Finite Group DatabasebetaFinite Group DatabasebetaOrder 16758.
132 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C19 | C2×C32 |
| LCS | G ▷ C72×C19 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C19⋊C3×#3861 | C2 | semidirect |
| 2. #19150297 | C3 | semidirect |
| 3. #19150296 | C3 | semidirect |
| 4. C19×#64423 | C3 | semidirect |
| 5. C7×D7×C19⋊C3 | C3 | semidirect |
| 6. #14584116 | C2×C3 | semidirect |
| 7. #14584115 | C2×C3 | semidirect |
| 8. C19×#3861 | C2×C3 | semidirect |
| 9. C72×C19⋊C3 | C2×C3 | semidirect |
| 12. C7×D7×C19 | C32 | semidirect |
| 16. C72×C19 | C2×C32 | semidirect |
| 25. D7×C19⋊C3 | ΓL(1,𝔽8) | semidirect |
| 34. C7×C19⋊C3 | AGL(1,𝔽7) | semidirect |
| 35. C7×C19⋊C3 | C2×ΓL(1,𝔽8) | semidirect |
| 43. #64423 | C19⋊C3 | direct (N × H) |
| 45. D7×C19 | C3×ΓL(1,𝔽8) | semidirect |
| 50. #3861 | C2×C19⋊C3 | semidirect |
| 52. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 53. C7×C19 | C2×C3×ΓL(1,𝔽8) | semidirect |
| 60. C7×D7 | C3×C19⋊C3 | semidirect |
| 67. C19⋊C3 | #64423 | direct (N × H) |
| 68. C72 | C2×C3×C19⋊C3 | semidirect |
| 90. C19 | C3×#64423 | semidirect |
| 93. D7 | ΓL(1,𝔽8)×C19⋊C3 | semidirect |
| 96. C7 | AGL(1,𝔽7)×C19⋊C3 | semidirect |
| 97. C7 | C2×ΓL(1,𝔽8)×C19⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C19⋊C3×#3861 | 2 |
| #19150297 | 3 |
| #19150296 | 3 |
| C19×#64423 | 3 |
| C7×D7×C19⋊C3 | 3 |
| C2×ΓL(1,𝔽8)×C19⋊C3 | 7 |
| AGL(1,𝔽7)×C19⋊C3 | 7 |
| C3×#64423 | 19 |