← #20283132
Subgroups — #20283132
Order 6784.
564 conjugacy classes of subgroups.
The subgroup lattice has 172 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×M16×D53 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#13133174 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #16810580 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #16810577 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C4×#11719917 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | M16×Dic53 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4×M16×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×C4×D53 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#11719917 | C22 | 1 | 2 | yes | — |
| 10 | C2×#11719912 | C22 | 1 | 4 | yes | — |
| 11 | M16×D53 | C4 | 1 | 16 | yes | complemented |
| 12 | C2×C8×D53 | C22 | 1 | 4 | yes | — |
| 13 | #13133174 | C4 | 1 | 4 | yes | — |
| 14 | #13133174 | C22 | 1 | 2 | yes | — |
| 15 | #13133174 | C22 | 1 | 2 | yes | — |
| 16 | C2×#11719919 | C4 | 1 | 2 | yes | — |
| 17 | C2×#11719921 | C4 | 1 | 2 | yes | — |
| 18 | C2×C4×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 19 | #13133128 | C22 | 1 | 2 | yes | complemented |
| 20 | #13133127 | C22 | 1 | 4 | yes | complemented |
| 21 | C4×#88322 | C22 | 1 | 2 | yes | complemented |
| 22 | C8×Dic53 | C22 | 1 | 4 | yes | complemented |
| 23 | C2×M16×C53 | C22 | 1 | 2 | yes | — |
| 24 | C2×C42×C53 | C22 | 1 | 1 | yes | characteristic |
| 25 | #97×C53 | C22 | 1 | 2 | yes | complemented |
| 26 | C4×C8×C53 | C22 | 1 | 2 | yes | complemented |
| 27 | C23×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×C4×C53 | C2×C4 | 1 | 2 | yes | — |
| 29 | C22×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 30 | C22×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 31 | C22×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×D53 | C2×C4 | 1 | 2 | yes | — |
| 33 | C2×C4×D53 | C2×C4 | 1 | 8 | yes | — |
| 34 | C2×C4×D53 | C23 | 1 | 2 | yes | — |
| 35 | M16×C53 | C2×C4 | 1 | 8 | yes | — |
| 36 | C2×C8×C53 | C23 | 1 | 4 | yes | — |
| 37 | C42×C53 | C2×C4 | 1 | 2 | yes | — |
| 38 | C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 39 | C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 40 | #11719921 | C2×C4 | 1 | 4 | yes | — |
| 41 | #11719921 | C2×C4 | 1 | 4 | yes | — |
| 42 | #11719919 | C2×C4 | 1 | 4 | yes | — |
| 43 | #11719919 | C2×C4 | 1 | 4 | yes | — |
| 44 | C4×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 45 | C4×Dic53 | C23 | 1 | 2 | yes | — |
| 46 | #11719917 | C2×C4 | 1 | 8 | yes | — |
| 47 | C2×#88322 | C23 | 1 | 4 | yes | — |
| 48 | #11719912 | C2×C4 | 1 | 16 | yes | complemented |
| 49 | C8×D53 | C2×C4 | 1 | 16 | yes | complemented |
| 50 | C23×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 51 | C22×D53 | — | 4 | 2 | no | — |
| 52 | C22×D53 | C42 | 1 | 4 | yes | — |
| 53 | C22×D53 | C22×C4 | 1 | 2 | yes | — |
| 54 | C2×C4×C53 | — | 4 | 2 | no | — |
| 55 | C2×C4×C53 | C22×C4 | 1 | 2 | yes | — |
| 56 | C2×C4×C53 | C22×C4 | 1 | 2 | yes | — |
| 57 | C2×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C2×C4×C53 | C22×C4 | 1 | 4 | yes | — |
| 59 | C2×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 60 | C2×Dic53 | — | 4 | 2 | no | — |
| 61 | C2×Dic53 | C42 | 1 | 4 | yes | — |
| 62 | C2×Dic53 | C22×C4 | 1 | 4 | yes | — |
| 63 | C2×Dic53 | C22×C4 | 1 | 2 | yes | — |
| 64 | C4×D53 | C42 | 1 | 8 | yes | — |
| 65 | C4×D53 | C22×C4 | 1 | 4 | yes | — |
| 66 | C4×D53 | C22×C4 | 1 | 4 | yes | — |
| 67 | C8×C53 | C22×C4 | 1 | 8 | yes | — |
| 68 | #88322 | C22×C4 | 1 | 8 | yes | — |
| 69 | C22×C53 | — | 2 | 2 | no | — |
| 70 | C22×C53 | C2×C42 | 1 | 2 | yes | — |
| 71 | C22×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×D53 | — | 2 | 4 | no | — |
| 73 | C2×D53 | — | 4 | 4 | no | — |
| 74 | C2×D53 | C2×C42 | 1 | 4 | yes | — |
| 75 | C4×C53 | — | 2 | 2 | no | — |
| 76 | C4×C53 | — | 2 | 2 | no | — |
| 77 | C4×C53 | C2×C42 | 1 | 2 | yes | — |
| 78 | C4×C53 | C23×C4 | 1 | 2 | yes | — |
| 79 | Dic53 | — | 2 | 4 | no | — |
| 80 | Dic53 | C2×C42 | 1 | 4 | yes | — |
| 81 | #2873 | — | 53 | 1 | no | maximal, Sylow |
| 82 | C2×C53 | — | 2 | 2 | no | — |
| 83 | C2×C53 | C22×C42 | 1 | 1 | yes | characteristic |
| 84 | C2×C53 | #568 | 1 | 2 | yes | — |
| 85 | D53 | — | 2 | 4 | no | — |
| 86 | C22×M16 | — | 53 | 2 | no | — |
| 87 | C2×#117 | — | 53 | 1 | no | — |
| 88 | #405 | — | 53 | 4 | no | — |
| 89 | #405 | — | 53 | 4 | no | — |
| 90 | C4×M16 | — | 53 | 2 | no | — |
| 91 | C4×M16 | — | 53 | 1 | no | — |
| 92 | C4×M16 | D53 | 1 | 1 | yes | characteristic, complemented |
| 93 | C53 | #2873 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 94 | C23×C4 | — | 53 | 1 | no | — |
| 95 | C2×M16 | — | 53 | 2 | no | — |
| 96 | C2×M16 | — | 53 | 16 | no | — |
| 97 | C2×M16 | — | 53 | 4 | no | — |
| 98 | C2×M16 | C2×D53 | 1 | 2 | yes | — |
| 99 | C22×C8 | — | 53 | 4 | no | — |
| 100 | #117 | — | 53 | 4 | no | — |
| 101 | #117 | — | 53 | 2 | no | — |
| 102 | #117 | — | 53 | 2 | no | — |
| 103 | C2×C4⋊C4 | — | 53 | 2 | no | — |
| 104 | C2×C22⋊C4 | — | 53 | 2 | no | — |
| 105 | C2×C42 | — | 53 | 1 | no | — |
| 106 | C2×C42 | C2×D53 | 1 | 1 | yes | characteristic |
| 107 | #97 | — | 53 | 4 | no | — |
| 108 | #97 | — | 53 | 2 | no | — |
| 109 | #97 | C2×D53 | 1 | 2 | yes | complemented |
| 110 | C4×C8 | — | 53 | 4 | no | — |
| 111 | C4×C8 | — | 53 | 2 | no | — |
| 112 | C4×C8 | C2×D53 | 1 | 2 | yes | complemented |
| 113 | C24 | — | 53 | 1 | no | — |
| 114 | C22×C4 | — | 53 | 1 | no | — |
| 115 | C22×C4 | — | 53 | 2 | no | — |
| 116 | C22×C4 | — | 53 | 8 | no | — |
| 117 | C22×C4 | — | 53 | 2 | no | — |
| 118 | C22×C4 | — | 53 | 2 | no | — |
| 119 | C22×C4 | C4×D53 | 1 | 2 | yes | — |
| 120 | C22×C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 121 | M16 | — | 53 | 8 | no | — |
| 122 | M16 | — | 53 | 16 | no | — |
| 123 | M16 | C4×D53 | 1 | 8 | yes | — |
| 124 | C2×C8 | — | 53 | 4 | no | — |
| 125 | C2×C8 | — | 53 | 16 | no | — |
| 126 | C2×C8 | C22×D53 | 1 | 4 | yes | — |
| 127 | C4⋊C4 | — | 53 | 4 | no | — |
| 128 | C4⋊C4 | — | 53 | 4 | no | — |
| 129 | C22⋊C4 | — | 53 | 4 | no | — |
| 130 | C22⋊C4 | — | 53 | 4 | no | — |
| 131 | C42 | — | 53 | 2 | no | — |
| 132 | C42 | — | 53 | 2 | no | — |
| 133 | C42 | C4×D53 | 1 | 2 | yes | — |
| 134 | C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 135 | C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 136 | C23 | — | 53 | 2 | no | — |
| 137 | C23 | — | 53 | 4 | no | — |
| 138 | C23 | — | 212 | 2 | no | — |
| 139 | C23 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 140 | C2×C4 | — | 4 | 2 | no | — |
| 141 | C2×C4 | — | 53 | 4 | no | — |
| 142 | C2×C4 | — | 53 | 2 | no | — |
| 143 | C2×C4 | — | 53 | 8 | no | — |
| 144 | C2×C4 | — | 53 | 4 | no | — |
| 145 | C2×C4 | — | 53 | 4 | no | — |
| 146 | C2×C4 | — | 53 | 4 | no | — |
| 147 | C2×C4 | — | 212 | 2 | no | — |
| 148 | C2×C4 | C2×C4×D53 | 1 | 2 | yes | — |
| 149 | C2×C4 | C2×C4×D53 | 1 | 2 | yes | — |
| 150 | C2×C4 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 151 | C2×C4 | C2×C4×D53 | 1 | 4 | yes | — |
| 152 | C2×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 153 | C8 | — | 53 | 8 | no | — |
| 154 | C8 | C2×C4×D53 | 1 | 8 | yes | — |
| 155 | C22 | — | 2 | 2 | no | — |
| 156 | C22 | — | 53 | 4 | no | — |
| 157 | C22 | — | 106 | 4 | no | — |
| 158 | C22 | — | 212 | 4 | no | — |
| 159 | C22 | C42×D53 | 1 | 2 | yes | — |
| 160 | C22 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 161 | C4 | — | 2 | 2 | no | — |
| 162 | C4 | — | 2 | 2 | no | — |
| 163 | C4 | — | 53 | 4 | no | — |
| 164 | C4 | — | 106 | 4 | no | — |
| 165 | C4 | C42×D53 | 1 | 2 | yes | — |
| 166 | C4 | C22×C4×D53 | 1 | 2 | yes | — |
| 167 | C2 | — | 2 | 2 | no | — |
| 168 | C2 | — | 106 | 4 | no | — |
| 169 | C2 | C2×C42×D53 | 1 | 1 | yes | characteristic |
| 170 | C2 | #16811389 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Finite Group Databasebeta