Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×Q16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Dic28 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C7×Q8 | — | 2 | 1 | no | — |
| 5 | Dic14 | — | 2 | 1 | no | — |
| 6 | C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 7 | Q32 | — | 7 | 1 | no | maximal, Sylow |
| 8 | C4×C7 | — | 4 | 1 | no | — |
| 9 | C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 10 | Dic7 | — | 4 | 1 | no | — |
| 11 | Q16 | — | 7 | 1 | no | — |
| 12 | Q16 | D7 | 1 | 1 | yes | characteristic |
| 13 | C16 | — | 7 | 1 | no | — |
| 14 | C2×C7 | D8 | 1 | 1 | yes | characteristic |
| 15 | Q8 | — | 2 | 1 | no | — |
| 16 | Q8 | — | 14 | 1 | no | — |
| 17 | C8 | C2×D7 | 1 | 1 | yes | characteristic |
| 18 | C7 | Q32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | C4 | — | 4 | 1 | no | — |
| 20 | C4 | — | 28 | 1 | no | — |
| 21 | C4 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 22 | C2 | #1141 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #1141 |
| commutator, cosocle | C7×C8 | C22 |
| Frattini | C8 | C2×D7 |
| Fitting | C7×Q16 | C2 |
| socle, prime radical | C2×C7 | D8 |
| hypercenter | Q16 | D7 |
| hypocenter | C7 | Q32 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 18. C7 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Q16 | 2 |
| Dic28 | 2 |
| C7⋊C16 | 2 |
| Q32 | 7 |