Finite Group DatabasebetaFinite Group DatabasebetaOrder 272.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D4×C17 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D68 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #3622 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×C17 | — | 2 | 1 | no | — |
| 5 | C2×D17 | — | 2 | 1 | no | — |
| 6 | C4×C17 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C17 | — | 4 | 1 | no | — |
| 8 | C2×C17 | D4 | 1 | 1 | yes | characteristic |
| 9 | D17 | — | 4 | 1 | no | — |
| 10 | C17 | D8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | D8 | — | 17 | 1 | no | maximal, Sylow |
| 12 | D4 | — | 17 | 1 | no | — |
| 13 | D4 | D17 | 1 | 1 | yes | characteristic, complemented |
| 14 | C8 | — | 17 | 1 | no | — |
| 15 | C22 | — | 2 | 1 | no | — |
| 16 | C22 | — | 34 | 1 | no | — |
| 17 | C4 | C2×D17 | 1 | 1 | yes | characteristic |
| 18 | C2 | — | 4 | 1 | no | — |
| 19 | C2 | — | 68 | 1 | no | — |
| 20 | C2 | C22⋊D17 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D17 |
| commutator, cosocle | C4×C17 | C22 |
| Frattini | C4 | C2×D17 |
| Fitting | D4×C17 | C2 |
| socle | C2×C17 | D4 |
| hypercenter | D4 | D17 |
| hypocenter | C17 | D8 |
| LCS | G ▷ C4×C17 ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×C17 | C2 | semidirect |
| 2. D68 | C2 | semidirect |
| 3. #3622 | C2 | semidirect |
| 10. C17 | D8 | semidirect |
| 13. D4 | D17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C17 | 2 |
| D68 | 2 |
| #3622 | 2 |
| D8 | 17 |