Finite Group DatabasebetaFinite Group DatabasebetaOrder 2368.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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×D37 |
| commutator, socle, cosocle | C22×C37 | C24 |
| Fitting | #126×C37 | C2 |
| prime radical | C2×C22⋊D37 | C22 |
| hypercenter | #126 | D37 |
| hypocenter | C37 | #563 |
| LCS | G ▷ C22×C37 ▷ C37 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11924536 | C2 | semidirect |
| 2. #11924535 | C2 | semidirect |
| 3. #11924531 | C2 | semidirect |
| 4. #11924530 | C2 | semidirect |
| 5. #11924529 | C2 | semidirect |
| 6. #11924527 | C2 | semidirect |
| 7. #11924526 | C2 | semidirect |
| 11. #11924516 | C2 | semidirect |
| 12. #11924514 | C2 | semidirect |
| 13. #11924513 | C2 | semidirect |
| 14. #11924513 | C2 | semidirect |
| 15. #126×C37 | C2 | semidirect |
| 98. C37 | #563 | semidirect |
| 104. #126 | D37 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11924536 | 2 |
| #11924535 | 2 |
| #11924531 | 2 |
| #11924530 | 2 |
| #11924529 | 2 |
| #11924527 | 2 |
| #11924526 | 2 |
| #11924522 | 2 |
| #11924520 | 2 |
| #11924518 | 2 |
| #11924516 | 2 |
| #11924514 | 2 |
| #11924513 | 2 |
| #126×C37 | 2 |
| #563 | 37 |