Finite Group DatabasebetaFinite Group DatabasebetaOrder 1715.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 7−1+2 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 2 | C5×C72 | C7 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×C49 | C7 | 1 | 7 | yes | maximal, complemented |
| 4 | C72 | C5×C7 | 1 | 1 | yes | characteristic |
| 5 | C49 | C5×C7 | 1 | 7 | yes | complemented |
| 6 | C5×C7 | — | 7 | 1 | no | — |
| 7 | C5×C7 | C72 | 1 | 1 | yes | characteristic |
| 8 | C7 | — | 7 | 1 | no | — |
| 9 | C7 | C5×C72 | 1 | 1 | yes | characteristic |
| 10 | C5 | 7−1+2 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C72 |
| commutator, Frattini, cosocle | C7 | C5×C72 |
| prime radical | C5×C72 | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 7−1+2 | C5 | direct (N × H) |
| 3. C5×C49 | C7 | semidirect |
| 5. C49 | C5×C7 | semidirect |
| 10. C5 | 7−1+2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 7−1+2 | 5 |
| C5×C72 | 7 |
| C5×C49 | 7 |