Finite Group DatabasebetaFinite Group DatabasebetaOrder 8224.
76 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#14523348 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 2 | C22×#11759967 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14523348 | C22 | 1 | 4 | yes | direct product factor |
| 4 | C2×#11759967 | C4 | 1 | 3 | yes | — |
| 5 | C2×#11759967 | C22 | 1 | 3 | yes | — |
| 6 | C22×D257 | C4 | 1 | 1 | yes | characteristic |
| 7 | C22×C257 | C8 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×D257 | C8 | 1 | 3 | yes | complemented |
| 9 | C2×D257 | C2×C4 | 1 | 3 | yes | — |
| 10 | #11759967 | C2×C4 | 1 | 3 | yes | — |
| 11 | #11759967 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C257 | C2×C8 | 1 | 3 | yes | complemented |
| 13 | D257 | C2×C8 | 1 | 3 | yes | complemented |
| 14 | D257 | C22×C4 | 1 | 1 | yes | characteristic |
| 15 | C257 | C22×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C22×C8 | — | 257 | 1 | no | maximal, Sylow |
| 17 | C22×C4 | — | 257 | 1 | no | — |
| 18 | C2×C8 | — | 257 | 6 | no | — |
| 19 | C23 | — | 257 | 1 | no | — |
| 20 | C2×C4 | — | 257 | 3 | no | — |
| 21 | C2×C4 | — | 257 | 3 | no | — |
| 22 | C8 | — | 257 | 4 | no | — |
| 23 | C22 | — | 257 | 3 | no | — |
| 24 | C22 | — | 257 | 3 | no | — |
| 25 | C22 | #14523348 | 1 | 1 | yes | characteristic, direct product factor |
| 26 | C4 | — | 257 | 1 | no | — |
| 27 | C4 | — | 257 | 3 | no | — |
| 28 | C2 | — | 257 | 1 | no | — |
| 29 | C2 | — | 257 | 3 | no | — |
| 30 | C2 | C2×#14523348 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #14523348 |
| commutator, hypocenter | C257 | C22×C8 |
| Frattini | C1 | G |
| Fitting, socle | C22×C257 | C8 |
| LCS | G ▷ C257 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14523348 | C2 | direct (N × H) |
| 3. #14523348 | C22 | direct (N × H) |
| 7. C22×C257 | C8 | semidirect |
| 8. C2×D257 | C8 | semidirect |
| 12. C2×C257 | C2×C8 | semidirect |
| 13. D257 | C2×C8 | semidirect |
| 15. C257 | C22×C8 | semidirect |
| 25. C22 | #14523348 | direct (N × H) |
| 30. C2 | C2×#14523348 | direct (N × H) |