Finite Group DatabasebetaFinite Group DatabasebetaOrder 58968.
67 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | AΓL(1,𝔽27) | — | 28 | 1 | no | maximal |
| 2 | PΣL(2,𝔽27) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | PGL(2,𝔽27) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | PSL(2,𝔽27) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 5 | (C33⋊C13)⋊C3 | — | 28 | 1 | no | Hall |
| 6 | AGL(1,𝔽27) | — | 28 | 1 | no | — |
| 7 | C33⋊C13 | — | 28 | 1 | no | — |
| 8 | C4⋊AGL(1,𝔽7) | — | 351 | 1 | no | maximal |
| 9 | #4175 | — | 364 | 1 | no | — |
| 10 | C2×D13⋊C3 | — | 378 | 1 | no | maximal |
| 11 | C2×AGL(1,𝔽7) | — | 351 | 1 | no | — |
| 12 | C2×AGL(1,𝔽7) | — | 351 | 1 | no | — |
| 13 | C4×ΓL(1,𝔽8) | — | 351 | 1 | no | — |
| 14 | AΣL(1,𝔽27) | — | 364 | 1 | no | Sylow |
| 15 | C2×C13⋊C3 | — | 378 | 1 | no | — |
| 16 | D13⋊C3 | — | 378 | 1 | no | — |
| 17 | D13⋊C3 | — | 378 | 1 | no | — |
| 18 | C3×S4 | — | 819 | 1 | no | maximal |
| 19 | D28 | — | 351 | 1 | no | Hall |
| 20 | C3⋊(C3⋊D3) | — | 28 | 1 | no | — |
| 21 | (C3⋊D3)⋊C3 | — | 1092 | 1 | no | — |
| 22 | C2×D13 | — | 378 | 1 | no | — |
| 23 | C2×ΓL(1,𝔽8) | — | 351 | 1 | no | — |
| 24 | AGL(1,𝔽7) | — | 702 | 1 | no | — |
| 25 | AGL(1,𝔽7) | — | 702 | 1 | no | — |
| 26 | C13⋊C3 | — | 378 | 1 | no | — |
| 27 | C3×A4 | — | 819 | 1 | no | — |
| 28 | C2×D7 | — | 351 | 1 | no | — |
| 29 | C2×D7 | — | 351 | 1 | no | — |
| 30 | C4×C7 | — | 351 | 1 | no | — |
| 31 | C33 | — | 28 | 1 | no | — |
| 32 | 3−1+2 | — | 728 | 1 | no | — |
| 33 | H(3,𝔽3) | — | 364 | 1 | no | — |
| 34 | C2×C13 | — | 378 | 1 | no | — |
| 35 | D13 | — | 378 | 1 | no | — |
| 36 | D13 | — | 378 | 1 | no | — |
| 37 | S4 | — | 819 | 1 | no | — |
| 38 | C3×D4 | — | 2457 | 1 | no | — |
| 39 | ΓL(1,𝔽8) | — | 351 | 1 | no | — |
| 40 | C3⋊D3 | — | 1092 | 1 | no | — |
| 41 | C3×D3 | — | 3276 | 1 | no | — |
| 42 | C2×C7 | — | 351 | 1 | no | — |
| 43 | D7 | — | 702 | 1 | no | — |
| 44 | D7 | — | 702 | 1 | no | — |
| 45 | C13 | — | 378 | 1 | no | Sylow |
| 46 | C22×C3 | — | 819 | 1 | no | — |
| 47 | C22×C3 | — | 2457 | 1 | no | — |
| 48 | A4 | — | 819 | 1 | no | — |
| 49 | A4 | — | 1638 | 1 | no | — |
| 50 | C3×C4 | — | 2457 | 1 | no | — |
| 51 | C32 | — | 1092 | 1 | no | — |
| 52 | C32 | — | 364 | 1 | no | — |
| 53 | C9 | — | 2184 | 1 | no | — |
| 54 | D4 | — | 2457 | 1 | no | Sylow |
| 55 | C7 | — | 351 | 1 | no | Sylow |
| 56 | C2×C3 | — | 4914 | 1 | no | — |
| 57 | C2×C3 | — | 2457 | 1 | no | — |
| 58 | D3 | — | 3276 | 1 | no | — |
| 59 | C22 | — | 819 | 1 | no | — |
| 60 | C22 | — | 2457 | 1 | no | — |
| 61 | C4 | — | 351 | 1 | no | — |
| 62 | C3 | — | 819 | 1 | no | — |
| 63 | C3 | — | 364 | 1 | no | — |
| 64 | C2 | — | 351 | 1 | no | — |
| 65 | C2 | — | 378 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | PSL(2,𝔽27) | C2×C3 |
| LCS | G ▷ PSL(2,𝔽27) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. PΣL(2,𝔽27) | C2 | semidirect |
| 3. PGL(2,𝔽27) | C3 | semidirect |
| 4. PSL(2,𝔽27) | C2×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| AΓL(1,𝔽27) | 28 |
| PΣL(2,𝔽27) | 2 |
| PGL(2,𝔽27) | 3 |
| C4⋊AGL(1,𝔽7) | 351 |
| C2×D13⋊C3 | 378 |
| C3×S4 | 819 |