Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
81 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2104⋊C7 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×#2104 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C22×AGL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 4 | #2104 | — | 7 | 1 | no | — |
| 5 | #2104 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#342 | — | 7 | 1 | no | — |
| 7 | C2×AGL(1,𝔽8) | — | 8 | 2 | no | — |
| 8 | C2×AGL(1,𝔽8) | — | 8 | 1 | no | — |
| 9 | C22×C22⋊C4 | — | 7 | 1 | no | — |
| 10 | #342 | — | 7 | 1 | no | — |
| 11 | #342 | — | 7 | 3 | no | — |
| 12 | AGL(1,𝔽8) | — | 8 | 1 | no | — |
| 13 | C25 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 14 | C23×C4 | — | 14 | 1 | no | — |
| 15 | C2×C22⋊C4 | — | 7 | 1 | no | — |
| 16 | C2×C22⋊C4 | — | 7 | 1 | no | — |
| 17 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 18 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 19 | C2×C22⋊C4 | — | 14 | 1 | no | — |
| 20 | C22×C7 | — | 64 | 1 | no | — |
| 21 | C24 | — | 7 | 1 | no | — |
| 22 | C24 | — | 7 | 1 | no | — |
| 23 | C24 | — | 8 | 2 | no | — |
| 24 | C24 | C2×AGL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 25 | C22×C4 | — | 14 | 1 | no | — |
| 26 | C22×C4 | — | 14 | 1 | no | — |
| 27 | C22×C4 | — | 28 | 1 | no | — |
| 28 | C22×C4 | — | 28 | 1 | no | — |
| 29 | C22×C4 | — | 56 | 2 | no | — |
| 30 | C22⋊C4 | — | 14 | 2 | no | — |
| 31 | C22⋊C4 | — | 14 | 2 | no | — |
| 32 | C22⋊C4 | — | 28 | 2 | no | — |
| 33 | C2×C7 | — | 64 | 1 | no | — |
| 34 | C2×C7 | — | 64 | 2 | no | — |
| 35 | C23 | — | 7 | 1 | no | — |
| 36 | C23 | — | 7 | 2 | no | — |
| 37 | C23 | — | 7 | 1 | no | — |
| 38 | C23 | — | 7 | 1 | no | — |
| 39 | C23 | — | 8 | 1 | no | — |
| 40 | C23 | — | 28 | 2 | no | — |
| 41 | C23 | — | 56 | 1 | no | — |
| 42 | C2×C4 | — | 28 | 1 | no | — |
| 43 | C2×C4 | — | 28 | 1 | no | — |
| 44 | C2×C4 | — | 56 | 2 | no | — |
| 45 | C2×C4 | — | 56 | 2 | no | — |
| 46 | C2×C4 | — | 56 | 1 | no | — |
| 47 | C2×C4 | — | 56 | 1 | no | — |
| 48 | C7 | — | 64 | 1 | no | Sylow |
| 49 | C22 | — | 7 | 1 | no | — |
| 50 | C22 | — | 7 | 1 | no | — |
| 51 | C22 | — | 14 | 2 | no | — |
| 52 | C22 | — | 28 | 1 | no | — |
| 53 | C22 | — | 28 | 1 | no | — |
| 54 | C22 | — | 28 | 2 | no | — |
| 55 | C22 | #401⋊C7 | 1 | 1 | yes | characteristic |
| 56 | C4 | — | 56 | 1 | no | — |
| 57 | C4 | — | 56 | 1 | no | — |
| 58 | C2 | — | 14 | 1 | no | — |
| 59 | C2 | — | 14 | 1 | no | — |
| 60 | C2 | #2104⋊C7 | 1 | 2 | yes | direct product factor |
| 61 | C2 | C2×#401⋊C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #401⋊C7 |
| commutator, cosocle, hypocenter | #2104 | C2×C7 |
| Frattini | C24 | C2×AGL(1,𝔽8) |
| Fitting | C2×#2104 | C7 |
| LCS | G ▷ #2104 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2104⋊C7 | C2 | direct (N × H) |
| 2. C2×#2104 | C7 | semidirect |
| 5. #2104 | C2×C7 | semidirect |
| 60. C2 | #2104⋊C7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2104⋊C7 | 2 |
| C2×#2104 | 7 |
| C22×AGL(1,𝔽8) | 8 |