Finite Group DatabasebetaFinite Group DatabasebetaOrder 56.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C7 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D7 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic7 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×C7 | — | 2 | 1 | no | — |
| 5 | C2×C7 | C22 | 1 | 1 | yes | characteristic |
| 6 | D7 | — | 2 | 1 | no | — |
| 7 | D4 | — | 7 | 1 | no | maximal, Sylow |
| 8 | C7 | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C22 | — | 7 | 1 | no | — |
| 10 | C22 | D7 | 1 | 1 | yes | characteristic, complemented |
| 11 | C4 | — | 7 | 1 | no | — |
| 12 | C2 | — | 2 | 1 | no | — |
| 13 | C2 | — | 14 | 1 | no | — |
| 14 | C2 | C2×D7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7 |
| commutator, socle, cosocle | C2×C7 | C22 |
| Fitting | C22×C7 | C2 |
| hypercenter | C22 | D7 |
| hypocenter | C7 | D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7 | C2 | semidirect |
| 2. C2×D7 | C2 | semidirect |
| 3. Dic7 | C2 | semidirect |
| 8. C7 | D4 | semidirect |
| 10. C22 | D7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7 | 2 |
| C2×D7 | 2 |
| Dic7 | 2 |
| D4 | 7 |