Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
1284 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 222 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C67 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C23×C67 ◁ C22×Q8×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(3,𝔽4)×C67 | C2 | semidirect |
| 2. C2×#454×C67 | C2 | semidirect |
| 3. C67×#2202 | C2 | direct (N × H) |
| 4. C2×#355×C67 | C2 | semidirect |
| 5. C2×2−1+4×C67 | C22 | semidirect |
| 8. H(3,𝔽4)×C67 | C22 | semidirect |
| 13. #454×C67 | C22 | semidirect |
| 15. C2×#101×C67 | C22 | semidirect |
| 16. #355×C67 | C22 | semidirect |
| 20. 2−1+4×C67 | C23 | semidirect |
| 45. #101×C67 | C23 | semidirect |
| 134. C2×#2202 | C67 | direct (N × H) |
| 139. C2×C67 | #2202 | direct (N × H) |
| 141. C2×H(3,𝔽4) | C2×C67 | semidirect |
| 142. C2×#454 | C2×C67 | semidirect |
| 143. #2202 | C2×C67 | direct (N × H) |
| 144. C2×#355 | C2×C67 | semidirect |
| 145. C67 | C2×#2202 | direct (N × H) |
| 146. C2×2−1+4 | C22×C67 | semidirect |
| 149. H(3,𝔽4) | C22×C67 | semidirect |
| 154. #454 | C22×C67 | semidirect |
| 156. C2×#101 | C22×C67 | semidirect |
| 157. #355 | C22×C67 | semidirect |
| 161. 2−1+4 | C23×C67 | semidirect |
| 186. #101 | C23×C67 | semidirect |
| 279. C2 | C67×#2202 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽4)×C67 | 2 |
| C2×#454×C67 | 2 |
| C67×#2202 | 2 |
| C2×#355×C67 | 2 |
| C2×#2202 | 67 |