Finite Group DatabasebetaFinite Group DatabasebetaOrder 210.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C5×C7 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×D5 | — | 3 | 1 | no | maximal, Hall |
| 3 | D3×C7 | — | 5 | 1 | no | maximal, Hall |
| 4 | C5×C7 | D3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | D15 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C3×C7 | D5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C3×C5 | C2×C7 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | C2×C7 | — | 15 | 1 | no | Hall |
| 9 | D5 | — | 3 | 1 | no | Hall |
| 10 | C7 | D15 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 11 | D3 | — | 5 | 1 | no | Hall |
| 12 | C5 | D3×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C3 | C7×D5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 15 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D15 |
| commutator, cosocle, hypocenter | C3×C5 | C2×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C7 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×C7 | C2 | semidirect |
| 4. C5×C7 | D3 | semidirect |
| 5. D15 | C7 | direct (N × H) |
| 6. C3×C7 | D5 | semidirect |
| 7. C3×C5 | C2×C7 | semidirect |
| 10. C7 | D15 | direct (N × H) |
| 12. C5 | D3×C7 | semidirect |
| 13. C3 | C7×D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×C7 | 2 |
| C7×D5 | 3 |
| D3×C7 | 5 |
| D15 | 7 |