Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
896 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#564 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 2 | C2×C7×#125 | C2 | 1 | 15 | yes | maximal |
| 3 | C7×#125 | C22 | 1 | 120 | yes | — |
| 4 | C2×C7×C4⋊C4 | C22 | 1 | 30 | yes | — |
| 5 | C2×C42×C7 | C22 | 1 | 5 | yes | — |
| 6 | C2×#564 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C22×C4×C7 | C23 | 1 | 15 | yes | — |
| 8 | C7×C4⋊C4 | C23 | 1 | 120 | yes | — |
| 9 | C42×C7 | C23 | 1 | 20 | yes | — |
| 10 | #564 | C2×C7 | 1 | 16 | yes | direct product factor |
| 11 | C2×#125 | C2×C7 | 1 | 15 | yes | — |
| 12 | C23×C7 | C24 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×C7 | — | 2 | 30 | no | — |
| 14 | C2×C4×C7 | C24 | 1 | 30 | yes | — |
| 15 | #125 | C22×C7 | 1 | 120 | yes | — |
| 16 | C2×C4⋊C4 | C22×C7 | 1 | 30 | yes | — |
| 17 | C2×C42 | C22×C7 | 1 | 5 | yes | — |
| 18 | C22×C7 | 2−1+4 | 1 | 6 | yes | — |
| 19 | C22×C7 | C25 | 1 | 1 | yes | characteristic |
| 20 | C4×C7 | — | 2 | 30 | no | — |
| 21 | C22×C4 | C23×C7 | 1 | 15 | yes | — |
| 22 | C4⋊C4 | C23×C7 | 1 | 120 | yes | — |
| 23 | C42 | C23×C7 | 1 | 20 | yes | — |
| 24 | C2×C7 | #564 | 1 | 4 | yes | direct product factor |
| 25 | C2×C7 | C2×2−1+4 | 1 | 3 | yes | — |
| 26 | C23 | C24×C7 | 1 | 1 | yes | characteristic |
| 27 | C2×C4 | — | 2 | 30 | no | — |
| 28 | C2×C4 | C24×C7 | 1 | 30 | yes | — |
| 29 | C7 | C2×#564 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 30 | C22 | C7×2−1+4 | 1 | 6 | yes | — |
| 31 | C22 | C25×C7 | 1 | 1 | yes | characteristic |
| 32 | C4 | — | 2 | 30 | no | — |
| 33 | C2 | C7×#564 | 1 | 4 | yes | direct product factor |
| 34 | C2 | C2×C7×2−1+4 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#564 | C2 | direct (N × H) |
| 6. C2×#564 | C7 | direct (N × H) |
| 10. #564 | C2×C7 | direct (N × H) |
| 24. C2×C7 | #564 | direct (N × H) |
| 29. C7 | C2×#564 | direct (N × H) |
| 33. C2 | C7×#564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#564 | 2 |
| C2×C7×#125 | 2 |
| C2×#564 | 7 |