Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
516 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C53×#401 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 2 | C2×#95×C53 | C2 | 1 | 7 | yes | maximal |
| 3 | C23×C4×C53 | C22 | 1 | 7 | yes | — |
| 4 | #95×C53 | C22 | 1 | 28 | yes | — |
| 5 | C24×C53 | C23 | 1 | 1 | yes | characteristic |
| 6 | C22×C4×C53 | — | 2 | 42 | no | — |
| 7 | C22×C4×C53 | C23 | 1 | 14 | yes | — |
| 8 | C23×C53 | #41 | 1 | 14 | yes | — |
| 9 | C23×C53 | C24 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×C53 | — | 2 | 42 | no | — |
| 11 | C2×C4×C53 | — | 4 | 28 | no | — |
| 12 | C22×C53 | #126 | 1 | 28 | yes | — |
| 13 | C22×C53 | C2×#41 | 1 | 7 | yes | — |
| 14 | C4×C53 | — | 4 | 14 | no | — |
| 15 | C2×#401 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 16 | C2×C53 | #401 | 1 | 8 | yes | direct product factor |
| 17 | C2×C53 | C2×#126 | 1 | 7 | yes | — |
| 18 | #401 | C2×C53 | 1 | 8 | yes | direct product factor |
| 19 | C2×#95 | C2×C53 | 1 | 7 | yes | — |
| 20 | C53 | C2×#401 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 21 | C23×C4 | C22×C53 | 1 | 7 | yes | — |
| 22 | #95 | C22×C53 | 1 | 28 | yes | — |
| 23 | C24 | C23×C53 | 1 | 1 | yes | characteristic |
| 24 | C22×C4 | — | 2 | 42 | no | — |
| 25 | C22×C4 | C23×C53 | 1 | 14 | yes | — |
| 26 | C23 | #41×C53 | 1 | 14 | yes | — |
| 27 | C23 | C24×C53 | 1 | 1 | yes | characteristic |
| 28 | C2×C4 | — | 2 | 42 | no | — |
| 29 | C2×C4 | — | 4 | 28 | no | — |
| 30 | C22 | #126×C53 | 1 | 28 | yes | — |
| 31 | C22 | C2×#41×C53 | 1 | 7 | yes | — |
| 32 | C4 | — | 4 | 14 | no | — |
| 33 | C2 | C53×#401 | 1 | 8 | yes | direct product factor |
| 34 | C2 | C2×#126×C53 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C53 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#401 | C2 | direct (N × H) |
| 15. C2×#401 | C53 | direct (N × H) |
| 16. C2×C53 | #401 | direct (N × H) |
| 18. #401 | C2×C53 | direct (N × H) |
| 20. C53 | C2×#401 | direct (N × H) |
| 33. C2 | C53×#401 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#401 | 2 |
| C2×#95×C53 | 2 |
| C2×#401 | 53 |