Finite Group DatabasebetaFinite Group DatabasebetaOrder 31213.
70 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C73×C13 | C7 | 1 | 1 | yes | maximal, characteristic |
| 2 | C13×7−1+2 | C7 | 1 | 7 | yes | maximal, complemented |
| 3 | #14558877 | C13 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 4 | C72×C13 | — | 7 | 1 | no | — |
| 5 | C72×C13 | — | 7 | 7 | no | — |
| 6 | C72×C13 | C72 | 1 | 1 | yes | characteristic |
| 7 | C13×C49 | — | 7 | 7 | no | — |
| 8 | C73 | C7×C13 | 1 | 1 | yes | characteristic |
| 9 | 7−1+2 | C7×C13 | 1 | 7 | yes | complemented |
| 10 | C7×C13 | — | 7 | 7 | no | — |
| 11 | C7×C13 | — | 7 | 1 | no | — |
| 12 | C7×C13 | H(3,𝔽7) | 1 | 1 | yes | characteristic |
| 13 | C72 | — | 7 | 7 | no | — |
| 14 | C72 | — | 7 | 1 | no | — |
| 15 | C72 | C72×C13 | 1 | 1 | yes | characteristic |
| 16 | C49 | — | 7 | 7 | no | — |
| 17 | C13 | #14558877 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 18 | C7 | — | 7 | 7 | no | — |
| 19 | C7 | — | 7 | 1 | no | — |
| 20 | C7 | C13×H(3,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C7×C13 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C72×C13 |
| prime radical | C73×C13 | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7×C13 ◁ C72×C13 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C13×7−1+2 | C7 | semidirect |
| 3. #14558877 | C13 | direct (N × H) |
| 9. 7−1+2 | C7×C13 | semidirect |
| 17. C13 | #14558877 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C73×C13 | 7 |
| C13×7−1+2 | 7 |
| #14558877 | 13 |