Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1728 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D97 |
| commutator, cosocle | C22×C97 | C25 |
| Frattini | C22 | C24×D97 |
| Fitting | C2×#127×C97 | C2 |
| socle | C23×C97 | C24 |
| hypercenter | C2×#127 | D97 |
| hypocenter | C97 | C2×#532 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13095510 | C2 | semidirect |
| 2. #20261283 | C2 | direct (N × H) |
| 3. C2×#15690810 | C2 | semidirect |
| 4. C2×#15690797 | C2 | semidirect |
| 5. C2×#15690781 | C2 | semidirect |
| 6. C2×D4×Dic97 | C2 | semidirect |
| 7. C2×C42×D97 | C2 | semidirect |
| 8. C2×#127×C97 | C2 | semidirect |
| 10. C2×#13095510 | C22 | semidirect |
| 16. #15690810 | C22 | semidirect |
| 17. #15690797 | C22 | semidirect |
| 18. #15690781 | C22 | semidirect |
| 19. D4×Dic97 | C22 | semidirect |
| 24. C42×D97 | C22 | semidirect |
| 27. #127×C97 | C22 | semidirect |
| 41. C2×C4×D97 | D4 | semidirect |
| 66. C2×C4×C97 | #41 | semidirect |
| 78. C4×D97 | C2×D4 | semidirect |
| 90. C4×C97 | C2×#41 | semidirect |
| 95. C2×C97 | #532 | semidirect |
| 100. C97 | C2×#532 | semidirect |
| 104. C2×#127 | D97 | semidirect |
| 119. #127 | C2×D97 | semidirect |
| 177. C2×C4 | #13095510 | semidirect |
| 193. C4 | C2×#13095510 | semidirect |
| 196. C2 | #20261283 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13095510 | 2 |
| #20261283 | 2 |
| C2×#15690810 | 2 |
| C2×#15690797 | 2 |
| C2×#15690781 | 2 |
| C2×D4×Dic97 | 2 |
| C2×C42×D97 | 2 |
| C2×#127×C97 | 2 |
| C2×#532 | 97 |