Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
62 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×M16 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C7×C8 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C7×Q8 | C4 | 1 | 1 | yes | characteristic |
| 5 | C7×D4 | C4 | 1 | 3 | yes | — |
| 6 | C2×C4×C7 | C22 | 1 | 3 | yes | — |
| 7 | C7×C8 | C22 | 1 | 3 | yes | — |
| 8 | C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 9 | #131 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 10 | C22×C7 | C2×C4 | 1 | 3 | yes | — |
| 11 | C4×C7 | C2×C4 | 1 | 3 | yes | — |
| 12 | C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 13 | #41 | C2×C7 | 1 | 1 | yes | characteristic |
| 14 | M16 | C2×C7 | 1 | 3 | yes | complemented |
| 15 | C2×C8 | C2×C7 | 1 | 3 | yes | complemented |
| 16 | C2×C7 | — | 2 | 3 | no | — |
| 17 | C2×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 18 | Q8 | C4×C7 | 1 | 1 | yes | characteristic |
| 19 | D4 | C4×C7 | 1 | 3 | yes | — |
| 20 | C2×C4 | C22×C7 | 1 | 3 | yes | — |
| 21 | C8 | C22×C7 | 1 | 3 | yes | — |
| 22 | C8 | C22×C7 | 1 | 1 | yes | characteristic |
| 23 | C7 | #131 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 24 | C22 | C2×C4×C7 | 1 | 3 | yes | — |
| 25 | C4 | C2×C4×C7 | 1 | 3 | yes | — |
| 26 | C4 | C23×C7 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C22×C4×C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C22 |
| commutator | C2 | C22×C4×C7 |
| Frattini, cosocle | C4 | C23×C7 |
| socle | C2×C7 | C22×C4 |
| prime radical | C7×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×M16 | C2 | semidirect |
| 3. C2×C7×C8 | C2 | semidirect |
| 9. #131 | C7 | direct (N × H) |
| 14. M16 | C2×C7 | semidirect |
| 15. C2×C8 | C2×C7 | semidirect |
| 23. C7 | #131 | 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 |
| C7×M16 | 2 |
| C2×C7×C8 | 2 |
| #131 | 7 |