Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1036 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 203 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, socle, cosocle | C23×C97 | C24 |
| Fitting | #393×C97 | C2 |
| prime radical | C24×D97 | C22 |
| hypercenter | #393 | D97 |
| hypocenter | C97 | C2×#393 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×D97 | C2 | semidirect |
| 3. #20260648 | C2 | semidirect |
| 4. #20260647 | C2 | semidirect |
| 5. #20260646 | C2 | semidirect |
| 6. #95×D97 | C2 | semidirect |
| 7. #393×C97 | C2 | semidirect |
| 9. C22×Dic194 | C22 | semidirect |
| 17. #15690717 | C22 | semidirect |
| 18. #15690717 | C22 | semidirect |
| 19. C22×Q8×C97 | C22 | semidirect |
| 21. #95×C97 | C22 | semidirect |
| 24. C23×D97 | Q8 | semidirect |
| 46. C23×C97 | C2×Q8 | semidirect |
| 55. C22×D97 | C2×Q8 | semidirect |
| 91. D97 | #393 | direct (N × H) |
| 93. C97 | C2×#393 | semidirect |
| 99. #393 | D97 | direct (N × H) |
| 104. C22×Q8 | C2×D97 | semidirect |
| 114. #95 | C2×D97 | semidirect |
| 148. C23 | Q8×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D97 | 2 |
| C2×C22⋊C4×D97 | 2 |
| #20260648 | 2 |
| #20260647 | 2 |
| #20260646 | 2 |
| #95×D97 | 2 |
| #393×C97 | 2 |
| C2×#393 | 97 |