Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 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 | #124×C197 | C2 |
| prime radical | C2×#15691357 | C2 |
| hypercenter | #124 | D197 |
| hypocenter | C197 | #536 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q8×D197 | C2 | semidirect |
| 3. #20263326 | C2 | semidirect |
| 4. #20263324 | C2 | semidirect |
| 5. #20263313 | C2 | semidirect |
| 6. #20263308 | C2 | semidirect |
| 7. #20263303 | C2 | semidirect |
| 8. #20263283 | C2 | semidirect |
| 9. #124×C197 | C2 | semidirect |
| 63. C197 | #536 | semidirect |
| 70. #124 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691357 | 2 |
| C2×Q8×D197 | 2 |
| #20263326 | 2 |
| #20263324 | 2 |
| #20263313 | 2 |
| #20263308 | 2 |
| #20263303 | 2 |
| #20263283 | 2 |
| #124×C197 | 2 |
| #536 | 197 |