Finite Group DatabasebetaFinite Group DatabasebetaOrder 16807.
3718 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #14558883 | C7 | 1 | 343 | yes | maximal, complemented |
| 2 | C7×7−1+2 | C7 | 1 | 56 | yes | maximal, complemented |
| 3 | C7×H(3,𝔽7) | C7 | 1 | 1 | yes | maximal, characteristic |
| 4 | C73 | C72 | 1 | 8 | yes | — |
| 5 | 7−1+2 | C72 | 1 | 49 | yes | — |
| 6 | 7−1+2 | C72 | 1 | 2352 | yes | complemented |
| 7 | H(3,𝔽7) | C72 | 1 | 49 | yes | — |
| 8 | C7×C49 | C72 | 1 | 392 | yes | complemented |
| 9 | C72 | — | 49 | 8 | no | — |
| 10 | C72 | C73 | 1 | 56 | yes | — |
| 11 | C72 | C73 | 1 | 1 | yes | characteristic |
| 12 | C49 | C73 | 1 | 343 | yes | — |
| 13 | C7 | — | 7 | 1 | no | — |
| 14 | C7 | — | 7 | 56 | no | — |
| 15 | C7 | C74 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C7 | C74 |
| prime radical | C7×H(3,𝔽7) | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14558883 | C7 | semidirect |
| 2. C7×7−1+2 | C7 | semidirect |
| 6. 7−1+2 | C72 | semidirect |
| 8. C7×C49 | C72 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14558883 | 7 |
| C7×7−1+2 | 7 |
| C7×H(3,𝔽7) | 7 |