Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D197 |
| commutator, socle, cosocle | C22×C197 | C24 |
| Fitting | #121×C197 | C2 |
| hypercenter | #121 | D197 |
| hypocenter | C197 | #537 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691357 | C2 | semidirect |
| 2. #20263304 | C2 | semidirect |
| 3. #20263303 | C2 | semidirect |
| 4. #20263300 | C2 | semidirect |
| 5. #20263299 | C2 | semidirect |
| 6. #20263298 | C2 | semidirect |
| 7. #20263298 | C2 | semidirect |
| 8. C2×#15691337 | C2 | semidirect |
| 9. #121×C197 | C2 | semidirect |
| 40. C2×Dic197 | D4 | semidirect |
| 75. C197 | #537 | semidirect |
| 84. #121 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691357 | 2 |
| #20263304 | 2 |
| #20263303 | 2 |
| #20263300 | 2 |
| #20263299 | 2 |
| #20263298 | 2 |
| C2×#15691337 | 2 |
| #121×C197 | 2 |
| #537 | 197 |