Finite Group DatabasebetaFinite Group DatabasebetaOrder 784.
90 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72×Q8 | C2 | 1 | 2 | yes | maximal |
| 2 | C72×C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C4×C72 | — | 2 | 2 | no | — |
| 4 | C4×C72 | C22 | 1 | 1 | yes | characteristic |
| 5 | C7×Q16 | C7 | 1 | 8 | yes | maximal, direct product factor |
| 6 | C2×C72 | D4 | 1 | 1 | yes | characteristic |
| 7 | C7×Q8 | C2×C7 | 1 | 16 | yes | — |
| 8 | C7×C8 | C2×C7 | 1 | 8 | yes | — |
| 9 | C72 | Q16 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C4×C7 | — | 2 | 16 | no | — |
| 11 | C4×C7 | C22×C7 | 1 | 8 | yes | — |
| 12 | Q16 | C72 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C2×C7 | C7×D4 | 1 | 8 | yes | — |
| 14 | Q8 | C2×C72 | 1 | 2 | yes | — |
| 15 | C8 | C2×C72 | 1 | 1 | yes | characteristic |
| 16 | C7 | C7×Q16 | 1 | 8 | yes | direct product factor |
| 17 | C4 | — | 2 | 2 | no | — |
| 18 | C4 | C22×C72 | 1 | 1 | yes | characteristic |
| 19 | C2 | C72×D4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C72 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C72 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C72 ◁ C4×C72 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C7×Q16 | C7 | direct (N × H) |
| 9. C72 | Q16 | direct (N × H) |
| 12. Q16 | C72 | direct (N × H) |
| 16. C7 | C7×Q16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C72×Q8 | 2 |
| C72×C8 | 2 |
| C7×Q16 | 7 |