Finite Group DatabasebetaFinite Group DatabasebetaOrder 31164.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D53×#3862 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×C7⋊#6157 | — | 3 | 1 | no | maximal, Hall |
| 3 | C53×#3862 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C72×D53 | D3 | 1 | 1 | yes | characteristic |
| 5 | C72×C53 | Dic3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | ΓL(1,𝔽8)×D53 | — | 14 | 1 | no | — |
| 7 | C7⋊#6157 | — | 3 | 1 | no | — |
| 8 | C7×#6157 | — | 21 | 1 | no | — |
| 9 | ΓL(1,𝔽8)×C53 | — | 14 | 1 | no | — |
| 10 | C7×D53 | — | 2 | 1 | no | — |
| 11 | C7×D53 | — | 3 | 1 | no | — |
| 12 | C7×D53 | — | 3 | 1 | no | — |
| 13 | C3⋊#6157 | — | 49 | 1 | no | maximal, Hall |
| 14 | #10596868 | — | 53 | 1 | no | maximal, Hall |
| 15 | C7×C53 | — | 2 | 1 | no | — |
| 16 | C7×C53 | — | 3 | 1 | no | — |
| 17 | C7×C53 | — | 3 | 1 | no | — |
| 18 | C3×D53 | — | 49 | 1 | no | — |
| 19 | C2×#3862 | — | 53 | 1 | no | — |
| 20 | #6157 | — | 21 | 1 | no | Hall |
| 21 | C7×Dic7 | — | 159 | 1 | no | Hall |
| 22 | C3×C53 | — | 49 | 1 | no | Hall |
| 23 | #3862 | #6157 | 1 | 1 | yes | characteristic, complemented, Hall |
| 24 | D53 | #64418 | 1 | 1 | yes | characteristic |
| 25 | C2×C72 | — | 53 | 1 | no | — |
| 26 | C53 | #10596868 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 27 | C72 | C3⋊#6157 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 28 | C2×ΓL(1,𝔽8) | — | 742 | 1 | no | — |
| 29 | C4×C7 | — | 1113 | 1 | no | — |
| 30 | Dic7 | — | 159 | 1 | no | — |
| 31 | ΓL(1,𝔽8) | — | 14 | 1 | no | — |
| 32 | C2×C7 | — | 106 | 1 | no | — |
| 33 | C2×C7 | — | 159 | 1 | no | — |
| 34 | C2×C7 | — | 159 | 1 | no | — |
| 35 | Dic3 | — | 2597 | 1 | no | Hall |
| 36 | C7 | — | 2 | 1 | no | — |
| 37 | C7 | — | 3 | 1 | no | — |
| 38 | C7 | — | 3 | 1 | no | — |
| 39 | C2×C3 | — | 2597 | 1 | no | — |
| 40 | C4 | — | 1113 | 1 | no | Sylow |
| 41 | C3 | — | 49 | 1 | no | Sylow |
| 42 | C2 | — | 53 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C53×#3862 | C4 |
| Fitting, socle | C72×C53 | Dic3 |
| prime radical, cosocle | D53×#3862 | C2 |
| LCS | G ▷ C53×#3862 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#3862 | C4 | semidirect |
| 5. C72×C53 | Dic3 | semidirect |
| 23. #3862 | #6157 | semidirect |
| 26. C53 | #10596868 | semidirect |
| 27. C72 | C3⋊#6157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D53×#3862 | 2 |
| C7×C7⋊#6157 | 3 |
| C3⋊#6157 | 49 |
| #10596868 | 53 |