Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
146 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#41 | C2 | 1 | 10 | yes | maximal, complemented |
| 2 | C2×C7×Q8 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | C7×Q8 | C22 | 1 | 10 | yes | — |
| 4 | C7×D4 | C22 | 1 | 10 | yes | — |
| 5 | C2×C4×C7 | C22 | 1 | 15 | yes | — |
| 6 | 2−1+4 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C22×C7 | C23 | 1 | 5 | yes | — |
| 8 | C4×C7 | C23 | 1 | 10 | yes | — |
| 9 | #41 | C2×C7 | 1 | 10 | yes | complemented |
| 10 | C2×Q8 | C2×C7 | 1 | 5 | yes | complemented |
| 11 | C2×C7 | — | 2 | 5 | no | — |
| 12 | C2×C7 | C24 | 1 | 1 | yes | characteristic |
| 13 | Q8 | C22×C7 | 1 | 10 | yes | — |
| 14 | D4 | C22×C7 | 1 | 10 | yes | — |
| 15 | C2×C4 | C22×C7 | 1 | 15 | yes | — |
| 16 | C7 | 2−1+4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 17 | C22 | C23×C7 | 1 | 5 | yes | — |
| 18 | C4 | C23×C7 | 1 | 10 | yes | — |
| 19 | C2 | — | 2 | 5 | no | — |
| 20 | C2 | C24×C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#41 | C2 | semidirect |
| 2. C2×C7×Q8 | C2 | semidirect |
| 6. 2−1+4 | C7 | direct (N × H) |
| 9. #41 | C2×C7 | semidirect |
| 10. C2×Q8 | C2×C7 | semidirect |
| 16. C7 | 2−1+4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#41 | 2 |
| C2×C7×Q8 | 2 |
| 2−1+4 | 7 |