Editing Conway group (section)

==Sublattice groups==
Conway and Thompson found that four recently discovered sporadic simple groups, described in conference proceedings {{harv|Brauer|Sah|1969}}, were isomorphic to subgroups or quotients of subgroups of Co<sub>0</sub>.

Conway himself employed a notation for stabilizers of points and subspaces where he prefixed a dot. Exceptional were '''.0''' and '''.1''', being Co<sub>0</sub> and Co<sub>1</sub>. For integer {{nowrap|'''n''' ≥ 2}} let '''.n''' denote the stabilizer of a point of type '''n''' (see above) in the Leech lattice.

Conway then named stabilizers of planes defined by triangles having the origin as a vertex. Let '''.hkl''' be the pointwise stabilizer of a triangle with edges (differences of vertices) of types '''h''', '''k''' and '''l'''. The triangle is commonly called an '''h-k-l triangle'''.  In the simplest cases Co<sub>0</sub> is transitive on the points or triangles in question and stabilizer groups are defined up to conjugacy.

Conway identified '''.322''' with the '''[[McLaughlin group (mathematics)|McLaughlin group]]''' McL (order {{val|fmt=commas|898,128,000}}) and '''.332''' with the '''[[Higman–Sims group]]''' HS (order {{val|fmt=commas|44,352,000}}); both of these had recently been discovered.

Here is a table<ref>Conway & Sloane (1999), p. 291</ref><ref>Griess (1998), p. 126</ref> of some sublattice groups:

{| class="wikitable" style="margin:1em auto;"
|-
! Name
! Order
! Structure
! Example vertices
|-
| •2 || 2<sup>18</sup> 3<sup>6</sup> 5<sup>3</sup> 7 11 23 || Co<sub>2</sub> || (−3, 1<sup>23</sup>)
|-
| •3 || 2<sup>10</sup> 3<sup>7</sup> 5<sup>3</sup> 7 11 23 || Co<sub>3</sub> || (5, 1<sup>23</sup>)
|-
| •4 || 2<sup>18</sup> 3<sup>2</sup> 5 7 11 23 || 2<sup>11</sup>:M<sub>23</sub> ||(8, 0<sup>23</sup>)
|-
| •222 || 2<sup>15</sup> 3<sup>6</sup> 5 7 11 || PSU<sub>6</sub>(2) ≈ [[Fischer group#3-transposition groups|Fi<sub>21</sub>]] || (4, −4, 0<sup>22</sup>), (0, −4, 4, 0<sup>21</sup>)
|-
| •322 || 2<sup>7</sup> 3<sup>6</sup> 5<sup>3</sup> 7 11 || McL || (5, 1<sup>23</sup>),(4, 4, 0<sup>22</sup>)
|-
| •332 || 2<sup>9</sup> 3<sup>2</sup> 5<sup>3</sup> 7 11 || HS || (5, 1<sup>23</sup>), (4, −4, 0<sup>22</sup>)
|-
| •333 || 2<sup>4</sup> 3<sup>7</sup> 5 11 || 3<sup>5</sup> M<sub>11</sub> || (5, 1<sup>23</sup>), (0, 2<sup>12</sup>, 0<sup>11</sup>)
|-
| •422 || 2<sup>17</sup> 3<sup>2</sup> 5 7 11 || 2<sup>10</sup>:M<sub>22</sub> || (8, 0<sup>23</sup>), (4, 4, 0<sup>22</sup>)
|-
| •432 || 2<sup>7</sup> 3<sup>2</sup> 5 7 11 23 ||M<sub>23</sub> || (8, 0<sup>23</sup>), (5, 1<sup>23</sup>)
|-
| •433 || 2<sup>10</sup> 3<sup>2</sup> 5 7 || 2<sup>4</sup>.A<sub>8</sub> || (8, 0<sup>23</sup>), (4, 2<sup>7</sup>, −2, 0<sup>15</sup>)
|-
| •442 || 2<sup>12</sup> 3<sup>2</sup> 5 7 || 2<sup>1+8</sup>.A<sub>7</sub> || (8, 0<sup>23</sup>), (6,  −2<sup>7</sup>, 0<sup>16</sup>)
|-
| •443 || 2<sup>7</sup> 3<sup>2</sup> 5 7 || M<sub>21</sub>:2 ≈ PSL<sub>3</sub>(4):2 || (8, 0<sup>23</sup>), (5, −3, −3, 1<sup>21</sup>)
|}