Finite Group DatabasebetaFinite Group DatabasebetaOrder 343.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72 | C7 | 1 | 1 | yes | maximal, characteristic |
| 2 | C49 | C7 | 1 | 7 | yes | maximal, direct product factor |
| 3 | C7 | C49 | 1 | 7 | yes | direct product factor |
| 4 | C7 | C72 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C7 | C72 |
| socle, prime radical | C72 | C7 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C49 | C7 | direct (N × H) |
| 3. C7 | C49 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C72 | 7 |
| C49 | 7 |