Finite Group DatabasebetaFinite Group DatabasebetaOrder 32732.
60 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C72×C167 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C72×C167 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | C7×Dic167 | C7 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×C7×C167 | C2×C7 | 1 | 8 | yes | — |
| 5 | C7×C167 | C4×C7 | 1 | 8 | yes | complemented |
| 6 | Dic167 | C72 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 7 | C2×C167 | C2×C72 | 1 | 1 | yes | characteristic |
| 8 | C4×C72 | — | 167 | 1 | no | maximal, Hall |
| 9 | C167 | C4×C72 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2×C72 | D167 | 1 | 1 | yes | characteristic |
| 11 | C72 | Dic167 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | C4×C7 | — | 167 | 8 | no | — |
| 13 | C2×C7 | C7×D167 | 1 | 8 | yes | — |
| 14 | C7 | C7×Dic167 | 1 | 8 | yes | direct product factor |
| 15 | C4 | — | 167 | 1 | no | Sylow |
| 16 | C2 | C72×D167 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C72 | D167 |
| commutator, hypocenter | C167 | C4×C72 |
| Frattini | C2 | C72×D167 |
| Fitting, socle, prime radical | C2×C72×C167 | C2 |
| cosocle | C2×C167 | C2×C72 |
| LCS | G ▷ C167 | |
| UCS | C1 ◁ C2×C72 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C72×C167 | C4 | semidirect |
| 3. C7×Dic167 | C7 | direct (N × H) |
| 5. C7×C167 | C4×C7 | semidirect |
| 6. Dic167 | C72 | direct (N × H) |
| 9. C167 | C4×C72 | semidirect |
| 11. C72 | Dic167 | direct (N × H) |
| 14. C7 | C7×Dic167 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C72×C167 | 2 |
| C7×Dic167 | 7 |
| C4×C72 | 167 |