Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
196 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16810378 | C2 | 1 | 6 | yes | maximal |
| 2 | C53×#376 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×#88322 | C22 | 1 | 4 | yes | — |
| 4 | C2×C42×C53 | C22 | 1 | 3 | yes | — |
| 5 | #95×C53 | C4 | 1 | 4 | yes | — |
| 6 | C22×C4×C53 | C2×C4 | 1 | 6 | yes | — |
| 7 | C22×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 8 | C42×C53 | — | 2 | 6 | no | — |
| 9 | C2×#88322 | — | 2 | 12 | no | — |
| 10 | C23×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C53 | — | 2 | 6 | no | — |
| 12 | C2×C4×C53 | — | 2 | 12 | no | — |
| 13 | C2×C4×C53 | #41 | 1 | 6 | yes | — |
| 14 | #88322 | — | 4 | 4 | no | — |
| 15 | C22×C53 | #117 | 1 | 3 | yes | — |
| 16 | C22×C53 | #131 | 1 | 4 | yes | — |
| 17 | C4×C53 | — | 4 | 6 | no | — |
| 18 | C4×C53 | #126 | 1 | 4 | yes | — |
| 19 | #1840 | — | 53 | 1 | no | maximal, Sylow |
| 20 | C2×C53 | #383 | 1 | 1 | yes | characteristic |
| 21 | C2×C53 | #433 | 1 | 6 | yes | — |
| 22 | #376 | D53 | 1 | 1 | yes | characteristic |
| 23 | #336 | — | 53 | 6 | no | — |
| 24 | C53 | #1840 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 25 | C22×C8 | — | 53 | 4 | no | — |
| 26 | C2×C42 | C2×D53 | 1 | 3 | yes | — |
| 27 | #95 | Dic53 | 1 | 4 | yes | — |
| 28 | C22×C4 | C2×Dic53 | 1 | 6 | yes | — |
| 29 | C22×C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 30 | C2×C8 | — | 106 | 12 | no | — |
| 31 | C42 | — | 2 | 6 | no | — |
| 32 | C23 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 33 | C2×C4 | — | 2 | 6 | no | — |
| 34 | C2×C4 | — | 2 | 12 | no | — |
| 35 | C2×C4 | #11719945 | 1 | 6 | yes | — |
| 36 | C8 | — | 212 | 4 | no | — |
| 37 | C22 | #13133176 | 1 | 3 | yes | — |
| 38 | C22 | #13133224 | 1 | 4 | yes | — |
| 39 | C4 | — | 4 | 6 | no | — |
| 40 | C4 | #13133211 | 1 | 4 | yes | — |
| 41 | C2 | #16810491 | 1 | 1 | yes | characteristic |
| 42 | C2 | #16810684 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D53 |
| commutator, socle | C23×C53 | C22×C4 |
| Fitting | C53×#376 | C2 |
| hypercenter | #376 | D53 |
| hypocenter | C53 | #1840 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 24. C53 | #1840 | semidirect |