Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
688 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 213 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, prime radical | C2×#120×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| hypercenter | C2×#120 | D97 |
| hypocenter | C97 | #2375 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20260638 | C2 | semidirect |
| 2. #20260628 | C2 | semidirect |
| 3. #20260615 | C2 | semidirect |
| 4. #20260591 | C2 | semidirect |
| 6. C23×Dic97 | C22 | semidirect |
| 7. C2×C4⋊Dic97 | C22 | semidirect |
| 10. C2×C22⋊Dic97 | C22 | semidirect |
| 11. C2×C4×Dic97 | C22 | semidirect |
| 12. #15690717 | C22 | semidirect |
| 15. #120×C97 | C4 | semidirect |
| 27. C24×C97 | C2×C4 | semidirect |
| 34. C22⋊C4×C97 | C2×C4 | semidirect |
| 77. C97 | #2375 | semidirect |
| 86. #120 | Dic97 | semidirect |
| 96. C24 | C2×Dic97 | semidirect |
| 110. C22⋊C4 | C2×Dic97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20260638 | 2 |
| #20260628 | 2 |
| #20260615 | 2 |
| #20260591 | 2 |
| C2×#120×C97 | 2 |
| #2375 | 97 |