Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
4032 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 4032 conjugacy classes of subgroups — too many to draw. Only its 370 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C67 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C67 |
| prime radical | C22×H(5,𝔽2)×C67 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C25×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#458×C67 | C2 | semidirect |
| 3. C67×#2405 | C2 | semidirect |
| 4. C67×#1790 | C2 | semidirect |
| 5. C67×#1781 | C2 | semidirect |
| 12. C2×#123×C67 | C22 | semidirect |
| 13. C22×C22⋊C4×C67 | C22 | semidirect |
| 15. C2×#99×C67 | C22 | semidirect |
| 17. #386×C67 | C22 | semidirect |
| 20. #379×C67 | C22 | semidirect |
| 43. C23×C4×C67 | C23 | semidirect |
| 44. C23×C4×C67 | C23 | semidirect |
| 53. C2×C22⋊C4×C67 | C23 | semidirect |
| 173. C22×C67 | #458 | semidirect |
| 180. #32900 | C67 | direct (N × H) |
| 189. C2×#458 | C2×C67 | semidirect |
| 190. #2405 | C2×C67 | semidirect |
| 191. #1790 | C2×C67 | semidirect |
| 192. #1781 | C2×C67 | semidirect |
| 193. C67 | #32900 | direct (N × H) |
| 200. C2×#123 | C22×C67 | semidirect |
| 201. C22×C22⋊C4 | C22×C67 | semidirect |
| 203. C2×#99 | C22×C67 | semidirect |
| 205. #386 | C22×C67 | semidirect |
| 208. #379 | C22×C67 | semidirect |
| 231. C23×C4 | C23×C67 | semidirect |
| 232. C23×C4 | C23×C67 | semidirect |
| 241. C2×C22⋊C4 | C23×C67 | semidirect |
| 361. C22 | #458×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(5,𝔽2)×C67 | 2 |
| C2×#458×C67 | 2 |
| C67×#2405 | 2 |
| C67×#1790 | 2 |
| C67×#1781 | 2 |
| #32900 | 67 |