Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
1330 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C67 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C67 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C67 ◁ C2×C42×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×#496×C67 | C2 | semidirect |
| 3. C67×#1655 | C2 | direct (N × H) |
| 4. C2×#329×C67 | C2 | semidirect |
| 8. H(3,𝔽4)×C67 | C22 | semidirect |
| 9. C2×#127×C67 | C22 | semidirect |
| 14. #496×C67 | C22 | semidirect |
| 15. C2×#97×C67 | C22 | semidirect |
| 16. #329×C67 | C22 | semidirect |
| 28. #127×C67 | C23 | semidirect |
| 41. #97×C67 | C23 | semidirect |
| 103. C2×#1655 | C67 | direct (N × H) |
| 106. C2×C67 | #1655 | direct (N × H) |
| 108. C2×H(3,𝔽4) | C2×C67 | semidirect |
| 109. C2×#496 | C2×C67 | semidirect |
| 110. #1655 | C2×C67 | direct (N × H) |
| 111. C2×#329 | C2×C67 | semidirect |
| 112. C67 | C2×#1655 | direct (N × H) |
| 116. H(3,𝔽4) | C22×C67 | semidirect |
| 117. C2×#127 | C22×C67 | semidirect |
| 122. #496 | C22×C67 | semidirect |
| 123. C2×#97 | C22×C67 | semidirect |
| 124. #329 | C22×C67 | semidirect |
| 136. #127 | C23×C67 | semidirect |
| 149. #97 | C23×C67 | semidirect |
| 213. C2 | C67×#1655 | 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×#496×C67 | 2 |
| C67×#1655 | 2 |
| C2×#329×C67 | 2 |
| C2×#1655 | 67 |