Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×D8 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C7×C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7×D4 | — | 2 | 2 | no | — |
| 4 | C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 5 | D16 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 6 | C22×C7 | — | 4 | 2 | no | — |
| 7 | C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 8 | D8 | C2×C7 | 1 | 2 | yes | complemented |
| 9 | C16 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C7 | — | 8 | 2 | no | — |
| 11 | C2×C7 | D8 | 1 | 1 | yes | characteristic |
| 12 | D4 | — | 2 | 2 | no | — |
| 13 | C8 | C22×C7 | 1 | 1 | yes | characteristic |
| 14 | C7 | D16 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 15 | C22 | — | 4 | 2 | no | — |
| 16 | C4 | C7×D4 | 1 | 1 | yes | characteristic |
| 17 | C2 | — | 8 | 2 | no | — |
| 18 | C2 | C7×D8 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D8 |
| commutator, Frattini, cosocle | C8 | C22×C7 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×D8 | C2 | semidirect |
| 2. C7×C16 | C2 | semidirect |
| 5. D16 | C7 | direct (N × H) |
| 8. D8 | C2×C7 | semidirect |
| 9. C16 | C2×C7 | semidirect |
| 14. C7 | D16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×D8 | 2 |
| C7×C16 | 2 |
| D16 | 7 |