← C23×C3×C7×C64 #72189293
Subgroups — C23×C3×C7×C64 #72189293
Order 10752.
1288 conjugacy classes of subgroups.
The subgroup lattice has 160 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C23×C3×C7×C32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×C3×C7×C64 | C2 | 1 | 14 | yes | maximal, direct product factor |
| 3 | C23×C7×C64 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C23×C3×C7×C16 | C4 | 1 | 1 | yes | characteristic |
| 5 | C22×C3×C7×C32 | C4 | 1 | 7 | yes | — |
| 6 | C22×C3×C7×C32 | C22 | 1 | 7 | yes | — |
| 7 | C2×C3×C7×C64 | C22 | 1 | 28 | yes | direct product factor |
| 8 | C23×C3×C64 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 9 | C23×C7×C32 | C2×C3 | 1 | 1 | yes | characteristic |
| 10 | C22×C7×C64 | C2×C3 | 1 | 14 | yes | direct product factor |
| 11 | C23×C3×C7×C8 | C8 | 1 | 1 | yes | characteristic |
| 12 | C22×C3×C7×C16 | C8 | 1 | 7 | yes | — |
| 13 | C22×C3×C7×C16 | C2×C4 | 1 | 7 | yes | — |
| 14 | C2×C3×C7×C32 | C2×C4 | 1 | 21 | yes | — |
| 15 | C2×C3×C7×C32 | C23 | 1 | 7 | yes | — |
| 16 | C3×C7×C64 | C23 | 1 | 8 | yes | direct product factor |
| 17 | C23×C7×C16 | C3×C4 | 1 | 1 | yes | characteristic |
| 18 | C22×C7×C32 | C3×C4 | 1 | 7 | yes | — |
| 19 | C22×C7×C32 | C22×C3 | 1 | 7 | yes | — |
| 20 | C2×C7×C64 | C22×C3 | 1 | 28 | yes | direct product factor |
| 21 | C23×C3×C32 | C2×C7 | 1 | 1 | yes | characteristic |
| 22 | C22×C3×C64 | C2×C7 | 1 | 14 | yes | direct product factor |
| 23 | C23×C3×C4×C7 | C16 | 1 | 1 | yes | characteristic |
| 24 | C22×C3×C7×C8 | C16 | 1 | 7 | yes | — |
| 25 | C22×C3×C7×C8 | C2×C8 | 1 | 7 | yes | — |
| 26 | C2×C3×C7×C16 | C2×C8 | 1 | 21 | yes | — |
| 27 | C2×C3×C7×C16 | C22×C4 | 1 | 7 | yes | — |
| 28 | C3×C7×C32 | C22×C4 | 1 | 7 | yes | — |
| 29 | C3×C7×C32 | C24 | 1 | 1 | yes | characteristic |
| 30 | C23×C64 | C3×C7 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 31 | C23×C7×C8 | C3×C8 | 1 | 1 | yes | characteristic |
| 32 | C22×C7×C16 | C3×C8 | 1 | 7 | yes | — |
| 33 | C22×C7×C16 | C2×C3×C4 | 1 | 7 | yes | — |
| 34 | C2×C7×C32 | C2×C3×C4 | 1 | 21 | yes | — |
| 35 | C2×C7×C32 | C23×C3 | 1 | 7 | yes | — |
| 36 | C7×C64 | C23×C3 | 1 | 8 | yes | direct product factor |
| 37 | C23×C3×C16 | C4×C7 | 1 | 1 | yes | characteristic |
| 38 | C22×C3×C32 | C4×C7 | 1 | 7 | yes | — |
| 39 | C22×C3×C32 | C22×C7 | 1 | 7 | yes | — |
| 40 | C2×C3×C64 | C22×C7 | 1 | 28 | yes | direct product factor |
| 41 | C24×C3×C7 | C32 | 1 | 1 | yes | characteristic |
| 42 | C22×C3×C4×C7 | C32 | 1 | 7 | yes | — |
| 43 | C22×C3×C4×C7 | C2×C16 | 1 | 7 | yes | — |
| 44 | C2×C3×C7×C8 | C2×C16 | 1 | 21 | yes | — |
| 45 | C2×C3×C7×C8 | C22×C8 | 1 | 7 | yes | — |
| 46 | C3×C7×C16 | C22×C8 | 1 | 7 | yes | — |
| 47 | C3×C7×C16 | C23×C4 | 1 | 1 | yes | characteristic |
| 48 | C23×C32 | C2×C3×C7 | 1 | 1 | yes | characteristic |
| 49 | C22×C64 | C2×C3×C7 | 1 | 14 | yes | direct product factor |
| 50 | C23×C4×C7 | C3×C16 | 1 | 1 | yes | characteristic |
| 51 | C22×C7×C8 | C3×C16 | 1 | 7 | yes | — |
| 52 | C22×C7×C8 | C2×C3×C8 | 1 | 7 | yes | — |
| 53 | C2×C7×C16 | C2×C3×C8 | 1 | 21 | yes | — |
| 54 | C2×C7×C16 | C22×C3×C4 | 1 | 7 | yes | — |
| 55 | C7×C32 | C22×C3×C4 | 1 | 7 | yes | — |
| 56 | C7×C32 | C24×C3 | 1 | 1 | yes | characteristic |
| 57 | C23×C3×C8 | C7×C8 | 1 | 1 | yes | characteristic |
| 58 | C22×C3×C16 | C7×C8 | 1 | 7 | yes | — |
| 59 | C22×C3×C16 | C2×C4×C7 | 1 | 7 | yes | — |
| 60 | C2×C3×C32 | C2×C4×C7 | 1 | 21 | yes | — |
| 61 | C2×C3×C32 | C23×C7 | 1 | 7 | yes | — |
| 62 | C3×C64 | C23×C7 | 1 | 8 | yes | direct product factor |
| 63 | C23×C3×C7 | C64 | 1 | 8 | yes | direct product factor |
| 64 | C23×C3×C7 | C2×C32 | 1 | 7 | yes | — |
| 65 | C2×C3×C4×C7 | C2×C32 | 1 | 21 | yes | — |
| 66 | C2×C3×C4×C7 | C22×C16 | 1 | 7 | yes | — |
| 67 | C3×C7×C8 | C22×C16 | 1 | 7 | yes | — |
| 68 | C3×C7×C8 | C23×C8 | 1 | 1 | yes | characteristic |
| 69 | C23×C16 | C3×C4×C7 | 1 | 1 | yes | characteristic |
| 70 | C22×C32 | C3×C4×C7 | 1 | 7 | yes | — |
| 71 | C22×C32 | C22×C3×C7 | 1 | 7 | yes | — |
| 72 | C2×C64 | C22×C3×C7 | 1 | 28 | yes | direct product factor |
| 73 | C24×C7 | C3×C32 | 1 | 1 | yes | characteristic |
| 74 | C22×C4×C7 | C3×C32 | 1 | 7 | yes | — |
| 75 | C22×C4×C7 | C2×C3×C16 | 1 | 7 | yes | — |
| 76 | C2×C7×C8 | C2×C3×C16 | 1 | 21 | yes | — |
| 77 | C2×C7×C8 | C22×C3×C8 | 1 | 7 | yes | — |
| 78 | C7×C16 | C22×C3×C8 | 1 | 7 | yes | — |
| 79 | C7×C16 | C23×C3×C4 | 1 | 1 | yes | characteristic |
| 80 | C23×C3×C4 | C7×C16 | 1 | 1 | yes | characteristic |
| 81 | C22×C3×C8 | C7×C16 | 1 | 7 | yes | — |
| 82 | C22×C3×C8 | C2×C7×C8 | 1 | 7 | yes | — |
| 83 | C2×C3×C16 | C2×C7×C8 | 1 | 21 | yes | — |
| 84 | C2×C3×C16 | C22×C4×C7 | 1 | 7 | yes | — |
| 85 | C3×C32 | C22×C4×C7 | 1 | 7 | yes | — |
| 86 | C3×C32 | C24×C7 | 1 | 1 | yes | characteristic |
| 87 | C22×C3×C7 | C2×C64 | 1 | 28 | yes | direct product factor |
| 88 | C22×C3×C7 | C22×C32 | 1 | 7 | yes | — |
| 89 | C3×C4×C7 | C22×C32 | 1 | 7 | yes | — |
| 90 | C3×C4×C7 | C23×C16 | 1 | 1 | yes | characteristic |
| 91 | C23×C8 | C3×C7×C8 | 1 | 1 | yes | characteristic |
| 92 | C22×C16 | C3×C7×C8 | 1 | 7 | yes | — |
| 93 | C22×C16 | C2×C3×C4×C7 | 1 | 7 | yes | — |
| 94 | C2×C32 | C2×C3×C4×C7 | 1 | 21 | yes | — |
| 95 | C2×C32 | C23×C3×C7 | 1 | 7 | yes | — |
| 96 | C64 | C23×C3×C7 | 1 | 8 | yes | direct product factor |
| 97 | C23×C7 | C3×C64 | 1 | 8 | yes | direct product factor |
| 98 | C23×C7 | C2×C3×C32 | 1 | 7 | yes | — |
| 99 | C2×C4×C7 | C2×C3×C32 | 1 | 21 | yes | — |
| 100 | C2×C4×C7 | C22×C3×C16 | 1 | 7 | yes | — |
| 101 | C7×C8 | C22×C3×C16 | 1 | 7 | yes | — |
| 102 | C7×C8 | C23×C3×C8 | 1 | 1 | yes | characteristic |
| 103 | C24×C3 | C7×C32 | 1 | 1 | yes | characteristic |
| 104 | C22×C3×C4 | C7×C32 | 1 | 7 | yes | — |
| 105 | C22×C3×C4 | C2×C7×C16 | 1 | 7 | yes | — |
| 106 | C2×C3×C8 | C2×C7×C16 | 1 | 21 | yes | — |
| 107 | C2×C3×C8 | C22×C7×C8 | 1 | 7 | yes | — |
| 108 | C3×C16 | C22×C7×C8 | 1 | 7 | yes | — |
| 109 | C3×C16 | C23×C4×C7 | 1 | 1 | yes | characteristic |
| 110 | C2×C3×C7 | C22×C64 | 1 | 14 | yes | direct product factor |
| 111 | C2×C3×C7 | C23×C32 | 1 | 1 | yes | characteristic |
| 112 | C23×C4 | C3×C7×C16 | 1 | 1 | yes | characteristic |
| 113 | C22×C8 | C3×C7×C16 | 1 | 7 | yes | — |
| 114 | C22×C8 | C2×C3×C7×C8 | 1 | 7 | yes | — |
| 115 | C2×C16 | C2×C3×C7×C8 | 1 | 21 | yes | — |
| 116 | C2×C16 | C22×C3×C4×C7 | 1 | 7 | yes | — |
| 117 | C32 | C22×C3×C4×C7 | 1 | 7 | yes | — |
| 118 | C32 | C24×C3×C7 | 1 | 1 | yes | characteristic |
| 119 | C22×C7 | C2×C3×C64 | 1 | 28 | yes | direct product factor |
| 120 | C22×C7 | C22×C3×C32 | 1 | 7 | yes | — |
| 121 | C4×C7 | C22×C3×C32 | 1 | 7 | yes | — |
| 122 | C4×C7 | C23×C3×C16 | 1 | 1 | yes | characteristic |
| 123 | C23×C3 | C7×C64 | 1 | 8 | yes | direct product factor |
| 124 | C23×C3 | C2×C7×C32 | 1 | 7 | yes | — |
| 125 | C2×C3×C4 | C2×C7×C32 | 1 | 21 | yes | — |
| 126 | C2×C3×C4 | C22×C7×C16 | 1 | 7 | yes | — |
| 127 | C3×C8 | C22×C7×C16 | 1 | 7 | yes | — |
| 128 | C3×C8 | C23×C7×C8 | 1 | 1 | yes | characteristic |
| 129 | C3×C7 | C23×C64 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 130 | C24 | C3×C7×C32 | 1 | 1 | yes | characteristic |
| 131 | C22×C4 | C3×C7×C32 | 1 | 7 | yes | — |
| 132 | C22×C4 | C2×C3×C7×C16 | 1 | 7 | yes | — |
| 133 | C2×C8 | C2×C3×C7×C16 | 1 | 21 | yes | — |
| 134 | C2×C8 | C22×C3×C7×C8 | 1 | 7 | yes | — |
| 135 | C16 | C22×C3×C7×C8 | 1 | 7 | yes | — |
| 136 | C16 | C23×C3×C4×C7 | 1 | 1 | yes | characteristic |
| 137 | C2×C7 | C22×C3×C64 | 1 | 14 | yes | direct product factor |
| 138 | C2×C7 | C23×C3×C32 | 1 | 1 | yes | characteristic |
| 139 | C22×C3 | C2×C7×C64 | 1 | 28 | yes | direct product factor |
| 140 | C22×C3 | C22×C7×C32 | 1 | 7 | yes | — |
| 141 | C3×C4 | C22×C7×C32 | 1 | 7 | yes | — |
| 142 | C3×C4 | C23×C7×C16 | 1 | 1 | yes | characteristic |
| 143 | C23 | C3×C7×C64 | 1 | 8 | yes | direct product factor |
| 144 | C23 | C2×C3×C7×C32 | 1 | 7 | yes | — |
| 145 | C2×C4 | C2×C3×C7×C32 | 1 | 21 | yes | — |
| 146 | C2×C4 | C22×C3×C7×C16 | 1 | 7 | yes | — |
| 147 | C8 | C22×C3×C7×C16 | 1 | 7 | yes | — |
| 148 | C8 | C23×C3×C7×C8 | 1 | 1 | yes | characteristic |
| 149 | C7 | C23×C3×C64 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 150 | C2×C3 | C22×C7×C64 | 1 | 14 | yes | direct product factor |
| 151 | C2×C3 | C23×C7×C32 | 1 | 1 | yes | characteristic |
| 152 | C22 | C2×C3×C7×C64 | 1 | 28 | yes | direct product factor |
| 153 | C22 | C22×C3×C7×C32 | 1 | 7 | yes | — |
| 154 | C4 | C22×C3×C7×C32 | 1 | 7 | yes | — |
| 155 | C4 | C23×C3×C7×C16 | 1 | 1 | yes | characteristic |
| 156 | C3 | C23×C7×C64 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 157 | C2 | C22×C3×C7×C64 | 1 | 14 | yes | direct product factor |
| 158 | C2 | C23×C3×C7×C32 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta