Finite Group DatabasebetaFinite Group DatabasebetaOrder 272.
134 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×C17 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×D17 | C2 | 1 | 14 | yes | maximal, direct product factor |
| 3 | C22×C17 | C22 | 1 | 7 | yes | complemented |
| 4 | C2×D17 | C22 | 1 | 28 | yes | direct product factor |
| 5 | C2×C17 | C23 | 1 | 7 | yes | complemented |
| 6 | D17 | C23 | 1 | 8 | yes | direct product factor |
| 7 | C17 | C24 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C24 | — | 17 | 1 | no | maximal, Sylow |
| 9 | C23 | — | 17 | 14 | no | — |
| 10 | C23 | D17 | 1 | 1 | yes | characteristic, direct product factor |
| 11 | C22 | — | 17 | 28 | no | — |
| 12 | C22 | C2×D17 | 1 | 7 | yes | direct product factor |
| 13 | C2 | — | 17 | 8 | no | — |
| 14 | C2 | C22×D17 | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D17 |
| commutator, cosocle, hypocenter | C17 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C17 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C17 | C2 | semidirect |
| 2. C22×D17 | C2 | direct (N × H) |
| 3. C22×C17 | C22 | semidirect |
| 4. C2×D17 | C22 | direct (N × H) |
| 5. C2×C17 | C23 | semidirect |
| 6. D17 | C23 | direct (N × H) |
| 7. C17 | C24 | semidirect |
| 10. C23 | D17 | direct (N × H) |
| 12. C22 | C2×D17 | direct (N × H) |
| 14. C2 | C22×D17 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C17 | 2 |
| C22×D17 | 2 |
| C24 | 17 |