Finite Group DatabasebetaFinite Group DatabasebetaOrder 8416.
748 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×D263 | C2 | 1 | 30 | yes | maximal, direct product factor |
| 2 | C24×C263 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C22×D263 | C22 | 1 | 140 | yes | direct product factor |
| 4 | C23×C263 | C22 | 1 | 15 | yes | complemented |
| 5 | C22×C263 | C23 | 1 | 35 | yes | complemented |
| 6 | C2×D263 | C23 | 1 | 120 | yes | direct product factor |
| 7 | C2×C263 | C24 | 1 | 15 | yes | complemented |
| 8 | D263 | C24 | 1 | 16 | yes | direct product factor |
| 9 | C263 | C25 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C25 | — | 263 | 1 | no | maximal, Sylow |
| 11 | C24 | — | 263 | 30 | no | — |
| 12 | C24 | D263 | 1 | 1 | yes | characteristic, direct product factor |
| 13 | C23 | — | 263 | 140 | no | — |
| 14 | C23 | C2×D263 | 1 | 15 | yes | direct product factor |
| 15 | C22 | — | 263 | 120 | no | — |
| 16 | C22 | C22×D263 | 1 | 35 | yes | direct product factor |
| 17 | C2 | — | 263 | 16 | no | — |
| 18 | C2 | C23×D263 | 1 | 15 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D263 |
| commutator, hypocenter | C263 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C24×C263 | C2 |
| LCS | G ▷ C263 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D263 | C2 | direct (N × H) |
| 2. C24×C263 | C2 | semidirect |
| 3. C22×D263 | C22 | direct (N × H) |
| 4. C23×C263 | C22 | semidirect |
| 5. C22×C263 | C23 | semidirect |
| 6. C2×D263 | C23 | direct (N × H) |
| 7. C2×C263 | C24 | semidirect |
| 8. D263 | C24 | direct (N × H) |
| 9. C263 | C25 | semidirect |
| 12. C24 | D263 | direct (N × H) |
| 14. C23 | C2×D263 | direct (N × H) |
| 16. C22 | C22×D263 | direct (N × H) |
| 18. C2 | C23×D263 | direct (N × H) |