Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
112 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×D97 |
| commutator, socle | C2×C97 | C22×C16 |
| Fitting | M64×C97 | C2 |
| prime radical | C22×D97 | C16 |
| cosocle | C16×C97 | C23 |
| hypercenter | M64 | D97 |
| hypocenter | C97 | C2×M64 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C16 ◁ M64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20260569 | C2 | semidirect |
| 3. #20260568 | C2 | semidirect |
| 4. C32×D97 | C2 | semidirect |
| 5. M64×C97 | C2 | semidirect |
| 12. #15690716 | C22 | semidirect |
| 14. C32×C97 | C22 | semidirect |
| 48. D97 | M64 | direct (N × H) |
| 50. C97 | C2×M64 | semidirect |
| 54. M64 | D97 | direct (N × H) |
| 64. C32 | C2×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C16×D97 | 2 |
| #20260569 | 2 |
| #20260568 | 2 |
| C32×D97 | 2 |
| M64×C97 | 2 |
| C2×M64 | 97 |