Finite Group DatabasebetaFinite Group DatabasebetaOrder 8224.
236 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×#11759967 | C2 | 1 | 14 | yes | maximal, direct product factor |
| 2 | C23×D257 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#11759967 | C22 | 1 | 28 | yes | direct product factor |
| 4 | C22×D257 | C4 | 1 | 7 | yes | complemented |
| 5 | C22×D257 | C22 | 1 | 7 | yes | — |
| 6 | C23×C257 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C22×C257 | C2×C4 | 1 | 7 | yes | complemented |
| 8 | C2×D257 | C2×C4 | 1 | 21 | yes | complemented |
| 9 | C2×D257 | C23 | 1 | 7 | yes | — |
| 10 | #11759967 | C23 | 1 | 8 | yes | direct product factor |
| 11 | C2×C257 | C22×C4 | 1 | 7 | yes | complemented |
| 12 | D257 | C22×C4 | 1 | 7 | yes | complemented |
| 13 | D257 | C24 | 1 | 1 | yes | characteristic |
| 14 | C257 | C23×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C23×C4 | — | 257 | 1 | no | maximal, Sylow |
| 16 | C24 | — | 257 | 1 | no | — |
| 17 | C22×C4 | — | 257 | 14 | no | — |
| 18 | C23 | — | 257 | 7 | no | — |
| 19 | C23 | — | 257 | 7 | no | — |
| 20 | C23 | #11759967 | 1 | 1 | yes | characteristic, direct product factor |
| 21 | C2×C4 | — | 257 | 28 | no | — |
| 22 | C22 | — | 257 | 7 | no | — |
| 23 | C22 | — | 257 | 21 | no | — |
| 24 | C22 | C2×#11759967 | 1 | 7 | yes | direct product factor |
| 25 | C4 | — | 257 | 8 | no | — |
| 26 | C2 | — | 257 | 1 | no | — |
| 27 | C2 | — | 257 | 7 | no | — |
| 28 | C2 | C22×#11759967 | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #11759967 |
| commutator, hypocenter | C257 | C23×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C257 | C4 |
| LCS | G ▷ C257 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#11759967 | C2 | direct (N × H) |
| 3. C2×#11759967 | C22 | direct (N × H) |
| 4. C22×D257 | C4 | semidirect |
| 6. C23×C257 | C4 | semidirect |
| 7. C22×C257 | C2×C4 | semidirect |
| 8. C2×D257 | C2×C4 | semidirect |
| 10. #11759967 | C23 | direct (N × H) |
| 11. C2×C257 | C22×C4 | semidirect |
| 12. D257 | C22×C4 | semidirect |
| 14. C257 | C23×C4 | semidirect |
| 20. C23 | #11759967 | direct (N × H) |
| 24. C22 | C2×#11759967 | direct (N × H) |
| 28. C2 | C22×#11759967 | direct (N × H) |