Finite Group DatabasebetaFinite Group DatabasebetaOrder 13104.
124 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×PSL(2,𝔽13) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×PSL(2,𝔽13) | — | 3 | 1 | no | — |
| 3 | C3×PSL(2,𝔽13) | — | 4 | 1 | no | maximal |
| 4 | PSL(2,𝔽13) | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 5 | A4×D13⋊C3 | — | 14 | 1 | no | maximal |
| 6 | A4×C13⋊C3 | — | 14 | 1 | no | — |
| 7 | D13⋊A4 | — | 14 | 2 | no | — |
| 8 | A4×D13 | — | 14 | 1 | no | — |
| 9 | C22×D13⋊C3 | — | 14 | 1 | no | — |
| 10 | C3×D13⋊C3 | — | 56 | 1 | no | — |
| 11 | A4×D7 | — | 78 | 1 | no | maximal |
| 12 | C22⋊(C13⋊C3) | — | 14 | 2 | no | — |
| 13 | A4×C13 | — | 14 | 1 | no | — |
| 14 | C22×C13⋊C3 | — | 14 | 1 | no | — |
| 15 | C2×D13⋊C3 | — | 42 | 1 | no | — |
| 16 | C2×D13⋊C3 | — | 42 | 1 | no | — |
| 17 | C2×D3×A4 | — | 91 | 1 | no | maximal, Hall |
| 18 | A42 | — | 91 | 1 | no | maximal, Hall |
| 19 | C3×C13⋊C3 | — | 56 | 1 | no | Hall |
| 20 | C22×D13 | — | 14 | 1 | no | — |
| 21 | C7×A4 | — | 78 | 1 | no | — |
| 22 | C3×D13 | — | 56 | 1 | no | — |
| 23 | C2×C13⋊C3 | — | 42 | 1 | no | — |
| 24 | D13⋊C3 | — | 14 | 1 | no | — |
| 25 | D13⋊C3 | — | 42 | 1 | no | — |
| 26 | D13⋊C3 | — | 56 | 2 | no | — |
| 27 | C2×C3×A4 | — | 91 | 1 | no | — |
| 28 | D3×A4 | — | 91 | 2 | no | — |
| 29 | C22×D7 | — | 78 | 1 | no | — |
| 30 | C22×C13 | — | 14 | 1 | no | — |
| 31 | C2×D13 | — | 42 | 1 | no | — |
| 32 | C2×D13 | — | 42 | 1 | no | — |
| 33 | C23×D3 | — | 91 | 1 | no | — |
| 34 | C22⋊A4 | — | 91 | 2 | no | — |
| 35 | C22×A4 | — | 91 | 1 | no | — |
| 36 | C22×A4 | — | 91 | 1 | no | — |
| 37 | C3×D7 | — | 312 | 1 | no | — |
| 38 | C3×C13 | — | 56 | 1 | no | — |
| 39 | C13⋊C3 | — | 14 | 1 | no | — |
| 40 | C13⋊C3 | — | 56 | 2 | no | — |
| 41 | C2×C3×D3 | — | 364 | 1 | no | — |
| 42 | C3×A4 | — | 91 | 1 | no | — |
| 43 | C3×A4 | — | 364 | 1 | no | — |
| 44 | C22×C7 | — | 78 | 1 | no | — |
| 45 | C2×D7 | — | 234 | 1 | no | — |
| 46 | C2×D7 | — | 234 | 1 | no | — |
| 47 | C2×C13 | — | 42 | 1 | no | — |
| 48 | D13 | — | 14 | 1 | no | — |
| 49 | D13 | — | 42 | 1 | no | — |
| 50 | C23×C3 | — | 91 | 1 | no | — |
| 51 | C22×D3 | — | 273 | 1 | no | — |
| 52 | C22×D3 | — | 273 | 2 | no | — |
| 53 | C22×D3 | — | 273 | 1 | no | — |
| 54 | C22×D3 | — | 91 | 2 | no | — |
| 55 | C2×A4 | — | 273 | 1 | no | — |
| 56 | C2×A4 | — | 91 | 1 | no | — |
| 57 | C2×A4 | — | 182 | 1 | no | — |
| 58 | C3×C7 | — | 312 | 1 | no | — |
| 59 | C2×C32 | — | 364 | 1 | no | — |
| 60 | C3×D3 | — | 364 | 2 | no | — |
| 61 | C24 | — | 91 | 1 | no | Sylow |
| 62 | C2×C7 | — | 234 | 1 | no | — |
| 63 | D7 | — | 78 | 1 | no | — |
| 64 | D7 | — | 234 | 1 | no | — |
| 65 | C13 | — | 14 | 1 | no | Sylow |
| 66 | C22×C3 | — | 273 | 1 | no | — |
| 67 | C22×C3 | — | 273 | 1 | no | — |
| 68 | C22×C3 | — | 91 | 1 | no | — |
| 69 | C22×C3 | — | 364 | 1 | no | — |
| 70 | C2×D3 | — | 273 | 2 | no | — |
| 71 | C2×D3 | — | 273 | 2 | no | — |
| 72 | C2×D3 | — | 273 | 2 | no | — |
| 73 | C2×D3 | — | 273 | 1 | no | — |
| 74 | C2×D3 | — | 273 | 2 | no | — |
| 75 | C2×D3 | — | 91 | 1 | no | — |
| 76 | A4 | — | 1092 | 2 | no | — |
| 77 | A4 | — | 91 | 1 | no | — |
| 78 | A4 | — | 364 | 2 | no | — |
| 79 | A4 | — | 182 | 1 | no | — |
| 80 | A4 | PSL(2,𝔽13) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 81 | C32 | — | 364 | 1 | no | Sylow |
| 82 | C23 | — | 273 | 1 | no | — |
| 83 | C23 | — | 819 | 1 | no | — |
| 84 | C23 | — | 91 | 1 | no | — |
| 85 | C7 | — | 78 | 1 | no | Sylow |
| 86 | C2×C3 | — | 273 | 1 | no | — |
| 87 | C2×C3 | — | 273 | 1 | no | — |
| 88 | C2×C3 | — | 91 | 1 | no | — |
| 89 | C2×C3 | — | 364 | 1 | no | — |
| 90 | C2×C3 | — | 728 | 1 | no | — |
| 91 | D3 | — | 273 | 2 | no | — |
| 92 | D3 | — | 91 | 2 | no | — |
| 93 | C22 | — | 273 | 1 | no | — |
| 94 | C22 | — | 273 | 1 | no | — |
| 95 | C22 | — | 273 | 2 | no | — |
| 96 | C22 | — | 819 | 1 | no | — |
| 97 | C22 | — | 91 | 1 | no | — |
| 98 | C22 | C3×PSL(2,𝔽13) | 1 | 1 | yes | characteristic, complemented |
| 99 | C3 | — | 4 | 1 | no | — |
| 100 | C3 | — | 91 | 1 | no | — |
| 101 | C3 | — | 728 | 1 | no | — |
| 102 | C2 | — | 3 | 1 | no | — |
| 103 | C2 | — | 273 | 1 | no | — |
| 104 | C2 | — | 91 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C22×PSL(2,𝔽13) | C3 |
| Fitting, cosocle | C22 | C3×PSL(2,𝔽13) |
| LCS | G ▷ C22×PSL(2,𝔽13) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×PSL(2,𝔽13) | C3 | semidirect |
| 4. PSL(2,𝔽13) | A4 | direct (N × H) |
| 80. A4 | PSL(2,𝔽13) | direct (N × H) |
| 98. C22 | C3×PSL(2,𝔽13) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×PSL(2,𝔽13) | 3 |
| C3×PSL(2,𝔽13) | 4 |
| A4×D13⋊C3 | 14 |
| A4×D7 | 78 |
| C2×D3×A4 | 91 |
| A42 | 91 |