Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D199 |
| commutator, cosocle | C22×C199 | C24 |
| Frattini | C22 | C23×D199 |
| Fitting | C22×D4×C199 | C2 |
| socle | C23×C199 | C23 |
| hypercenter | C22×D4 | D199 |
| hypocenter | C199 | C2×#121 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊D199 | C2 | semidirect |
| 2. C22×C4×D199 | C2 | semidirect |
| 3. #20263926 | C2 | direct (N × H) |
| 4. C2×C4⋊Dic199 | C2 | semidirect |
| 5. C2×C22⋊Dic199 | C2 | semidirect |
| 6. C22×D4×C199 | C2 | semidirect |
| 9. C2×C22⋊D199 | C22 | semidirect |
| 13. C2×C4×D199 | C22 | semidirect |
| 14. C4⋊Dic199 | C22 | semidirect |
| 15. C22⋊Dic199 | C22 | semidirect |
| 18. C2×D4×C199 | C22 | semidirect |
| 22. C22×D199 | D4 | semidirect |
| 36. C2×C4×C199 | D4 | semidirect |
| 49. C2×D199 | C2×D4 | semidirect |
| 50. C4×C199 | C2×D4 | semidirect |
| 54. C2×C199 | #121 | semidirect |
| 59. C199 | C2×#121 | semidirect |
| 62. C22×D4 | D199 | semidirect |
| 72. C2×D4 | C2×D199 | semidirect |
| 96. C2×C4 | C22⋊D199 | semidirect |
| 112. C4 | C2×C22⋊D199 | semidirect |
| 115. C2 | #20263926 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊D199 | 2 |
| C22×C4×D199 | 2 |
| #20263926 | 2 |
| C2×C4⋊Dic199 | 2 |
| C2×C22⋊Dic199 | 2 |
| C22×D4×C199 | 2 |
| C2×#121 | 199 |