Finite Group DatabasebetaFinite Group DatabasebetaOrder 16758.
240 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 43 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. C19⋊C3×C7⋊D7 | C3 | semidirect |
| 3. #19150287 | C3 | semidirect |
| 4. #19150286 | C3 | semidirect |
| 5. C19×#64424 | 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 |
| 13. C19×C7⋊D7 | C32 | semidirect |
| 17. C72×C19 | C2×C32 | semidirect |
| 38. C7×C19⋊C3 | AGL(1,𝔽7) | semidirect |
| 39. C7×C19⋊C3 | AGL(1,𝔽7) | semidirect |
| 40. C7×C19⋊C3 | AGL(1,𝔽7) | semidirect |
| 48. #64424 | C19⋊C3 | direct (N × H) |
| 56. #3861 | C2×C19⋊C3 | semidirect |
| 57. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 58. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 59. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 67. C7⋊D7 | C3×C19⋊C3 | semidirect |
| 74. C19⋊C3 | #64424 | direct (N × H) |
| 75. C72 | C2×C3×C19⋊C3 | semidirect |
| 101. C19 | C3×#64424 | semidirect |
| 107. C7 | AGL(1,𝔽7)×C19⋊C3 | semidirect |
| 108. C7 | AGL(1,𝔽7)×C19⋊C3 | semidirect |
| 109. C7 | AGL(1,𝔽7)×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 |
| C19⋊C3×C7⋊D7 | 3 |
| #19150287 | 3 |
| #19150286 | 3 |
| C19×#64424 | 3 |
| AGL(1,𝔽7)×C19⋊C3 | 7 |
| C3×#64424 | 19 |