Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #15690781 |
| commutator, cosocle | C42×C97 | C23 |
| Frattini | C42 | C22×D97 |
| Fitting | #493×C97 | C2 |
| socle | C22×C97 | #121 |
| hypercenter | #493 | D97 |
| hypocenter | C97 | #1687 |
| LCS | G ▷ C42×C97 ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20260927 | C2 | semidirect |
| 2. #20260814 | C2 | semidirect |
| 3. #20260486 | C2 | semidirect |
| 4. #20260479 | C2 | semidirect |
| 5. #493×C97 | C2 | semidirect |
| 48. C8×C97 | D8 | semidirect |
| 69. C97 | #1687 | semidirect |
| 70. #493 | D97 | semidirect |
| 119. C8 | #13095496 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20260927 | 2 |
| #20260814 | 2 |
| #20260486 | 2 |
| #20260479 | 2 |
| #493×C97 | 2 |
| #1687 | 97 |