Finite Group DatabasebetaFinite Group DatabasebetaOrder 30492.
168 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 22 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 | C3×C7×C112 | C22×C3 |
| LCS | G ▷ C3×C7×C112 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊((C11⋊D11)⋊ΓL(1,𝔽8)) | C2 | semidirect |
| 2. C3⋊(D7⋊(C112⋊C3)) | C2 | semidirect |
| 3. C3×#19099620 | C2 | semidirect |
| 4. D21⋊(C11⋊D11) | C3 | semidirect |
| 6. C3×C7⋊(C112⋊C3) | C22 | semidirect |
| 7. C112×D21 | C2×C3 | semidirect |
| 8. C7×C3⋊(C11⋊D11) | C2×C3 | semidirect |
| 9. C3×C7⋊(C11⋊D11) | C2×C3 | semidirect |
| 13. #19099620 | D3 | semidirect |
| 17. C7⋊(C112⋊C3) | C2×D3 | semidirect |
| 18. C3×C7×C112 | C22×C3 | semidirect |
| 22. C7⋊(C11⋊D11) | C3×D3 | semidirect |
| 29. C7×C112 | C2×C3×D3 | semidirect |
| 30. C3⋊(C11⋊D11) | AGL(1,𝔽7) | semidirect |
| 41. C3×C112 | C2×AGL(1,𝔽7) | semidirect |
| 57. C112 | D3×AGL(1,𝔽7) | semidirect |
| 66. D21 | (C11⋊D11)⋊C3 | semidirect |
| 79. C3×C7 | C2×(C11⋊D11)⋊C3 | semidirect |
| 92. C7 | D3×(C11⋊D11)⋊C3 | semidirect |
| 103. C3 | D7⋊((C11⋊D11)⋊C3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊((C11⋊D11)⋊ΓL(1,𝔽8)) | 2 |
| C3⋊(D7⋊(C112⋊C3)) | 2 |
| C3×#19099620 | 2 |
| D21⋊(C11⋊D11) | 3 |
| D7⋊((C11⋊D11)⋊C3) | 3 |
| D3×(C11⋊D11)⋊C3 | 7 |
| D3×AGL(1,𝔽7) | 121 |