Finite Group DatabasebetaFinite Group DatabasebetaOrder 280.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×Dic5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C5×C7 | C4 | 1 | 1 | yes | characteristic |
| 3 | C7⋊C8 | — | 5 | 1 | no | maximal, Hall |
| 4 | #170 | — | 7 | 1 | no | maximal, Hall |
| 5 | C5×C7 | C8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C4×C7 | — | 5 | 1 | no | — |
| 7 | Dic5 | D7 | 1 | 1 | yes | characteristic |
| 8 | C2×C7 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 9 | C2×C5 | Dic7 | 1 | 1 | yes | characteristic |
| 10 | C8 | — | 35 | 1 | no | Sylow |
| 11 | C7 | #170 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C5 | C7⋊C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C4 | — | 5 | 1 | no | — |
| 14 | C2 | C7⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×C7 | C8 |
| Fitting, socle, prime radical | C2×C5×C7 | C4 |
| cosocle | C7×Dic5 | C2 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C5×C7 | C8 | semidirect |
| 11. C7 | #170 | semidirect |
| 12. C5 | C7⋊C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Dic5 | 2 |
| C7⋊C8 | 5 |
| #170 | 7 |