Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3966 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C43×C53 | C2 | 1 | 15 | yes | maximal |
| 2 | C22×C42×C53 | C22 | 1 | 35 | yes | — |
| 3 | C43×C53 | C4 | 1 | 120 | yes | direct product factor |
| 4 | C23×C4×C53 | C23 | 1 | 15 | yes | — |
| 5 | C2×C42×C53 | C2×C4 | 1 | 420 | yes | — |
| 6 | C24×C53 | C24 | 1 | 1 | yes | characteristic |
| 7 | C22×C4×C53 | C22×C4 | 1 | 210 | yes | — |
| 8 | C42×C53 | C42 | 1 | 560 | yes | direct product factor |
| 9 | C23×C53 | C23×C4 | 1 | 15 | yes | — |
| 10 | C2×C4×C53 | C2×C42 | 1 | 420 | yes | — |
| 11 | C44 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 12 | C22×C53 | C22×C42 | 1 | 35 | yes | — |
| 13 | C4×C53 | C43 | 1 | 120 | yes | direct product factor |
| 14 | C2×C43 | C2×C53 | 1 | 15 | yes | — |
| 15 | C2×C53 | C2×C43 | 1 | 15 | yes | — |
| 16 | C22×C42 | C22×C53 | 1 | 35 | yes | — |
| 17 | C43 | C4×C53 | 1 | 120 | yes | direct product factor |
| 18 | C53 | C44 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 19 | C23×C4 | C23×C53 | 1 | 15 | yes | — |
| 20 | C2×C42 | C2×C4×C53 | 1 | 420 | yes | — |
| 21 | C24 | C24×C53 | 1 | 1 | yes | characteristic |
| 22 | C22×C4 | C22×C4×C53 | 1 | 210 | yes | — |
| 23 | C42 | C42×C53 | 1 | 560 | yes | direct product factor |
| 24 | C23 | C23×C4×C53 | 1 | 15 | yes | — |
| 25 | C2×C4 | C2×C42×C53 | 1 | 420 | yes | — |
| 26 | C22 | C22×C42×C53 | 1 | 35 | yes | — |
| 27 | C4 | C43×C53 | 1 | 120 | yes | direct product factor |
| 28 | C2 | C2×C43×C53 | 1 | 15 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C24 | C24×C53 |
| socle, prime radical | C24×C53 | C24 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C43×C53 | C4 | direct (N × H) |
| 8. C42×C53 | C42 | direct (N × H) |
| 11. C44 | C53 | direct (N × H) |
| 13. C4×C53 | C43 | direct (N × H) |
| 17. C43 | C4×C53 | direct (N × H) |
| 18. C53 | C44 | direct (N × H) |
| 23. C42 | C42×C53 | direct (N × H) |
| 27. C4 | C43×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C43×C53 | 2 |
| C44 | 53 |