Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
740 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C22×C97 | C23×C4 |
| Fitting, prime radical | C2×#127×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| hypercenter | C2×#127 | D97 |
| hypocenter | C97 | #2328 |
| 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. C2×D4×Dic97 | C2 | semidirect |
| 2. #20260638 | C2 | semidirect |
| 3. #20260615 | C2 | semidirect |
| 4. #20260601 | C2 | semidirect |
| 5. #20260581 | C2 | semidirect |
| 18. #127×C97 | C4 | semidirect |
| 29. C4×Dic97 | D4 | semidirect |
| 34. C2×D4×C97 | C2×C4 | semidirect |
| 35. C2×D4×C97 | C2×C4 | semidirect |
| 38. C42×C97 | C2×C4 | semidirect |
| 79. C97 | #2328 | semidirect |
| 89. #127 | Dic97 | semidirect |
| 103. C2×D4 | C2×Dic97 | semidirect |
| 104. C2×D4 | C2×Dic97 | semidirect |
| 117. C42 | C2×Dic97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×Dic97 | 2 |
| #20260638 | 2 |
| #20260615 | 2 |
| #20260601 | 2 |
| #20260581 | 2 |
| C2×#127×C97 | 2 |
| #2328 | 97 |