Finite Group DatabasebetaFinite Group DatabasebetaOrder 8268.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C13×C53 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C13⋊C3×C53 | — | 4 | 1 | no | maximal, Hall |
| 3 | C2×C13×C53 | — | 3 | 1 | no | — |
| 4 | C13×C53 | A4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | A4×C53 | — | 13 | 1 | no | maximal, Hall |
| 6 | C22×C53 | C13⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C3×C53 | — | 52 | 1 | no | Hall |
| 8 | C22⋊(C13⋊C3) | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 9 | C2×C53 | — | 3 | 1 | no | — |
| 10 | C53 | C22⋊(C13⋊C3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 11 | C22×C13 | C3×C53 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | C13⋊C3 | — | 4 | 1 | no | Hall |
| 13 | C2×C13 | — | 3 | 1 | no | — |
| 14 | C13 | A4×C53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | A4 | — | 13 | 1 | no | Hall |
| 16 | C22 | C13⋊C3×C53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C3 | — | 52 | 1 | no | Sylow |
| 18 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C53 | C22⋊(C13⋊C3) |
| commutator, hypocenter | C22×C13 | C3×C53 |
| Frattini | C1 | G |
| Fitting, socle | C22×C13×C53 | C3 |
| LCS | G ▷ C22×C13 | |
| UCS | C1 ◁ C53 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C13×C53 | C3 | semidirect |
| 4. C13×C53 | A4 | semidirect |
| 6. C22×C53 | C13⋊C3 | semidirect |
| 8. C22⋊(C13⋊C3) | C53 | direct (N × H) |
| 10. C53 | C22⋊(C13⋊C3) | direct (N × H) |
| 11. C22×C13 | C3×C53 | semidirect |
| 14. C13 | A4×C53 | semidirect |
| 16. C22 | C13⋊C3×C53 | semidirect |