← #14278775
Subgroups — #14278775
Order 1920.
264 conjugacy classes of subgroups.
The subgroup lattice has 228 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #11751202 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C5×#812 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C3×#3944 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10598305 | — | 3 | 1 | no | maximal, Hall |
| 5 | C22×C3×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 6 | #91815 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×#6710 | C4 | 1 | 1 | yes | characteristic |
| 8 | #91099 | — | 2 | 1 | no | — |
| 9 | C8×Dic15 | — | 2 | 1 | no | — |
| 10 | C5×#812 | — | 2 | 1 | no | — |
| 11 | C5×#812 | C4 | 1 | 2 | yes | complemented |
| 12 | C2×C5×C3⋊C16 | — | 2 | 1 | no | — |
| 13 | C3×#3944 | — | 2 | 1 | no | — |
| 14 | C3×#3944 | — | 2 | 1 | no | — |
| 15 | C2×C3×#672 | C4 | 1 | 2 | yes | complemented |
| 16 | #67813 | — | 5 | 1 | no | maximal, Hall |
| 17 | C2×C5×M32 | — | 3 | 1 | no | — |
| 18 | #65390 | — | 3 | 1 | no | — |
| 19 | C2×#3944 | D3 | 1 | 1 | yes | characteristic |
| 20 | C22×C3×C4×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C2×C3×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C3×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C2×C3×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×C3×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×C3×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×C3×C5×C8 | Q8 | 1 | 1 | yes | characteristic |
| 27 | #6719 | C8 | 1 | 2 | yes | — |
| 28 | C4⋊Dic15 | — | 2 | 1 | no | — |
| 29 | C4×Dic15 | — | 2 | 1 | no | — |
| 30 | #6710 | — | 2 | 1 | no | — |
| 31 | #6710 | C8 | 1 | 2 | yes | — |
| 32 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 33 | C3×#672 | — | 2 | 2 | no | — |
| 34 | C5×C3⋊C16 | — | 2 | 2 | no | — |
| 35 | C2×C3×M32 | — | 5 | 1 | no | — |
| 36 | #5101 | — | 5 | 1 | no | — |
| 37 | C2×#812 | D5 | 1 | 1 | yes | characteristic |
| 38 | C22×C5×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | #4072 | — | 3 | 1 | no | — |
| 40 | C2×#681 | — | 3 | 1 | no | — |
| 41 | C5×M32 | — | 3 | 2 | no | — |
| 42 | C5×M32 | — | 6 | 1 | no | — |
| 43 | C2×C5×C16 | — | 6 | 1 | no | — |
| 44 | #3947 | — | 6 | 1 | no | — |
| 45 | C8×Dic5 | — | 6 | 1 | no | — |
| 46 | #3944 | — | 2 | 1 | no | — |
| 47 | #3944 | — | 2 | 1 | no | — |
| 48 | C2×#672 | Dic3 | 1 | 2 | yes | complemented |
| 49 | #1378 | — | 15 | 1 | no | Sylow |
| 50 | C23×C3×C5 | C2×C8 | 1 | 1 | yes | characteristic |
| 51 | C2×C3×C4×C5 | C42 | 1 | 1 | yes | characteristic |
| 52 | C2×C3×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | C2×C3×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C2×C3×C4×C5 | C2×C8 | 1 | 1 | yes | characteristic |
| 55 | C2×C3×C4×C5 | M16 | 1 | 1 | yes | characteristic |
| 56 | C2×C3×C4×C5 | M16 | 1 | 1 | yes | characteristic |
| 57 | C2×Dic15 | — | 2 | 2 | no | — |
| 58 | C3×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C3×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C3×C5×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C3×C5×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C3⋊#168 | — | 2 | 2 | no | — |
| 63 | C22×C3×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 64 | #926 | — | 5 | 1 | no | — |
| 65 | C2×#208 | — | 5 | 1 | no | — |
| 66 | C3×M32 | — | 10 | 1 | no | — |
| 67 | C3×M32 | — | 10 | 1 | no | — |
| 68 | C2×C3×C16 | — | 5 | 2 | no | — |
| 69 | #815 | — | 10 | 1 | no | — |
| 70 | C8×Dic3 | — | 10 | 1 | no | — |
| 71 | #812 | — | 2 | 1 | no | — |
| 72 | #812 | Dic5 | 1 | 2 | yes | complemented |
| 73 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 74 | C22×C4×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 75 | C2×C5×C8 | Dic6 | 1 | 1 | yes | characteristic |
| 76 | C2×C5×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 77 | C2×C5×C8 | D12 | 1 | 1 | yes | characteristic |
| 78 | C2×C5×C8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 79 | C2×C5×C8 | #67 | 1 | 1 | yes | characteristic |
| 80 | C2×C5×C8 | #67 | 1 | 1 | yes | characteristic |
| 81 | C22⋊Dic5 | — | 3 | 2 | no | — |
| 82 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 83 | C4×Dic5 | — | 6 | 1 | no | — |
| 84 | #681 | — | 3 | 2 | no | — |
| 85 | #681 | — | 6 | 1 | no | — |
| 86 | C2×#168 | — | 6 | 1 | no | — |
| 87 | C5×C16 | — | 6 | 2 | no | — |
| 88 | #672 | — | 2 | 2 | no | — |
| 89 | C2×M32 | — | 3 | 1 | no | — |
| 90 | C2×M32 | — | 5 | 1 | no | — |
| 91 | #405 | — | 15 | 1 | no | — |
| 92 | C22×C3×C5 | — | 4 | 1 | no | — |
| 93 | C22×C3×C5 | C4×C8 | 1 | 1 | yes | characteristic |
| 94 | C22×C3×C5 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 95 | C22×C3×C5 | C4⋊C8 | 1 | 1 | yes | characteristic |
| 96 | C3×C4×C5 | #95 | 1 | 1 | yes | characteristic |
| 97 | C3×C4×C5 | #97 | 1 | 1 | yes | characteristic |
| 98 | C3×C4×C5 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 99 | C3×C4×C5 | C4⋊C8 | 1 | 1 | yes | characteristic |
| 100 | Dic15 | — | 4 | 2 | no | — |
| 101 | C22×C3×C4 | C4×D5 | 1 | 1 | yes | characteristic |
| 102 | C2×C3×C8 | Dic10 | 1 | 1 | yes | characteristic |
| 103 | C2×C3×C8 | C4×D5 | 1 | 1 | yes | characteristic |
| 104 | C2×C3×C8 | D20 | 1 | 1 | yes | characteristic |
| 105 | C2×C3×C8 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 106 | C2×C3×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 107 | C2×C3×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 108 | #217 | — | 5 | 2 | no | — |
| 109 | C4⋊Dic3 | — | 10 | 1 | no | — |
| 110 | C4×Dic3 | — | 10 | 1 | no | — |
| 111 | #208 | — | 5 | 2 | no | — |
| 112 | #208 | — | 10 | 1 | no | — |
| 113 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 114 | C3×C16 | — | 10 | 2 | no | — |
| 115 | C3⋊C16 | — | 2 | 2 | no | — |
| 116 | C23×C5 | D3×C8 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C5 | D3×C8 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C5 | ΓL(1,𝔽25) | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C5 | ΓL(1,𝔽25) | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C5 | C4×Dic3 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C5 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 123 | C2×Dic5 | — | 6 | 2 | no | — |
| 124 | C5×C8 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 125 | C5×C8 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 126 | C5×C8 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 127 | C5×C8 | #217 | 1 | 1 | yes | characteristic |
| 128 | #168 | — | 6 | 2 | no | — |
| 129 | C2×M16 | — | 15 | 1 | no | — |
| 130 | C22×C8 | D3×D5 | 1 | 1 | yes | characteristic |
| 131 | #117 | — | 15 | 1 | no | — |
| 132 | M32 | — | 3 | 2 | no | — |
| 133 | M32 | — | 6 | 1 | no | — |
| 134 | M32 | — | 10 | 1 | no | — |
| 135 | M32 | — | 10 | 1 | no | — |
| 136 | C2×C16 | — | 5 | 2 | no | — |
| 137 | C2×C16 | — | 6 | 1 | no | — |
| 138 | #97 | — | 30 | 1 | no | — |
| 139 | C4×C8 | — | 30 | 1 | no | — |
| 140 | C2×C3×C5 | — | 2 | 1 | no | — |
| 141 | C2×C3×C5 | — | 2 | 1 | no | — |
| 142 | C2×C3×C5 | — | 2 | 1 | no | — |
| 143 | C2×C3×C5 | #336 | 1 | 1 | yes | characteristic |
| 144 | C23×C3 | C8×D5 | 1 | 1 | yes | characteristic |
| 145 | C2×C3×C4 | C8×D5 | 1 | 1 | yes | characteristic |
| 146 | C2×C3×C4 | #676 | 1 | 1 | yes | characteristic |
| 147 | C2×C3×C4 | #676 | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C4 | C4×Dic5 | 1 | 1 | yes | characteristic |
| 149 | C2×C3×C4 | #683 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×C4 | #685 | 1 | 1 | yes | characteristic |
| 151 | C2×Dic3 | — | 10 | 2 | no | — |
| 152 | C3×C8 | #683 | 1 | 1 | yes | characteristic |
| 153 | C3×C8 | C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 154 | C3×C8 | #685 | 1 | 1 | yes | characteristic |
| 155 | C3×C8 | C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 156 | C3⋊C8 | — | 10 | 2 | no | — |
| 157 | C22×C5 | — | 4 | 1 | no | — |
| 158 | C22×C5 | C8×Dic3 | 1 | 1 | yes | characteristic |
| 159 | C22×C5 | Dic3⋊C8 | 1 | 1 | yes | characteristic |
| 160 | C22×C5 | (C2×D3)⋊C8 | 1 | 1 | yes | characteristic |
| 161 | C4×C5 | Dic3⋊C8 | 1 | 1 | yes | characteristic |
| 162 | C4×C5 | #815 | 1 | 1 | yes | characteristic |
| 163 | C4×C5 | (C2×D3)⋊C8 | 1 | 1 | yes | characteristic |
| 164 | C4×C5 | C3⋊#95 | 1 | 1 | yes | characteristic |
| 165 | Dic5 | — | 12 | 2 | no | — |
| 166 | C22×C4 | #1200 | 1 | 1 | yes | characteristic |
| 167 | M16 | — | 15 | 2 | no | — |
| 168 | M16 | — | 30 | 1 | no | — |
| 169 | C2×C8 | — | 30 | 1 | no | — |
| 170 | C2×C8 | D5×Dic3 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | D3×Dic5 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | #1201 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | #1202 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | #1203 | 1 | 1 | yes | characteristic |
| 175 | C2×C8 | #1204 | 1 | 1 | yes | characteristic |
| 176 | C4⋊C4 | — | 30 | 1 | no | — |
| 177 | C22⋊C4 | — | 15 | 2 | no | — |
| 178 | C42 | — | 30 | 1 | no | — |
| 179 | C16 | — | 6 | 2 | no | — |
| 180 | C16 | — | 10 | 2 | no | — |
| 181 | C3×C5 | #1378 | 1 | 1 | yes | characteristic, complemented, Hall |
| 182 | C22×C3 | — | 4 | 1 | no | — |
| 183 | C22×C3 | C8×Dic5 | 1 | 1 | yes | characteristic |
| 184 | C22×C3 | #3946 | 1 | 1 | yes | characteristic |
| 185 | C22×C3 | #3952 | 1 | 1 | yes | characteristic |
| 186 | C3×C4 | #3946 | 1 | 1 | yes | characteristic |
| 187 | C3×C4 | #3947 | 1 | 1 | yes | characteristic |
| 188 | C3×C4 | #3952 | 1 | 1 | yes | characteristic |
| 189 | C3×C4 | #3963 | 1 | 1 | yes | characteristic |
| 190 | Dic3 | — | 20 | 2 | no | — |
| 191 | C2×C5 | — | 2 | 1 | no | — |
| 192 | C2×C5 | — | 2 | 1 | no | — |
| 193 | C2×C5 | — | 2 | 1 | no | — |
| 194 | C2×C5 | #4550 | 1 | 1 | yes | characteristic |
| 195 | C23 | #6648 | 1 | 1 | yes | characteristic |
| 196 | C2×C4 | — | 30 | 2 | no | — |
| 197 | C2×C4 | #6648 | 1 | 1 | yes | characteristic |
| 198 | C2×C4 | #6651 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | #6651 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | Dic3×Dic5 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | #6667 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | #6669 | 1 | 1 | yes | characteristic |
| 203 | C8 | — | 30 | 2 | no | — |
| 204 | C8 | #6665 | 1 | 1 | yes | characteristic |
| 205 | C8 | (C2×D3)⋊Dic5 | 1 | 1 | yes | characteristic |
| 206 | C8 | #6668 | 1 | 1 | yes | characteristic |
| 207 | C8 | Dic3⋊Dic5 | 1 | 1 | yes | characteristic |
| 208 | C2×C3 | — | 2 | 1 | no | — |
| 209 | C2×C3 | — | 2 | 1 | no | — |
| 210 | C2×C3 | — | 2 | 1 | no | — |
| 211 | C2×C3 | #64770 | 1 | 1 | yes | characteristic |
| 212 | C5 | #67813 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 213 | C22 | — | 4 | 1 | no | — |
| 214 | C22 | #90951 | 1 | 1 | yes | characteristic |
| 215 | C22 | #90957 | 1 | 1 | yes | characteristic |
| 216 | C22 | #90983 | 1 | 1 | yes | characteristic |
| 217 | C4 | — | 60 | 2 | no | — |
| 218 | C4 | #90954 | 1 | 1 | yes | characteristic |
| 219 | C4 | #90957 | 1 | 1 | yes | characteristic |
| 220 | C4 | #90983 | 1 | 1 | yes | characteristic |
| 221 | C4 | #90994 | 1 | 1 | yes | characteristic |
| 222 | C3 | #10598305 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 223 | C2 | — | 2 | 1 | no | — |
| 224 | C2 | — | 2 | 1 | no | — |
| 225 | C2 | — | 2 | 1 | no | — |
| 226 | C2 | #11745968 | 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