Finite Group DatabasebetaFinite Group DatabasebetaOrder 20160.
136 conjugacy classes of subgroups; only the 2 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 2 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 | AGL(3,𝔽2) | — | 15 | 2 | no | maximal |
| 2 | S6 | — | 28 | 1 | no | maximal |
| 3 | (#1020⋊C3)⋊C2 | — | 35 | 1 | no | maximal, Hall |
| 4 | ΓL(2,𝔽4) | — | 56 | 1 | no | maximal |
| 5 | A6 | — | 28 | 1 | no | — |
| 6 | A4⋊S4 | — | 35 | 1 | no | — |
| 7 | #1020⋊C3 | — | 35 | 2 | no | — |
| 8 | #5934 | — | 105 | 1 | no | — |
| 9 | #1020⋊C2 | — | 105 | 2 | no | — |
| 10 | C3×A5 | — | 56 | 1 | no | — |
| 11 | AΓL(1,𝔽8) | — | 120 | 2 | no | — |
| 12 | GL(3,𝔽2) | — | 120 | 2 | no | — |
| 13 | GL(3,𝔽2) | — | 120 | 1 | no | — |
| 14 | A42 | — | 35 | 1 | no | — |
| 15 | S5 | — | 168 | 1 | no | — |
| 16 | S5 | — | 168 | 1 | no | — |
| 17 | #1020 | — | 35 | 2 | no | — |
| 18 | #1020 | — | 105 | 2 | no | — |
| 19 | #997 | — | 105 | 1 | no | — |
| 20 | GL(2,ℤ/4) | — | 210 | 1 | no | — |
| 21 | #120⋊C3 | — | 105 | 2 | no | — |
| 22 | C3⋊S4 | — | 280 | 1 | no | — |
| 23 | GO+(4,𝔽2) | — | 280 | 1 | no | — |
| 24 | B(4,𝔽2) | — | 315 | 1 | no | Sylow |
| 25 | ΓL(1,𝔽16) | — | 336 | 1 | no | — |
| 26 | A5 | — | 56 | 1 | no | — |
| 27 | A5 | — | 168 | 1 | no | — |
| 28 | AGL(1,𝔽8) | — | 120 | 2 | no | — |
| 29 | C22⋊A4 | — | 35 | 2 | no | — |
| 30 | C22×A4 | — | 70 | 1 | no | — |
| 31 | C2×S4 | — | 420 | 2 | no | — |
| 32 | C2×S4 | — | 420 | 1 | no | — |
| 33 | C2×S4 | — | 210 | 1 | no | — |
| 34 | SL(2,ℤ/4) | — | 210 | 1 | no | — |
| 35 | C3×A4 | — | 280 | 1 | no | — |
| 36 | D32 | — | 280 | 1 | no | — |
| 37 | D32 | — | 280 | 1 | no | — |
| 38 | #157 | — | 280 | 1 | no | — |
| 39 | H(5,𝔽2) | — | 105 | 1 | no | — |
| 40 | #120 | — | 315 | 1 | no | — |
| 41 | #120 | — | 105 | 2 | no | — |
| 42 | #99 | — | 315 | 1 | no | — |
| 43 | #99 | — | 315 | 2 | no | — |
| 44 | C3×D5 | — | 336 | 1 | no | — |
| 45 | C2×A4 | — | 420 | 2 | no | — |
| 46 | C2×A4 | — | 420 | 1 | no | — |
| 47 | C2×A4 | — | 210 | 1 | no | — |
| 48 | S4 | — | 840 | 2 | no | — |
| 49 | S4 | — | 420 | 2 | no | — |
| 50 | S4 | — | 420 | 1 | no | — |
| 51 | S4 | — | 420 | 1 | no | — |
| 52 | S4 | — | 420 | 2 | no | — |
| 53 | S4 | — | 420 | 1 | no | — |
| 54 | #67 | — | 840 | 1 | no | — |
| 55 | SL(2,𝔽3) | — | 840 | 1 | no | — |
| 56 | ΓL(1,𝔽8) | — | 960 | 1 | no | — |
| 57 | AGL(1,𝔽5) | — | 1008 | 1 | no | — |
| 58 | C3⋊D3 | — | 280 | 1 | no | — |
| 59 | C3×D3 | — | 560 | 1 | no | — |
| 60 | C3×D3 | — | 560 | 1 | no | — |
| 61 | C24 | — | 35 | 1 | no | — |
| 62 | #41 | — | 630 | 1 | no | — |
| 63 | C2×D4 | — | 315 | 2 | no | — |
| 64 | C2×D4 | — | 315 | 1 | no | — |
| 65 | C2×D4 | — | 630 | 1 | no | — |
| 66 | C22⋊C4 | — | 315 | 2 | no | — |
| 67 | C22⋊C4 | — | 315 | 1 | no | — |
| 68 | C22⋊C4 | — | 630 | 1 | no | — |
| 69 | C3×C5 | — | 336 | 1 | no | — |
| 70 | C22×C3 | — | 280 | 1 | no | — |
| 71 | C2×D3 | — | 840 | 1 | no | — |
| 72 | C2×D3 | — | 1680 | 1 | no | — |
| 73 | A4 | — | 280 | 2 | no | — |
| 74 | A4 | — | 70 | 1 | no | — |
| 75 | A4 | — | 420 | 2 | no | — |
| 76 | A4 | — | 420 | 1 | no | — |
| 77 | Dic3 | — | 840 | 1 | no | — |
| 78 | D5 | — | 336 | 1 | no | — |
| 79 | C32 | — | 280 | 1 | no | Sylow |
| 80 | C23 | — | 15 | 2 | no | — |
| 81 | C23 | — | 315 | 1 | no | — |
| 82 | C23 | — | 105 | 1 | no | — |
| 83 | C23 | — | 210 | 1 | no | — |
| 84 | Q8 | — | 210 | 1 | no | — |
| 85 | D4 | — | 630 | 2 | no | — |
| 86 | D4 | — | 630 | 1 | no | — |
| 87 | D4 | — | 1260 | 1 | no | — |
| 88 | D4 | — | 1260 | 1 | no | — |
| 89 | C2×C4 | — | 315 | 1 | no | — |
| 90 | C2×C4 | — | 315 | 2 | no | — |
| 91 | C2×C4 | — | 630 | 1 | no | — |
| 92 | C7 | — | 960 | 1 | no | Sylow |
| 93 | C2×C3 | — | 840 | 1 | no | — |
| 94 | C2×C3 | — | 1680 | 1 | no | — |
| 95 | D3 | — | 560 | 1 | no | — |
| 96 | D3 | — | 560 | 1 | no | — |
| 97 | D3 | — | 1680 | 1 | no | — |
| 98 | C5 | — | 336 | 1 | no | Sylow |
| 99 | C22 | — | 315 | 1 | no | — |
| 100 | C22 | — | 70 | 1 | no | — |
| 101 | C22 | — | 105 | 2 | no | — |
| 102 | C22 | — | 630 | 1 | no | — |
| 103 | C22 | — | 420 | 1 | no | — |
| 104 | C4 | — | 630 | 1 | no | — |
| 105 | C4 | — | 1260 | 1 | no | — |
| 106 | C3 | — | 560 | 1 | no | — |
| 107 | C3 | — | 56 | 1 | no | — |
| 108 | C2 | — | 105 | 1 | no | — |
| 109 | C2 | — | 210 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| AGL(3,𝔽2) | 15 |
| S6 | 28 |
| (#1020⋊C3)⋊C2 | 35 |
| ΓL(2,𝔽4) | 56 |