Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
106 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D16×D97 |
| commutator, cosocle | C16×C97 | C23 |
| Frattini | C16 | C22×D97 |
| Fitting | SD64×C97 | C2 |
| socle | C2×C97 | C2×D16 |
| hypercenter | SD64 | D97 |
| hypocenter | C97 | #2262 |
| LCS | G ▷ C16×C97 ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261168 | C2 | semidirect |
| 2. #20261164 | C2 | semidirect |
| 3. #20260575 | C2 | semidirect |
| 4. #20260574 | C2 | semidirect |
| 5. #20260570 | C2 | semidirect |
| 6. C32×D97 | C2 | semidirect |
| 7. SD64×C97 | C2 | semidirect |
| 53. C97 | #2262 | semidirect |
| 58. SD64 | D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261168 | 2 |
| #20261164 | 2 |
| #20260575 | 2 |
| #20260574 | 2 |
| #20260570 | 2 |
| C32×D97 | 2 |
| SD64×C97 | 2 |
| #2262 | 97 |