Finite Group DatabasebetaFinite Group DatabasebetaOrder 7526.
8 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C53×C71 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C2×C71 | — | 53 | 1 | no | maximal, Hall |
| 3 | D53 | C71 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C71 | D53 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 5 | C53 | C2×C71 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C2 | — | 53 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C71 | D53 |
| commutator, hypocenter | C53 | C2×C71 |
| Frattini | C1 | G |
| Fitting, socle | C53×C71 | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C71 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×C71 | C2 | semidirect |
| 3. D53 | C71 | direct (N × H) |
| 4. C71 | D53 | direct (N × H) |
| 5. C53 | C2×C71 | semidirect |