Finite Group DatabasebetaFinite Group DatabasebetaOrder 6724.
60 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C41⋊D41 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C41×D41 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C412 | C22 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C2×D41 | — | 41 | 2 | no | maximal |
| 5 | C2×C41 | — | 41 | 2 | no | — |
| 6 | D41 | — | 41 | 2 | no | — |
| 7 | D41 | — | 82 | 20 | no | — |
| 8 | D41 | D41 | 1 | 2 | yes | direct product factor |
| 9 | C41 | — | 2 | 20 | no | — |
| 10 | C41 | C2×D41 | 1 | 2 | yes | complemented |
| 11 | C22 | — | 1681 | 1 | no | Sylow |
| 12 | C2 | — | 41 | 2 | no | — |
| 13 | C2 | — | 1681 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C412 | C22 |
| LCS | G ▷ C412 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C41⋊D41 | C2 | semidirect |
| 2. C41×D41 | C2 | semidirect |
| 3. C412 | C22 | semidirect |
| 8. D41 | D41 | direct (N × H) |
| 10. C41 | C2×D41 | semidirect |