Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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×D97 |
| commutator, socle, cosocle | C23×C97 | C24 |
| Fitting | C2×#127×C97 | C2 |
| hypercenter | C2×#127 | D97 |
| hypocenter | C97 | #2856 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690781 | C2 | semidirect |
| 2. C2×C42×D97 | C2 | semidirect |
| 3. #20260638 | C2 | semidirect |
| 4. #20260605 | C2 | semidirect |
| 5. C2×#127×C97 | C2 | semidirect |
| 22. C2×C4×D97 | D4 | semidirect |
| 31. C42×C97 | D4 | semidirect |
| 57. C2×D97 | #127 | semidirect |
| 58. C4×C97 | #121 | semidirect |
| 67. C97 | #2856 | semidirect |
| 68. C2×#127 | D97 | semidirect |
| 98. C42 | C22⋊D97 | semidirect |
| 127. C4 | #15690781 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690781 | 2 |
| C2×C42×D97 | 2 |
| #20260638 | 2 |
| #20260605 | 2 |
| C2×#127×C97 | 2 |
| #2856 | 97 |