Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1152 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 469 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×#125×C97 | C2 |
| socle | C23×C97 | C24 |
| hypercenter | C2×#125 | D97 |
| hypocenter | C97 | C2×#562 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261483 | C2 | direct (N × H) |
| 2. C2×#15690798 | C2 | semidirect |
| 3. C2×#15690795 | C2 | semidirect |
| 4. C2×#15690788 | C2 | semidirect |
| 5. C2×#15690785 | C2 | semidirect |
| 6. C2×#125×C97 | C2 | semidirect |
| 11. #15690798 | C22 | semidirect |
| 12. #15690795 | C22 | semidirect |
| 13. #15690788 | C22 | semidirect |
| 14. #15690785 | C22 | semidirect |
| 19. #125×C97 | C22 | semidirect |
| 71. C2×C97 | #562 | semidirect |
| 76. C97 | C2×#562 | semidirect |
| 78. C2×#125 | D97 | semidirect |
| 87. #125 | C2×D97 | semidirect |
| 148. C2 | #20261483 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261483 | 2 |
| C2×#15690798 | 2 |
| C2×#15690795 | 2 |
| C2×#15690788 | 2 |
| C2×#15690785 | 2 |
| C2×#125×C97 | 2 |
| C2×#562 | 97 |