Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×AGL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | AGL(1,𝔽8) | C4 | 1 | 1 | yes | characteristic, direct product factor |
| 3 | C23×C4 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | C4×C7 | — | 8 | 1 | no | maximal |
| 5 | C24 | C2×C7 | 1 | 1 | yes | characteristic |
| 6 | C22×C4 | — | 7 | 1 | no | — |
| 7 | C22×C4 | — | 7 | 1 | no | — |
| 8 | C2×C7 | — | 8 | 1 | no | — |
| 9 | C23 | — | 7 | 1 | no | — |
| 10 | C23 | — | 7 | 1 | no | — |
| 11 | C23 | C4×C7 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×C4 | — | 7 | 1 | no | — |
| 13 | C2×C4 | — | 7 | 3 | no | — |
| 14 | C7 | — | 8 | 1 | no | Sylow |
| 15 | C22 | — | 7 | 1 | no | — |
| 16 | C22 | — | 7 | 3 | no | — |
| 17 | C22 | — | 7 | 1 | no | — |
| 18 | C4 | — | 7 | 1 | no | — |
| 19 | C4 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 20 | C2 | — | 7 | 1 | no | — |
| 21 | C2 | — | 7 | 1 | no | — |
| 22 | C2 | C2×AGL(1,𝔽8) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | AGL(1,𝔽8) |
| commutator, hypocenter | C23 | C4×C7 |
| Frattini | C2 | C2×AGL(1,𝔽8) |
| Fitting | C23×C4 | C7 |
| socle, cosocle | C24 | C2×C7 |
| prime radical | C2×AGL(1,𝔽8) | C2 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. AGL(1,𝔽8) | C4 | direct (N × H) |
| 3. C23×C4 | C7 | semidirect |
| 11. C23 | C4×C7 | semidirect |
| 19. C4 | AGL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,𝔽8) | 2 |
| C23×C4 | 7 |
| C4×C7 | 8 |