Finite Group DatabasebetaFinite Group DatabasebetaOrder 272.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C4×C17 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×C4×C17 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×C17 | C22 | 1 | 1 | yes | characteristic |
| 4 | C4×C17 | — | 2 | 2 | no | — |
| 5 | C4×C17 | C4 | 1 | 2 | yes | complemented |
| 6 | C2×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 7 | C2×C17 | D4 | 1 | 1 | yes | characteristic |
| 8 | C2×C17 | Q8 | 1 | 1 | yes | characteristic |
| 9 | C17 | C4⋊C4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C4⋊C4 | C17 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 11 | C2×C4 | C2×C17 | 1 | 2 | yes | — |
| 12 | C2×C4 | C2×C17 | 1 | 1 | yes | characteristic |
| 13 | C22 | C22×C17 | 1 | 1 | yes | characteristic |
| 14 | C4 | — | 2 | 2 | no | — |
| 15 | C4 | C4×C17 | 1 | 2 | yes | complemented |
| 16 | C2 | C2×C4×C17 | 1 | 1 | yes | characteristic |
| 17 | C2 | D4×C17 | 1 | 1 | yes | characteristic |
| 18 | C2 | Q8×C17 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C17 | C22 |
| commutator | C2 | C2×C4×C17 |
| Frattini, cosocle | C22 | C22×C17 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4×C17 | C4 | semidirect |
| 9. C17 | C4⋊C4 | direct (N × H) |
| 10. C4⋊C4 | C17 | direct (N × H) |
| 15. C4 | C4×C17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C17 | 2 |
| C4⋊C4 | 17 |