Finite Group DatabasebetaFinite Group DatabasebetaOrder 56.
32 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 | 6 | yes | maximal, direct product factor |
| 3 | C2×C7 | C22 | 1 | 3 | yes | complemented |
| 4 | D7 | C22 | 1 | 4 | yes | direct product factor |
| 5 | C23 | — | 7 | 1 | no | maximal, Sylow |
| 6 | C7 | C23 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C22 | — | 7 | 6 | no | — |
| 8 | C22 | D7 | 1 | 1 | yes | characteristic, direct product factor |
| 9 | C2 | — | 7 | 4 | no | — |
| 10 | C2 | C2×D7 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7 |
| commutator, cosocle, hypocenter | C7 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ 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 | direct (N × H) |
| 3. C2×C7 | C22 | semidirect |
| 4. D7 | C22 | direct (N × H) |
| 6. C7 | C23 | semidirect |
| 8. C22 | D7 | direct (N × H) |
| 10. C2 | C2×D7 | direct (N × H) |
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 |
| C23 | 7 |