Finite Group DatabasebetaFinite Group DatabasebetaOrder 10768.
54 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#14582089 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 2 | C22×D673 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×C673 | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | C2×D673 | C4 | 1 | 3 | yes | complemented |
| 5 | C2×D673 | C22 | 1 | 3 | yes | — |
| 6 | #14582089 | C22 | 1 | 4 | yes | direct product factor |
| 7 | C2×C673 | C2×C4 | 1 | 3 | yes | complemented |
| 8 | D673 | C2×C4 | 1 | 3 | yes | complemented |
| 9 | D673 | C23 | 1 | 1 | yes | characteristic |
| 10 | C673 | C22×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C22×C4 | — | 673 | 1 | no | maximal, Sylow |
| 12 | C23 | — | 673 | 1 | no | — |
| 13 | C2×C4 | — | 673 | 6 | no | — |
| 14 | C22 | — | 673 | 3 | no | — |
| 15 | C22 | — | 673 | 3 | no | — |
| 16 | C22 | #14582089 | 1 | 1 | yes | characteristic, direct product factor |
| 17 | C4 | — | 673 | 4 | no | — |
| 18 | C2 | — | 673 | 1 | no | — |
| 19 | C2 | — | 673 | 3 | no | — |
| 20 | C2 | C2×#14582089 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #14582089 |
| commutator, hypocenter | C673 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C673 | C4 |
| prime radical | C22×D673 | C2 |
| cosocle | D673 | C23 |
| LCS | G ▷ C673 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14582089 | C2 | direct (N × H) |
| 3. C22×C673 | C4 | semidirect |
| 4. C2×D673 | C4 | semidirect |
| 6. #14582089 | C22 | direct (N × H) |
| 7. C2×C673 | C2×C4 | semidirect |
| 8. D673 | C2×C4 | semidirect |
| 10. C673 | C22×C4 | semidirect |
| 16. C22 | #14582089 | direct (N × H) |
| 20. C2 | C2×#14582089 | direct (N × H) |