Finite Group DatabasebetaFinite Group DatabasebetaOrder 16807.
4453 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×H(3,𝔽7) | C7 | 1 | 400 | yes | maximal, complemented |
| 2 | C73 | C72 | 1 | 400 | yes | complemented |
| 3 | H(3,𝔽7) | C72 | 1 | 2450 | yes | complemented |
| 4 | C72 | — | 49 | 400 | no | — |
| 5 | C72 | C73 | 1 | 400 | yes | — |
| 6 | C7 | — | 7 | 400 | no | — |
| 7 | C7 | C74 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C7 | C74 |
| 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. C7×H(3,𝔽7) | C7 | semidirect |
| 2. C73 | C72 | semidirect |
| 3. H(3,𝔽7) | C72 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(3,𝔽7) | 7 |