Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
560 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 185 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 | #391×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| prime radical | C22×D388 | C22 |
| hypercenter | #391 | D97 |
| hypocenter | C97 | #2372 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20260642 | C2 | semidirect |
| 3. #20260627 | C2 | semidirect |
| 4. #20260625 | C2 | semidirect |
| 6. C4×#13095488 | C2 | semidirect |
| 7. #20260582 | C2 | semidirect |
| 8. #20260580 | C2 | semidirect |
| 9. #391×C97 | C2 | semidirect |
| 12. #15690799 | C4 | semidirect |
| 46. #13095488 | C2×C4 | semidirect |
| 47. #13095488 | C2×C4 | semidirect |
| 51. C4×Dic97 | C2×C4 | semidirect |
| 54. C2×Q8×C97 | C2×C4 | semidirect |
| 124. C97 | #2372 | semidirect |
| 127. #391 | D97 | semidirect |
| 159. C2×Q8 | C4×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690799 | 2 |
| #20260642 | 2 |
| #20260627 | 2 |
| #20260625 | 2 |
| #20260613 | 2 |
| C4×#13095488 | 2 |
| #20260582 | 2 |
| #20260580 | 2 |
| #391×C97 | 2 |
| #2372 | 97 |