Finite Group DatabasebetaFinite Group DatabasebetaOrder 22336.
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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×D349 |
| commutator, socle, cosocle | C22×C349 | C24 |
| Fitting | #124×C349 | C2 |
| prime radical | C2×D4×C349 | C22 |
| hypercenter | #124 | D349 |
| hypocenter | C349 | #556 |
| LCS | G ▷ C22×C349 ▷ C349 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26013637 | C2 | semidirect |
| 2. #26013632 | C2 | semidirect |
| 3. #26013632 | C2 | semidirect |
| 5. #26013616 | C2 | semidirect |
| 6. Q8×Dic349 | C2 | semidirect |
| 7. C4×Dic698 | C2 | semidirect |
| 8. C4×Dic698 | C2 | semidirect |
| 10. #26013595 | C2 | semidirect |
| 69. C349 | #556 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26013637 | 2 |
| #26013632 | 2 |
| #26013626 | 2 |
| #26013616 | 2 |
| Q8×Dic349 | 2 |
| C4×Dic698 | 2 |
| D4×Dic349 | 2 |
| #26013595 | 2 |
| #124×C349 | 2 |
| #556 | 349 |