Finite Group DatabasebetaFinite Group DatabasebetaOrder 6824.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C853 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D853 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic853 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×C853 | — | 2 | 1 | no | — |
| 5 | C2×C853 | C22 | 1 | 1 | yes | characteristic |
| 6 | D853 | — | 2 | 1 | no | — |
| 7 | C853 | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | D4 | — | 853 | 1 | no | maximal, Sylow |
| 9 | C22 | — | 853 | 1 | no | — |
| 10 | C22 | D853 | 1 | 1 | yes | characteristic, complemented |
| 11 | C4 | — | 853 | 1 | no | — |
| 12 | C2 | — | 2 | 1 | no | — |
| 13 | C2 | — | 1706 | 1 | no | — |
| 14 | C2 | C2×D853 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D853 |
| commutator, socle | C2×C853 | C22 |
| Fitting | C22×C853 | C2 |
| hypercenter | C22 | D853 |
| hypocenter | C853 | D4 |
| LCS | G ▷ C2×C853 ▷ C853 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C853 | C2 | semidirect |
| 2. C2×D853 | C2 | semidirect |
| 3. Dic853 | C2 | semidirect |
| 7. C853 | D4 | semidirect |
| 10. C22 | D853 | semidirect |