Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
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×C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C23×#562 | C2 | semidirect |
| 2. C2×C23×#496 | C2 | semidirect |
| 3. C2×C23×#490 | C2 | semidirect |
| 4. C23×#1656 | C2 | direct (N × H) |
| 5. C2×C23×#330 | C2 | semidirect |
| 6. C2×C23×#329 | C2 | semidirect |
| 11. C23×#562 | C22 | semidirect |
| 19. C23×#496 | C22 | semidirect |
| 21. C23×#490 | C22 | semidirect |
| 25. C23×#330 | C22 | semidirect |
| 26. C23×#329 | C22 | semidirect |
| 109. C2×#1656 | C23 | direct (N × H) |
| 141. C2×#562 | C2×C23 | semidirect |
| 142. C2×#496 | C2×C23 | semidirect |
| 143. C2×#490 | C2×C23 | semidirect |
| 144. #1656 | C2×C23 | direct (N × H) |
| 145. C2×#330 | C2×C23 | semidirect |
| 146. C2×#329 | C2×C23 | semidirect |
| 167. #562 | C22×C23 | semidirect |
| 175. #496 | C22×C23 | semidirect |
| 177. #490 | C22×C23 | semidirect |
| 181. #330 | C22×C23 | semidirect |
| 182. #329 | C22×C23 | semidirect |
| 185. C2×C23 | #1656 | direct (N × H) |
| 228. C23 | C2×#1656 | direct (N × H) |
| 320. C2 | C23×#1656 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C23×#562 | 2 |
| C2×C23×#496 | 2 |
| C2×C23×#490 | 2 |
| C23×#1656 | 2 |
| C2×C23×#330 | 2 |
| C2×C23×#329 | 2 |
| C2×#1656 | 23 |