Finite Group DatabasebetaFinite Group DatabasebetaOrder 31124.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C31×C251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C31×C251 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | Dic251 | C31 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C2×C251 | C2×C31 | 1 | 1 | yes | characteristic |
| 5 | C251 | C4×C31 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C4×C31 | — | 251 | 1 | no | maximal, Hall |
| 7 | C2×C31 | D251 | 1 | 1 | yes | characteristic |
| 8 | C31 | Dic251 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C4 | — | 251 | 1 | no | Sylow |
| 10 | C2 | C31×D251 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C31 | D251 |
| commutator, hypocenter | C251 | C4×C31 |
| Frattini | C2 | C31×D251 |
| Fitting, socle, prime radical | C2×C31×C251 | C2 |
| cosocle | C2×C251 | C2×C31 |
| LCS | G ▷ C251 | |
| UCS | C1 ◁ C2×C31 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C31×C251 | C4 | semidirect |
| 3. Dic251 | C31 | direct (N × H) |
| 5. C251 | C4×C31 | semidirect |
| 8. C31 | Dic251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C31×C251 | 2 |
| Dic251 | 31 |
| C4×C31 | 251 |