Finite Group DatabasebetaFinite Group DatabasebetaOrder 20288.
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×D317 |
| commutator, socle, cosocle | C22×C317 | C24 |
| Fitting | C22⋊Q8×C317 | C2 |
| hypercenter | C22⋊Q8 | D317 |
| hypocenter | C317 | #562 |
| LCS | G ▷ C22×C317 ▷ C317 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23778770 | C2 | semidirect |
| 2. #23778769 | C2 | semidirect |
| 3. #23778766 | C2 | semidirect |
| 4. #23778765 | C2 | semidirect |
| 5. #23778765 | C2 | semidirect |
| 6. #23778763 | C2 | semidirect |
| 7. #23778762 | C2 | semidirect |
| 8. #23778760 | C2 | semidirect |
| 9. #23778758 | C2 | semidirect |
| 10. #23778756 | C2 | semidirect |
| 11. #23778755 | C2 | semidirect |
| 12. #23778754 | C2 | semidirect |
| 13. #23778753 | C2 | semidirect |
| 14. #23778753 | C2 | semidirect |
| 15. C22⋊Q8×C317 | C2 | semidirect |
| 109. C317 | #562 | semidirect |
| 121. C22⋊Q8 | D317 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778770 | 2 |
| #23778769 | 2 |
| #23778766 | 2 |
| #23778765 | 2 |
| #23778763 | 2 |
| #23778762 | 2 |
| #23778760 | 2 |
| #23778758 | 2 |
| #23778756 | 2 |
| #23778755 | 2 |
| #23778754 | 2 |
| #23778753 | 2 |
| C22⋊Q8×C317 | 2 |
| #562 | 317 |