Finite Group DatabasebetaFinite Group DatabasebetaOrder 18496.
2580 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C42×C172 | C2 | 1 | 7 | yes | maximal |
| 2 | C22×C4×C172 | C22 | 1 | 7 | yes | — |
| 3 | C42×C172 | C4 | 1 | 28 | yes | direct product factor |
| 4 | C23×C172 | C23 | 1 | 1 | yes | characteristic |
| 5 | C2×C4×C172 | C2×C4 | 1 | 42 | yes | — |
| 6 | C22×C172 | C22×C4 | 1 | 7 | yes | — |
| 7 | C4×C172 | C42 | 1 | 28 | yes | direct product factor |
| 8 | C43×C17 | C17 | 1 | 18 | yes | maximal, direct product factor |
| 9 | C2×C172 | C2×C42 | 1 | 7 | yes | — |
| 10 | C2×C42×C17 | C2×C17 | 1 | 126 | yes | — |
| 11 | C172 | C43 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | C22×C4×C17 | C22×C17 | 1 | 126 | yes | — |
| 13 | C42×C17 | C4×C17 | 1 | 504 | yes | direct product factor |
| 14 | C23×C17 | C23×C17 | 1 | 18 | yes | — |
| 15 | C2×C4×C17 | C2×C4×C17 | 1 | 756 | yes | — |
| 16 | C22×C17 | C22×C4×C17 | 1 | 126 | yes | — |
| 17 | C4×C17 | C42×C17 | 1 | 504 | yes | direct product factor |
| 18 | C43 | C172 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 19 | C2×C17 | C2×C42×C17 | 1 | 126 | yes | — |
| 20 | C2×C42 | C2×C172 | 1 | 7 | yes | — |
| 21 | C17 | C43×C17 | 1 | 18 | yes | direct product factor |
| 22 | C22×C4 | C22×C172 | 1 | 7 | yes | — |
| 23 | C42 | C4×C172 | 1 | 28 | yes | direct product factor |
| 24 | C23 | C23×C172 | 1 | 1 | yes | characteristic |
| 25 | C2×C4 | C2×C4×C172 | 1 | 42 | yes | — |
| 26 | C22 | C22×C4×C172 | 1 | 7 | yes | — |
| 27 | C4 | C42×C172 | 1 | 28 | yes | direct product factor |
| 28 | C2 | C2×C42×C172 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C23 | C23×C172 |
| socle, prime radical | C23×C172 | C23 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C42×C172 | C4 | direct (N × H) |
| 7. C4×C172 | C42 | direct (N × H) |
| 8. C43×C17 | C17 | direct (N × H) |
| 11. C172 | C43 | direct (N × H) |
| 13. C42×C17 | C4×C17 | direct (N × H) |
| 17. C4×C17 | C42×C17 | direct (N × H) |
| 18. C43 | C172 | direct (N × H) |
| 21. C17 | C43×C17 | direct (N × H) |
| 23. C42 | C4×C172 | direct (N × H) |
| 27. C4 | C42×C172 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×C172 | 2 |
| C43×C17 | 17 |