Finite Group DatabasebetaFinite Group DatabasebetaOrder 16758.
100 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 100 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×#3862 | C2 | semidirect |
| 2. #19150303 | C3 | semidirect |
| 3. #19150302 | C3 | semidirect |
| 4. C19×#64422 | C3 | semidirect |
| 5. C7×D7×C19⋊C3 | C3 | semidirect |
| 6. C19⋊#3862 | C2×C3 | semidirect |
| 7. C19⋊#3862 | C2×C3 | semidirect |
| 8. C19×#3862 | C2×C3 | semidirect |
| 9. C72×C19⋊C3 | C2×C3 | semidirect |
| 12. C7×D7×C19 | C32 | semidirect |
| 15. C72×C19 | C2×C32 | semidirect |
| 24. D7×C19⋊C3 | ΓL(1,𝔽8) | semidirect |
| 31. C7×C19⋊C3 | AGL(1,𝔽7) | semidirect |
| 32. C7×C19⋊C3 | C2×ΓL(1,𝔽8) | semidirect |
| 39. #64422 | C19⋊C3 | direct (N × H) |
| 41. D7×C19 | C3×ΓL(1,𝔽8) | semidirect |
| 46. #3862 | C2×C19⋊C3 | semidirect |
| 48. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 49. C7×C19 | C2×C3×ΓL(1,𝔽8) | semidirect |
| 56. C7×D7 | C3×C19⋊C3 | semidirect |
| 62. C19⋊C3 | #64422 | direct (N × H) |
| 63. C72 | C2×C3×C19⋊C3 | semidirect |
| 82. C19 | C3×#64422 | semidirect |
| 85. D7 | ΓL(1,𝔽8)×C19⋊C3 | semidirect |
| 88. C7 | AGL(1,𝔽7)×C19⋊C3 | semidirect |
| 89. 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×#3862 | 2 |
| #19150303 | 3 |
| #19150302 | 3 |
| C19×#64422 | 3 |
| C7×D7×C19⋊C3 | 3 |
| C2×ΓL(1,𝔽8)×C19⋊C3 | 7 |
| AGL(1,𝔽7)×C19⋊C3 | 7 |
| C3×#64422 | 19 |