Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
770 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C34×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×H(3,𝔽3)×C37 | C3 | semidirect |
| 2. C32×H(3,𝔽3)×C37 | C3 | semidirect |
| 3. C3×C37×AΣL(1,𝔽27) | C3 | semidirect |
| 4. C37×#6891 | C3 | semidirect |
| 7. C34×C37 | C32 | semidirect |
| 8. C3×3−1+2×C37 | C32 | semidirect |
| 37. H(3,𝔽3)×C37 | H(3,𝔽3) | semidirect |
| 39. #10624742 | C37 | direct (N × H) |
| 56. C32×C37 | AΣL(1,𝔽27) | semidirect |
| 60. C32×H(3,𝔽3) | C3×C37 | semidirect |
| 61. C32×H(3,𝔽3) | C3×C37 | semidirect |
| 62. C3×AΣL(1,𝔽27) | C3×C37 | semidirect |
| 63. #6891 | C3×C37 | semidirect |
| 74. C34 | C32×C37 | semidirect |
| 75. C3×3−1+2 | C32×C37 | semidirect |
| 85. C37 | #10624742 | direct (N × H) |
| 105. H(3,𝔽3) | H(3,𝔽3)×C37 | semidirect |
| 123. C32 | C37×AΣL(1,𝔽27) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×H(3,𝔽3)×C37 | 3 |
| C3×C37×AΣL(1,𝔽27) | 3 |
| C37×#6891 | 3 |
| #10624742 | 37 |