Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
656 conjugacy classes of subgroups; only the 245 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 245 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 | #382×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| prime radical | C23×D97 | C23 |
| hypercenter | #382 | D97 |
| hypocenter | C97 | #2285 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #382 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20260604 | C2 | semidirect |
| 4. #20260601 | C2 | semidirect |
| 6. #20260586 | C2 | semidirect |
| 7. #382×C97 | C2 | semidirect |
| 17. C42×D97 | C4 | semidirect |
| 62. C4×D97 | C4⋊C4 | semidirect |
| 76. Dic97 | #117 | semidirect |
| 83. C97 | #2285 | semidirect |
| 88. #382 | D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690768 | 2 |
| C2×C42×D97 | 2 |
| #20260604 | 2 |
| #20260601 | 2 |
| #20260594 | 2 |
| #20260586 | 2 |
| #382×C97 | 2 |
| #2285 | 97 |