Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
3422 conjugacy classes of subgroups; only the 245 normal subgroups are shown.
This group has 3422 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, socle, cosocle | C23×C97 | C24 |
| Fitting | #379×C97 | C2 |
| hypercenter | #379 | D97 |
| hypocenter | C97 | #2846 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×D97 | C2 | semidirect |
| 2. C2×#15690780 | C2 | semidirect |
| 3. C2×#15690779 | C2 | semidirect |
| 4. #20260590 | C2 | semidirect |
| 5. #379×C97 | C2 | semidirect |
| 9. C22×D388 | C22 | semidirect |
| 10. C22×C22⋊D97 | C22 | semidirect |
| 13. C2×C22⋊Dic97 | C22 | semidirect |
| 14. C2×#13095488 | C22 | semidirect |
| 16. C2×C22⋊C4×C97 | C22 | semidirect |
| 22. C23×D97 | D4 | semidirect |
| 24. C23×D97 | C23 | semidirect |
| 28. C22×Dic97 | C23 | semidirect |
| 32. C24×C97 | D4 | semidirect |
| 34. C22×C4×C97 | C23 | semidirect |
| 60. C22×C97 | #120 | semidirect |
| 68. C2×D97 | #120 | semidirect |
| 77. C97 | #2846 | semidirect |
| 82. #379 | D97 | semidirect |
| 93. C2×C22⋊C4 | C2×D97 | semidirect |
| 103. C24 | D388 | semidirect |
| 109. C22×C4 | C22×D97 | semidirect |
| 143. C22 | #15690780 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×D97 | 2 |
| C2×#15690780 | 2 |
| C2×#15690779 | 2 |
| #20260590 | 2 |
| #379×C97 | 2 |
| #2846 | 97 |