Finite Group DatabasebetaFinite Group DatabasebetaOrder 14272.
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×D223 |
| commutator, socle, cosocle | C22×C223 | C24 |
| Fitting | C22⋊Q8×C223 | C2 |
| hypercenter | C22⋊Q8 | D223 |
| hypocenter | C223 | #562 |
| LCS | G ▷ C22×C223 ▷ C223 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20297294 | C2 | semidirect |
| 2. #20297293 | C2 | semidirect |
| 3. #20297290 | C2 | semidirect |
| 4. #20297289 | C2 | semidirect |
| 5. #20297289 | C2 | semidirect |
| 6. #20297287 | C2 | semidirect |
| 7. #20297286 | C2 | semidirect |
| 8. #20297284 | C2 | semidirect |
| 9. #20297282 | C2 | semidirect |
| 10. #20297280 | C2 | semidirect |
| 11. #20297279 | C2 | semidirect |
| 12. #20297278 | C2 | semidirect |
| 13. #20297277 | C2 | semidirect |
| 14. #20297277 | C2 | semidirect |
| 15. C22⋊Q8×C223 | C2 | semidirect |
| 109. C223 | #562 | semidirect |
| 121. C22⋊Q8 | D223 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20297294 | 2 |
| #20297293 | 2 |
| #20297290 | 2 |
| #20297289 | 2 |
| #20297287 | 2 |
| #20297286 | 2 |
| #20297284 | 2 |
| #20297282 | 2 |
| #20297280 | 2 |
| #20297279 | 2 |
| #20297278 | 2 |
| #20297277 | 2 |
| C22⋊Q8×C223 | 2 |
| #562 | 223 |