Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
46 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | SO(3,𝔽7) | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C5×AGL(1,𝔽7) | — | 8 | 1 | no | maximal |
| 4 | GL(3,𝔽2) | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 5 | C5×S4 | — | 14 | 1 | no | — |
| 6 | C5×ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 7 | C5×D8 | — | 21 | 1 | no | maximal, Hall |
| 8 | C5×D7 | — | 8 | 1 | no | — |
| 9 | C2×C5×D3 | — | 28 | 1 | no | maximal |
| 10 | C5×A4 | — | 14 | 1 | no | — |
| 11 | AGL(1,𝔽7) | — | 8 | 1 | no | — |
| 12 | C5×D4 | — | 21 | 1 | no | — |
| 13 | C5×D4 | — | 21 | 1 | no | — |
| 14 | C5×C8 | — | 21 | 1 | no | — |
| 15 | C5×C7 | — | 8 | 1 | no | Hall |
| 16 | C2×C3×C5 | — | 28 | 1 | no | — |
| 17 | C5×D3 | — | 28 | 1 | no | — |
| 18 | C5×D3 | — | 28 | 1 | no | — |
| 19 | S4 | — | 14 | 1 | no | — |
| 20 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 21 | C22×C5 | — | 14 | 1 | no | — |
| 22 | C22×C5 | — | 42 | 1 | no | — |
| 23 | C4×C5 | — | 21 | 1 | no | — |
| 24 | D8 | — | 21 | 1 | no | Sylow |
| 25 | C3×C5 | — | 28 | 1 | no | Hall |
| 26 | D7 | — | 8 | 1 | no | — |
| 27 | C2×D3 | — | 28 | 1 | no | — |
| 28 | A4 | — | 14 | 1 | no | — |
| 29 | C2×C5 | — | 21 | 1 | no | — |
| 30 | C2×C5 | — | 28 | 1 | no | — |
| 31 | D4 | — | 21 | 1 | no | — |
| 32 | D4 | — | 21 | 1 | no | — |
| 33 | C8 | — | 21 | 1 | no | — |
| 34 | C7 | — | 8 | 1 | no | Sylow |
| 35 | C2×C3 | — | 28 | 1 | no | — |
| 36 | D3 | — | 28 | 1 | no | — |
| 37 | D3 | — | 28 | 1 | no | — |
| 38 | C5 | SO(3,𝔽7) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 39 | C22 | — | 14 | 1 | no | — |
| 40 | C22 | — | 42 | 1 | no | — |
| 41 | C4 | — | 21 | 1 | no | — |
| 42 | C3 | — | 28 | 1 | no | Sylow |
| 43 | C2 | — | 21 | 1 | no | — |
| 44 | C2 | — | 28 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C5 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | GL(3,𝔽2) | C2×C5 |
| Frattini | C1 | G |
| socle | C5×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×GL(3,𝔽2) | C2 | semidirect |
| 2. SO(3,𝔽7) | C5 | direct (N × H) |
| 4. GL(3,𝔽2) | C2×C5 | semidirect |
| 38. C5 | SO(3,𝔽7) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×GL(3,𝔽2) | 2 |
| SO(3,𝔽7) | 5 |
| C5×AGL(1,𝔽7) | 8 |
| C5×D8 | 21 |
| C2×C5×D3 | 28 |