Finite Group DatabasebetaFinite Group DatabasebetaOrder 27712.
330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D433 |
| commutator, socle, cosocle | C22×C433 | C24 |
| Frattini | C22 | C23×D433 |
| Fitting | #117×C433 | C2 |
| hypercenter | #117 | D433 |
| hypocenter | C433 | #525 |
| LCS | G ▷ C22×C433 ▷ C433 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#20294097 | C2 | semidirect |
| 2. C22×C4×D433 | C2 | semidirect |
| 3. #28357204 | C2 | semidirect |
| 4. #28357200 | C2 | semidirect |
| 5. #28357195 | C2 | semidirect |
| 6. #28357190 | C2 | semidirect |
| 7. C4×D1732 | C2 | semidirect |
| 8. #117×C433 | C2 | semidirect |
| 11. C2×Dic866 | C22 | semidirect |
| 13. C2×D1732 | C22 | semidirect |
| 19. C2×C4×D433 | C22 | semidirect |
| 20. C4⋊Dic433 | C22 | semidirect |
| 22. #20294074 | C22 | semidirect |
| 24. C4⋊C4×C433 | C22 | semidirect |
| 26. C42×C433 | C22 | semidirect |
| 42. C2×C4×C433 | D4 | semidirect |
| 54. C2×D433 | #41 | semidirect |
| 65. C433 | #525 | semidirect |
| 74. #117 | D433 | semidirect |
| 88. C4⋊C4 | C2×D433 | semidirect |
| 92. C42 | C2×D433 | semidirect |
| 108. C2×C4 | D1732 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20294097 | 2 |
| C22×C4×D433 | 2 |
| #28357204 | 2 |
| #28357200 | 2 |
| #28357195 | 2 |
| #28357190 | 2 |
| C4×D1732 | 2 |
| #117×C433 | 2 |
| #525 | 433 |