← #14278752
Subgroups — #14278752
Order 1920.
280 conjugacy classes of subgroups.
The subgroup lattice has 232 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#91227 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#3998 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#90953 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10598701 | — | 3 | 1 | no | maximal, Hall |
| 5 | C22×C3⋊#168 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×C4×C5×Dic3 | C4 | 1 | 1 | yes | characteristic |
| 7 | C22×C3×#168 | C22 | 1 | 1 | yes | characteristic |
| 8 | #91227 | — | 2 | 1 | no | — |
| 9 | #91227 | C4 | 1 | 2 | yes | complemented |
| 10 | C2×C3⋊#674 | — | 2 | 1 | no | — |
| 11 | C3×#3998 | — | 2 | 1 | no | — |
| 12 | C3×#3998 | C4 | 1 | 2 | yes | complemented |
| 13 | C2×C3×#674 | — | 2 | 1 | no | — |
| 14 | #90953 | — | 2 | 1 | no | — |
| 15 | #90953 | — | 2 | 1 | no | — |
| 16 | #67812 | — | 5 | 1 | no | maximal, Hall |
| 17 | C2×#3998 | — | 3 | 1 | no | — |
| 18 | C2×#3998 | D3 | 1 | 1 | yes | characteristic |
| 19 | C2×#3935 | — | 3 | 1 | no | — |
| 20 | C22×C3×C4×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C22×C5×Dic3 | C8 | 1 | 2 | yes | — |
| 22 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 23 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 24 | C2×C3⋊#168 | C8 | 1 | 2 | yes | — |
| 25 | C4×C5×Dic3 | — | 2 | 1 | no | — |
| 26 | C4×C5×Dic3 | — | 2 | 1 | no | — |
| 27 | C2×C3×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×C3×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C3×#168 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×C3×#168 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C3×#168 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×C3×#168 | Q8 | 1 | 1 | yes | characteristic |
| 33 | C3⋊#674 | — | 2 | 2 | no | — |
| 34 | C3×#674 | — | 2 | 2 | no | — |
| 35 | C2×C3×M32 | — | 5 | 1 | no | — |
| 36 | C2×#815 | — | 5 | 1 | no | — |
| 37 | C2×#812 | — | 5 | 1 | no | — |
| 38 | C2×C42×C5 | — | 3 | 1 | no | — |
| 39 | C22×#168 | — | 3 | 1 | no | — |
| 40 | C22×#168 | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | #3998 | — | 2 | 1 | no | — |
| 42 | #3998 | — | 3 | 2 | no | — |
| 43 | #3998 | — | 6 | 1 | no | — |
| 44 | #3998 | Dic3 | 1 | 2 | yes | complemented |
| 45 | C2×#674 | — | 2 | 1 | no | — |
| 46 | C2×#674 | — | 6 | 1 | no | — |
| 47 | #3935 | — | 6 | 1 | no | — |
| 48 | #3935 | — | 6 | 1 | no | — |
| 49 | #1377 | — | 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×C5×Dic3 | — | 2 | 2 | no | — |
| 58 | C2×C5×Dic3 | — | 4 | 2 | no | — |
| 59 | C3⋊#168 | — | 2 | 2 | no | — |
| 60 | C3×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C3×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C3×#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C3×#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C22×C3×C8 | — | 5 | 1 | no | — |
| 65 | C2×C4×Dic3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 66 | C22×C3⋊C8 | — | 5 | 1 | no | — |
| 67 | C3×M32 | — | 5 | 2 | no | — |
| 68 | C3×M32 | — | 10 | 1 | no | — |
| 69 | C2×C3×C16 | — | 10 | 1 | no | — |
| 70 | #815 | — | 10 | 1 | no | — |
| 71 | #815 | — | 10 | 1 | no | — |
| 72 | #812 | — | 5 | 2 | no | — |
| 73 | #812 | — | 10 | 1 | no | — |
| 74 | C2×C3⋊C16 | — | 10 | 1 | no | — |
| 75 | C22×C4×C5 | — | 3 | 2 | no | — |
| 76 | C22×C4×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 77 | C42×C5 | — | 6 | 1 | no | — |
| 78 | C42×C5 | — | 6 | 1 | no | — |
| 79 | C2×#168 | — | 3 | 2 | no | — |
| 80 | C2×#168 | — | 6 | 1 | no | — |
| 81 | C2×#168 | — | 6 | 1 | no | — |
| 82 | C2×#168 | Dic6 | 1 | 1 | yes | characteristic |
| 83 | C2×#168 | C4×D3 | 1 | 1 | yes | characteristic |
| 84 | C2×#168 | D12 | 1 | 1 | yes | characteristic |
| 85 | C2×#168 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 86 | C2×#168 | #67 | 1 | 1 | yes | characteristic |
| 87 | C2×#168 | #67 | 1 | 1 | yes | characteristic |
| 88 | #674 | — | 2 | 2 | no | — |
| 89 | #674 | — | 6 | 2 | no | — |
| 90 | C2×M32 | — | 5 | 1 | no | — |
| 91 | C2×M32 | — | 15 | 1 | no | — |
| 92 | C2×#97 | — | 15 | 1 | no | — |
| 93 | C22×C3×C5 | — | 2 | 2 | no | — |
| 94 | C22×C3×C5 | C4×C8 | 1 | 1 | yes | characteristic |
| 95 | C22×C3×C5 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 96 | C22×C3×C5 | C4⋊C8 | 1 | 1 | yes | characteristic |
| 97 | C3×C4×C5 | #95 | 1 | 1 | yes | characteristic |
| 98 | C3×C4×C5 | #97 | 1 | 1 | yes | characteristic |
| 99 | C3×C4×C5 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 100 | C3×C4×C5 | C4⋊C8 | 1 | 1 | yes | characteristic |
| 101 | C5×Dic3 | — | 4 | 2 | no | — |
| 102 | C22×C3×C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 103 | C22×Dic3 | #170 | 1 | 2 | yes | — |
| 104 | C2×C3×C8 | — | 5 | 1 | no | — |
| 105 | C2×C3×C8 | — | 5 | 1 | no | — |
| 106 | C2×C3×C8 | — | 5 | 1 | no | — |
| 107 | C2×C3×C8 | — | 5 | 1 | no | — |
| 108 | C2×C3×C8 | — | 5 | 1 | no | — |
| 109 | C2×C3×C8 | — | 5 | 1 | no | — |
| 110 | C4×Dic3 | — | 2 | 1 | no | — |
| 111 | C4×Dic3 | — | 2 | 1 | no | — |
| 112 | C2×C3⋊C8 | — | 5 | 2 | no | — |
| 113 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 114 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 115 | C3×C16 | — | 10 | 2 | no | — |
| 116 | C3⋊C16 | — | 10 | 2 | no | — |
| 117 | C23×C5 | D3×C8 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C5 | — | 6 | 2 | no | — |
| 119 | C2×C4×C5 | — | 12 | 2 | no | — |
| 120 | C2×C4×C5 | D3×C8 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C5 | ΓL(1,𝔽25) | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C5 | ΓL(1,𝔽25) | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C5 | C4×Dic3 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C5 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 126 | #168 | — | 6 | 2 | no | — |
| 127 | #168 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 128 | #168 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 129 | #168 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 130 | #168 | #217 | 1 | 1 | yes | characteristic |
| 131 | C22×C8 | — | 5 | 1 | no | — |
| 132 | C22×C8 | — | 15 | 1 | no | — |
| 133 | C2×C42 | — | 3 | 1 | no | — |
| 134 | M32 | — | 5 | 2 | no | — |
| 135 | M32 | — | 10 | 1 | no | — |
| 136 | M32 | — | 15 | 2 | no | — |
| 137 | M32 | — | 30 | 1 | no | — |
| 138 | C2×C16 | — | 10 | 1 | no | — |
| 139 | C2×C16 | — | 30 | 1 | no | — |
| 140 | #97 | — | 30 | 1 | no | — |
| 141 | #97 | — | 30 | 1 | no | — |
| 142 | C2×C3×C5 | — | 2 | 1 | no | — |
| 143 | C2×C3×C5 | — | 2 | 1 | no | — |
| 144 | C2×C3×C5 | #336 | 1 | 1 | yes | characteristic |
| 145 | C2×C3×C5 | #349 | 1 | 2 | yes | — |
| 146 | C23×C3 | C2×#170 | 1 | 1 | yes | characteristic |
| 147 | C2×C3×C4 | D5⋊C8 | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C4 | #700 | 1 | 1 | yes | characteristic |
| 149 | C2×C3×C4 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 150 | C2×C3×C4 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 151 | C2×C3×C4 | #704 | 1 | 1 | yes | characteristic |
| 152 | C2×C3×C4 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 153 | C2×Dic3 | — | 2 | 2 | no | — |
| 154 | C2×Dic3 | — | 4 | 2 | no | — |
| 155 | C3×C8 | — | 5 | 1 | no | — |
| 156 | C3×C8 | — | 5 | 1 | no | — |
| 157 | C3×C8 | — | 5 | 1 | no | — |
| 158 | C3×C8 | — | 5 | 1 | no | — |
| 159 | C3⋊C8 | — | 10 | 2 | no | — |
| 160 | C22×C5 | — | 2 | 2 | no | — |
| 161 | C22×C5 | C8×Dic3 | 1 | 1 | yes | characteristic |
| 162 | C22×C5 | Dic3⋊C8 | 1 | 1 | yes | characteristic |
| 163 | C22×C5 | (C2×D3)⋊C8 | 1 | 1 | yes | characteristic |
| 164 | C4×C5 | — | 12 | 2 | no | — |
| 165 | C4×C5 | Dic3⋊C8 | 1 | 1 | yes | characteristic |
| 166 | C4×C5 | #815 | 1 | 1 | yes | characteristic |
| 167 | C4×C5 | (C2×D3)⋊C8 | 1 | 1 | yes | characteristic |
| 168 | C4×C5 | C3⋊#95 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | — | 3 | 2 | no | — |
| 170 | C22×C4 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | — | 5 | 1 | no | — |
| 172 | C2×C8 | — | 5 | 1 | no | — |
| 173 | C2×C8 | — | 5 | 1 | no | — |
| 174 | C2×C8 | — | 5 | 1 | no | — |
| 175 | C2×C8 | — | 5 | 1 | no | — |
| 176 | C2×C8 | — | 5 | 1 | no | — |
| 177 | C2×C8 | — | 15 | 2 | no | — |
| 178 | C2×C8 | — | 30 | 1 | no | — |
| 179 | C2×C8 | — | 30 | 1 | no | — |
| 180 | C42 | — | 6 | 1 | no | — |
| 181 | C42 | — | 6 | 1 | no | — |
| 182 | C16 | — | 10 | 2 | no | — |
| 183 | C16 | — | 30 | 2 | no | — |
| 184 | C3×C5 | #1377 | 1 | 1 | yes | characteristic, complemented, Hall |
| 185 | C22×C3 | — | 2 | 2 | no | — |
| 186 | C22×C3 | C4×#170 | 1 | 1 | yes | characteristic |
| 187 | C22×C3 | C4⋊#170 | 1 | 1 | yes | characteristic |
| 188 | C22×C3 | C22⋊#170 | 1 | 1 | yes | characteristic |
| 189 | C3×C4 | #4002 | 1 | 1 | yes | characteristic |
| 190 | C3×C4 | #4003 | 1 | 1 | yes | characteristic |
| 191 | C3×C4 | #4004 | 1 | 1 | yes | characteristic |
| 192 | C3×C4 | (C2×AGL(1,𝔽5))⋊C4 | 1 | 1 | yes | characteristic |
| 193 | Dic3 | — | 4 | 2 | no | — |
| 194 | C2×C5 | — | 2 | 1 | no | — |
| 195 | C2×C5 | — | 2 | 1 | no | — |
| 196 | C2×C5 | #4514 | 1 | 2 | yes | — |
| 197 | C2×C5 | #4550 | 1 | 1 | yes | characteristic |
| 198 | C23 | D3×#170 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | — | 6 | 2 | no | — |
| 200 | C2×C4 | — | 12 | 2 | no | — |
| 201 | C2×C4 | Dic3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | (C2×D3)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | Dic3⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | #6738 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | #6739 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | #6740 | 1 | 1 | yes | characteristic |
| 207 | C8 | — | 5 | 1 | no | — |
| 208 | C8 | — | 5 | 1 | no | — |
| 209 | C8 | — | 5 | 1 | no | — |
| 210 | C8 | — | 5 | 1 | no | — |
| 211 | C8 | — | 30 | 2 | no | — |
| 212 | C2×C3 | — | 2 | 1 | no | — |
| 213 | C2×C3 | — | 2 | 1 | no | — |
| 214 | C2×C3 | #64912 | 1 | 2 | yes | — |
| 215 | C2×C3 | #64918 | 1 | 1 | yes | characteristic |
| 216 | C5 | #67812 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 217 | C22 | — | 2 | 2 | no | — |
| 218 | C22 | Dic3×#170 | 1 | 1 | yes | characteristic |
| 219 | C22 | (C2×D3)⋊#170 | 1 | 1 | yes | characteristic |
| 220 | C22 | Dic3⋊#170 | 1 | 1 | yes | characteristic |
| 221 | C4 | — | 12 | 2 | no | — |
| 222 | C4 | (C2×AGL(1,𝔽5))⋊Dic3 | 1 | 1 | yes | characteristic |
| 223 | C4 | #91169 | 1 | 1 | yes | characteristic |
| 224 | C4 | #91171 | 1 | 1 | yes | characteristic |
| 225 | C4 | #91177 | 1 | 1 | yes | characteristic |
| 226 | C3 | #10598701 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 227 | C2 | — | 2 | 1 | no | — |
| 228 | C2 | — | 2 | 1 | no | — |
| 229 | C2 | #11746548 | 1 | 2 | yes | — |
| 230 | C2 | #11746570 | 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