Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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 | #123×C197 | C2 |
| hypercenter | #123 | D197 |
| hypocenter | C197 | #547 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691358 | C2 | semidirect |
| 2. C2×#15691357 | C2 | semidirect |
| 3. #20263325 | C2 | semidirect |
| 4. #20263311 | C2 | semidirect |
| 5. #20263310 | C2 | semidirect |
| 6. #20263308 | C2 | semidirect |
| 7. #20263304 | C2 | semidirect |
| 8. #20263303 | C2 | semidirect |
| 9. #20263299 | C2 | semidirect |
| 10. #20263298 | C2 | semidirect |
| 11. #20263286 | C2 | semidirect |
| 12. #20263286 | C2 | semidirect |
| 13. C2×#15691337 | C2 | semidirect |
| 14. C4⋊C4×D197 | C2 | semidirect |
| 15. #123×C197 | C2 | semidirect |
| 61. C22⋊D197 | D4 | semidirect |
| 107. Dic197 | #41 | semidirect |
| 115. C197 | #547 | semidirect |
| 123. #123 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691358 | 2 |
| C2×#15691357 | 2 |
| #20263325 | 2 |
| #20263311 | 2 |
| #20263310 | 2 |
| #20263308 | 2 |
| #20263304 | 2 |
| #20263303 | 2 |
| #20263299 | 2 |
| #20263298 | 2 |
| #20263286 | 2 |
| C2×#15691337 | 2 |
| C4⋊C4×D197 | 2 |
| #123×C197 | 2 |
| #547 | 197 |