Finite Group DatabasebetaFinite Group DatabasebetaOrder 5292.
216 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72224669 |
| commutator, cosocle, hypocenter | C32×C72 | C22×C3 |
| Frattini | C3 | C2×C3⋊(ΓL(1,𝔽8)⋊D7) |
| Fitting | C2×C32×C72 | C2×C3 |
| socle | C2×C3×C72 | C3×D3 |
| LCS | G ▷ C32×C72 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×C7⋊H(3,𝔽3) | C2 | semidirect |
| 2. #72224669 | C2 | direct (N × H) |
| 3. C2×C3⋊(C3⋊(C7⋊D7)) | C3 | semidirect |
| 5. C7×C7⋊H(3,𝔽3) | C22 | semidirect |
| 6. C2×C32×C72 | C2×C3 | semidirect |
| 7. C3⋊(C3⋊(C7⋊D7)) | C2×C3 | semidirect |
| 9. C2×C3×C7×ΓL(1,𝔽8) | D3 | semidirect |
| 16. C32×C72 | C22×C3 | semidirect |
| 18. C3×C7×ΓL(1,𝔽8) | C2×D3 | semidirect |
| 21. C2×C7⋊H(3,𝔽3) | D7 | semidirect |
| 37. C7⋊H(3,𝔽3) | C2×D7 | semidirect |
| 45. C2×C32×C7 | AGL(1,𝔽7) | semidirect |
| 46. C2×C32×C7 | C3×D7 | semidirect |
| 52. C2×C3×ΓL(1,𝔽8) | D21 | semidirect |
| 55. C2×C72 | (C3⋊D3)⋊C3 | semidirect |
| 68. C32×C7 | C2×AGL(1,𝔽7) | semidirect |
| 69. C32×C7 | C2×C3×D7 | semidirect |
| 71. C3×ΓL(1,𝔽8) | C2×D21 | semidirect |
| 74. C72 | C2×(C3⋊D3)⋊C3 | semidirect |
| 111. C2×C32 | ΓL(1,𝔽8)⋊D7 | semidirect |
| 115. C2×C7 | #67369 | semidirect |
| 116. C2×C7 | #67385 | semidirect |
| 125. C32 | C2×ΓL(1,𝔽8)⋊D7 | semidirect |
| 127. C7 | C2×#67369 | semidirect |
| 128. C7 | C2×#67385 | semidirect |
| 142. C2 | #72224669 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×C7⋊H(3,𝔽3) | 2 |
| #72224669 | 2 |
| C2×C3⋊(C3⋊(C7⋊D7)) | 3 |
| C2×C3⋊(ΓL(1,𝔽8)⋊D7) | 3 |
| C2×#67385 | 7 |
| C2×#67369 | 7 |