Finite Group DatabasebetaFinite Group DatabasebetaOrder 24353.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72×C71 | C7 | 1 | 1 | yes | maximal, characteristic |
| 2 | C71⋊C49 | C7 | 1 | 7 | yes | maximal, direct product factor |
| 3 | C7×C71 | C49 | 1 | 7 | yes | complemented |
| 4 | C7×C71 | C72 | 1 | 1 | yes | characteristic |
| 5 | C7×C49 | — | 71 | 1 | no | maximal, Sylow |
| 6 | C71 | C7×C49 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C72 | C71⋊C7 | 1 | 1 | yes | characteristic |
| 8 | C49 | — | 71 | 7 | no | — |
| 9 | C7 | C71⋊C49 | 1 | 7 | yes | direct product factor |
| 10 | C7 | C7×C71⋊C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C72 | C71⋊C7 |
| commutator, hypocenter | C71 | C7×C49 |
| Frattini | C7 | C7×C71⋊C7 |
| Fitting, socle, prime radical | C72×C71 | C7 |
| cosocle | C7×C71 | C72 |
| LCS | G ▷ C71 | |
| UCS | C1 ◁ C72 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C71⋊C49 | C7 | direct (N × H) |
| 3. C7×C71 | C49 | semidirect |
| 6. C71 | C7×C49 | semidirect |
| 9. C7 | C71⋊C49 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C72×C71 | 7 |
| C71⋊C49 | 7 |
| C7×C49 | 71 |