Finite Group DatabasebetaFinite Group DatabasebetaOrder 23488.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 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×D367 |
| commutator, socle, cosocle | C22×C367 | C24 |
| Fitting | C22⋊Q8×C367 | C2 |
| hypercenter | C22⋊Q8 | D367 |
| hypocenter | C367 | #562 |
| LCS | G ▷ C22×C367 ▷ C367 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26044251 | C2 | semidirect |
| 2. #26044250 | C2 | semidirect |
| 3. #26044247 | C2 | semidirect |
| 4. #26044246 | C2 | semidirect |
| 5. #26044246 | C2 | semidirect |
| 6. #26044244 | C2 | semidirect |
| 7. #26044243 | C2 | semidirect |
| 8. #26044241 | C2 | semidirect |
| 9. #26044239 | C2 | semidirect |
| 10. #26044237 | C2 | semidirect |
| 11. #26044236 | C2 | semidirect |
| 12. #26044235 | C2 | semidirect |
| 13. #26044234 | C2 | semidirect |
| 14. #26044234 | C2 | semidirect |
| 15. C22⋊Q8×C367 | C2 | semidirect |
| 109. C367 | #562 | semidirect |
| 121. C22⋊Q8 | D367 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26044251 | 2 |
| #26044250 | 2 |
| #26044247 | 2 |
| #26044246 | 2 |
| #26044244 | 2 |
| #26044243 | 2 |
| #26044241 | 2 |
| #26044239 | 2 |
| #26044237 | 2 |
| #26044236 | 2 |
| #26044235 | 2 |
| #26044234 | 2 |
| C22⋊Q8×C367 | 2 |
| #562 | 367 |