7\cdot11\cdot23simple under the stated constraintsspecified object
246⋅320⋅59⋅76⋅112⋅133⋅17⋅19⋅23⋅29⋅31⋅41⋅47⋅59⋅712^{46}\cdot3^{20}\cdot5^{9}\cdot7^{6}\cdot11^{2}\cdot13^{3}\cdot17\cdot19\cdot23\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71simple under the stated constraintsspecified object
215⋅310⋅53⋅72⋅13⋅19⋅312^{15}\cdot3^{10}\cdot5^{3}\cdot7^{2}\cdot13\cdot19\cdot31simple under the stated constraintsspecified object

Sporadic simple group

ObjectOrderSimplicityKind of entry
221⋅39⋅54⋅72⋅11⋅13⋅232^{21}\cdot3^{9}\cdot5^{4}\cdot7^{2}\cdot11\cdot13\cdot23simple under the stated constraintsspecified object
218⋅36⋅53⋅7⋅11⋅232^{18}\cdot3^{6}\cdot5^{3}\cdot7\cdot11\cdot23simple under the stated constraintsspecified object
210⋅37⋅53⋅7⋅11⋅232^{10}\cdot3^{7}\cdot5^{3}\cdot7\cdot11\cdot23simple under the stated constraintsspecified object
29⋅32⋅53⋅7⋅112^{9}\cdot3^{2}\cdot5^{3}\cdot7\cdot11simple under the stated constraintsspecified object
23⋅3⋅5⋅7⋅11⋅192^{3}\cdot3\cdot5\cdot7\cdot11\cdot19simple under the stated constraintsspecified object
27⋅33⋅52⋅72^{7}\cdot3^{3}\cdot5^{2}\cdot7simple under the stated constraintsspecified object
27⋅35⋅5⋅17⋅192^{7}\cdot3^{5}\cdot5\cdot17\cdot19simple under the stated constraintsspecified object
221⋅33⋅5⋅7⋅113⋅23⋅29⋅31⋅37⋅432^{21}\cdot3^{3}\cdot5\cdot7\cdot11^{3}\cdot23\cdot29\cdot31\cdot37\cdot43simple under the stated constraintsspecified object
28⋅37⋅56⋅7⋅11⋅31⋅37⋅672^{8}\cdot3^{7}\cdot5^{6}\cdot7\cdot11\cdot31\cdot37\cdot67simple under the stated constraintsspecified object
27⋅36⋅53⋅7⋅112^{7}\cdot3^{6}\cdot5^{3}\cdot7\cdot11simple under the stated constraintsspecified object
29⋅34⋅5⋅73⋅11⋅19⋅312^{9}\cdot3^{4}\cdot5\cdot7^{3}\cdot11\cdot19\cdot31simple under the stated constraintsspecified object
214⋅33⋅53⋅7⋅13⋅292^{14}\cdot3^{3}\cdot5^{3}\cdot7\cdot13\cdot29simple under the stated constraintsspecified object
213⋅37⋅52⋅7⋅11⋅132^{13}\cdot3^{7}\cdot5^{2}\cdot7\cdot11\cdot13simple under the stated constraintsspecified object