Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1138 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×#562 | C2 | semidirect |
| 2. C2×C53×#496 | C2 | semidirect |
| 3. C2×C53×#490 | C2 | semidirect |
| 4. C53×#1656 | C2 | direct (N × H) |
| 5. C2×C53×#330 | C2 | semidirect |
| 6. C2×C53×#329 | C2 | semidirect |
| 11. C53×#562 | C22 | semidirect |
| 19. C53×#496 | C22 | semidirect |
| 21. C53×#490 | C22 | semidirect |
| 25. C53×#330 | C22 | semidirect |
| 26. C53×#329 | C22 | semidirect |
| 140. C2×#1656 | C53 | direct (N × H) |
| 157. C2×#562 | C2×C53 | semidirect |
| 158. C2×#496 | C2×C53 | semidirect |
| 159. C2×#490 | C2×C53 | semidirect |
| 160. #1656 | C2×C53 | direct (N × H) |
| 161. C2×#330 | C2×C53 | semidirect |
| 162. C2×#329 | C2×C53 | semidirect |
| 165. C2×C53 | #1656 | direct (N × H) |
| 172. #562 | C22×C53 | semidirect |
| 180. #496 | C22×C53 | semidirect |
| 182. #490 | C22×C53 | semidirect |
| 186. #330 | C22×C53 | semidirect |
| 187. #329 | C22×C53 | semidirect |
| 188. C53 | C2×#1656 | direct (N × H) |
| 320. C2 | C53×#1656 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#562 | 2 |
| C2×C53×#496 | 2 |
| C2×C53×#490 | 2 |
| C53×#1656 | 2 |
| C2×C53×#330 | 2 |
| C2×C53×#329 | 2 |
| C2×#1656 | 53 |