Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
2154 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C67 | C24 |
| commutator | C22 | C24×C4×C67 |
| Frattini, cosocle | C2×C4 | C25×C67 |
| socle | C23×C67 | C23×C4 |
| prime radical | C24×C67 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×M16×C67 | C2 | direct (N × H) |
| 3. C2×#446×C67 | C2 | semidirect |
| 4. C2×C8×Q8×C67 | C2 | semidirect |
| 18. #446×C67 | C22 | semidirect |
| 19. C8×Q8×C67 | C22 | semidirect |
| 34. C2×M16×C67 | Q8 | direct (N × H) |
| 62. C2×Q8×C67 | M16 | direct (N × H) |
| 80. M16×C67 | C2×Q8 | direct (N × H) |
| 82. C2×C8×C67 | C2×Q8 | semidirect |
| 100. Q8×C67 | C2×M16 | direct (N × H) |
| 121. C8×C67 | C22×Q8 | semidirect |
| 138. C2×Q8×M16 | C67 | direct (N × H) |
| 140. C2×C67 | Q8×M16 | direct (N × H) |
| 145. Q8×M16 | C2×C67 | direct (N × H) |
| 146. C2×#446 | C2×C67 | semidirect |
| 147. C2×C8×Q8 | C2×C67 | semidirect |
| 150. C67 | C2×Q8×M16 | direct (N × H) |
| 162. #446 | C22×C67 | semidirect |
| 163. C8×Q8 | C22×C67 | semidirect |
| 178. C2×M16 | Q8×C67 | direct (N × H) |
| 206. C2×Q8 | M16×C67 | direct (N × H) |
| 224. M16 | C2×Q8×C67 | direct (N × H) |
| 226. C2×C8 | C2×Q8×C67 | semidirect |
| 244. Q8 | C2×M16×C67 | direct (N × H) |
| 265. C8 | C22×Q8×C67 | semidirect |
| 283. C2 | Q8×M16×C67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×Q8×C67 | 2 |
| Q8×M16×C67 | 2 |
| C2×#446×C67 | 2 |
| C2×C8×Q8×C67 | 2 |
| C2×#423×C67 | 2 |
| C2×C4×M16×C67 | 2 |
| C2×Q8×M16 | 67 |