Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
1348 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D271 |
| commutator, socle, cosocle | C23×C271 | C24 |
| Fitting | C4×C22⋊C4×C271 | C2 |
| hypercenter | C4×C22⋊C4 | D271 |
| hypocenter | C271 | #2433 |
| LCS | G ▷ C23×C271 ▷ C271 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D1084 | C2 | semidirect |
| 2. C2×#22576157 | C2 | semidirect |
| 3. C2×#22576134 | C2 | semidirect |
| 4. C2×#22576130 | C2 | semidirect |
| 5. #31773807 | C2 | semidirect |
| 6. #31773799 | C2 | semidirect |
| 7. C4×C22⋊C4×C271 | C2 | semidirect |
| 13. C22×D1084 | C22 | semidirect |
| 14. C22×D1084 | C22 | semidirect |
| 19. C2×C4⋊Dic271 | C22 | semidirect |
| 20. C2×(C2×D271)⋊C4 | C22 | semidirect |
| 21. C2×(C2×D271)⋊C4 | C22 | semidirect |
| 24. C2×C42×C271 | C22 | semidirect |
| 25. #95×C271 | C22 | semidirect |
| 42. C2×D1084 | D4 | semidirect |
| 51. C22×C4×C271 | D4 | semidirect |
| 55. C22⋊C4×C271 | D4 | semidirect |
| 89. C2×C4×C271 | C2×D4 | semidirect |
| 93. C22×C271 | #127 | semidirect |
| 110. C4×C271 | #120 | semidirect |
| 119. C271 | #2433 | semidirect |
| 127. C4×C22⋊C4 | D271 | semidirect |
| 144. C2×C42 | C2×D271 | semidirect |
| 145. #95 | C2×D271 | semidirect |
| 168. C22×C4 | D1084 | semidirect |
| 175. C22⋊C4 | D1084 | semidirect |
| 209. C2×C4 | C2×D1084 | semidirect |
| 223. C22 | #22576157 | semidirect |
| 231. C4 | #22576130 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D1084 | 2 |
| C2×#22576157 | 2 |
| C2×#22576134 | 2 |
| C2×#22576130 | 2 |
| #31773807 | 2 |
| #31773799 | 2 |
| C4×C22⋊C4×C271 | 2 |
| #2433 | 271 |