← D5×#2276 #12257521
Subgroups — D5×#2276 #12257521
Order 1280.
3654 conjugacy classes of subgroups.
The subgroup lattice has 236 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4×Q8×D5 | C2 | 1 | 3 | yes | maximal |
| 2 | C5×#2276 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10614181 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C4×D5×C4⋊C4 | C2 | 1 | 9 | yes | maximal |
| 5 | #10605347 | C2 | 1 | 9 | yes | maximal, complemented |
| 6 | D5×#376 | C2 | 1 | 3 | yes | maximal |
| 7 | #10605114 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C22×Q8×D5 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×C5×Q8 | C22 | 1 | 3 | yes | — |
| 10 | C2×Q8×Dic5 | C22 | 1 | 3 | yes | — |
| 11 | C4×Q8×D5 | C4 | 1 | 48 | yes | complemented |
| 12 | C2×D5×C4⋊C4 | C22 | 1 | 9 | yes | — |
| 13 | C2×#4033 | C22 | 1 | 9 | yes | — |
| 14 | C2×C42×D5 | C22 | 1 | 6 | yes | — |
| 15 | C2×C42×D5 | C22 | 1 | 9 | yes | — |
| 16 | C2×C4×Dic10 | C22 | 1 | 9 | yes | — |
| 17 | C4×C5×C4⋊C4 | C22 | 1 | 9 | yes | — |
| 18 | C5×#376 | C22 | 1 | 3 | yes | — |
| 19 | C4⋊C4×Dic5 | C22 | 1 | 18 | yes | — |
| 20 | C4×C4⋊Dic5 | C22 | 1 | 9 | yes | — |
| 21 | #65221 | C22 | 1 | 9 | yes | — |
| 22 | C4×#683 | C22 | 1 | 18 | yes | — |
| 23 | D5×#95 | C22 | 1 | 1 | yes | characteristic |
| 24 | D5×#95 | C22 | 1 | 9 | yes | — |
| 25 | #64938 | C22 | 1 | 18 | yes | — |
| 26 | #64937 | C22 | 1 | 9 | yes | — |
| 27 | #64937 | C22 | 1 | 3 | yes | — |
| 28 | C2×#2276 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C22×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×Q8×D5 | C2×C4 | 1 | 24 | yes | — |
| 31 | C22×C4×D5 | C23 | 1 | 9 | yes | — |
| 32 | C22×C4×D5 | C23 | 1 | 3 | yes | — |
| 33 | C22×C4×D5 | C23 | 1 | 3 | yes | — |
| 34 | C22×Dic10 | C23 | 1 | 3 | yes | — |
| 35 | C4×C5×Q8 | C2×C4 | 1 | 24 | yes | complemented |
| 36 | C2×C5×C4⋊C4 | C23 | 1 | 9 | yes | — |
| 37 | C2×C42×C5 | C23 | 1 | 6 | yes | — |
| 38 | C2×C42×C5 | C23 | 1 | 9 | yes | — |
| 39 | Q8×Dic5 | C2×C4 | 1 | 24 | yes | complemented |
| 40 | C2×C4⋊Dic5 | C23 | 1 | 9 | yes | — |
| 41 | C2×#683 | C23 | 1 | 18 | yes | — |
| 42 | C2×C4×Dic5 | C23 | 1 | 18 | yes | — |
| 43 | C2×C4×Dic5 | C23 | 1 | 9 | yes | — |
| 44 | C2×C4×Dic5 | C23 | 1 | 9 | yes | — |
| 45 | C2×C4×Dic5 | C23 | 1 | 9 | yes | — |
| 46 | D5×C4⋊C4 | C2×C4 | 1 | 72 | yes | — |
| 47 | #4033 | C2×C4 | 1 | 72 | yes | complemented |
| 48 | C42×D5 | — | 4 | 12 | no | — |
| 49 | C42×D5 | C2×C4 | 1 | 72 | yes | — |
| 50 | C4×Dic10 | C2×C4 | 1 | 72 | yes | complemented |
| 51 | C5×#95 | C23 | 1 | 1 | yes | characteristic |
| 52 | C5×#95 | C23 | 1 | 9 | yes | — |
| 53 | #3963 | C23 | 1 | 3 | yes | — |
| 54 | #3963 | C23 | 1 | 9 | yes | — |
| 55 | #3963 | C23 | 1 | 18 | yes | — |
| 56 | C22×C4×Q8 | — | 5 | 3 | no | — |
| 57 | #2276 | — | 5 | 9 | no | — |
| 58 | #2276 | — | 5 | 3 | no | — |
| 59 | #2276 | — | 5 | 3 | no | — |
| 60 | #2276 | D5 | 1 | 1 | yes | characteristic, direct product factor |
| 61 | C2×C4×C4⋊C4 | — | 5 | 9 | no | — |
| 62 | C2×#376 | — | 5 | 3 | no | — |
| 63 | C23×D5 | C24 | 1 | 1 | yes | characteristic |
| 64 | C2×C5×Q8 | C22×C4 | 1 | 12 | yes | — |
| 65 | C22×C4×C5 | C24 | 1 | 9 | yes | — |
| 66 | C22×C4×C5 | C24 | 1 | 3 | yes | — |
| 67 | C22×C4×C5 | C24 | 1 | 3 | yes | — |
| 68 | C22×Dic5 | C24 | 1 | 3 | yes | — |
| 69 | C22×Dic5 | C24 | 1 | 9 | yes | — |
| 70 | C22×Dic5 | C24 | 1 | 3 | yes | — |
| 71 | Q8×D5 | C42 | 1 | 64 | yes | complemented |
| 72 | C2×C4×D5 | — | 2 | 18 | no | — |
| 73 | C2×C4×D5 | — | 2 | 12 | no | — |
| 74 | C2×C4×D5 | — | 2 | 18 | no | — |
| 75 | C2×C4×D5 | C22×C4 | 1 | 12 | yes | — |
| 76 | C2×C4×D5 | C22×C4 | 1 | 36 | yes | — |
| 77 | C2×C4×D5 | C22×C4 | 1 | 36 | yes | — |
| 78 | C2×Dic10 | C22×C4 | 1 | 36 | yes | — |
| 79 | C5×C4⋊C4 | C22×C4 | 1 | 36 | yes | — |
| 80 | C42×C5 | — | 4 | 6 | no | — |
| 81 | C42×C5 | C22×C4 | 1 | 36 | yes | — |
| 82 | C4⋊Dic5 | C22×C4 | 1 | 36 | yes | — |
| 83 | #683 | C22×C4 | 1 | 72 | yes | — |
| 84 | C4×Dic5 | — | 4 | 18 | no | — |
| 85 | C4×Dic5 | C22×C4 | 1 | 36 | yes | — |
| 86 | C4×Dic5 | C22×C4 | 1 | 36 | yes | — |
| 87 | C4×Dic5 | C22×C4 | 1 | 36 | yes | — |
| 88 | C23×Q8 | — | 5 | 1 | no | — |
| 89 | C2×C4×Q8 | — | 5 | 3 | no | — |
| 90 | C2×C4×Q8 | — | 5 | 48 | no | — |
| 91 | C2×C4×Q8 | — | 5 | 9 | no | — |
| 92 | C2×C4×Q8 | — | 5 | 9 | no | — |
| 93 | C2×C4×Q8 | C2×D5 | 1 | 3 | yes | — |
| 94 | C22×C4⋊C4 | — | 5 | 9 | no | — |
| 95 | C22×C42 | — | 5 | 9 | no | — |
| 96 | C22×C42 | — | 5 | 6 | no | — |
| 97 | C4×C4⋊C4 | — | 5 | 18 | no | — |
| 98 | C4×C4⋊C4 | — | 5 | 18 | no | — |
| 99 | C4×C4⋊C4 | — | 5 | 9 | no | — |
| 100 | C4×C4⋊C4 | — | 5 | 18 | no | — |
| 101 | C4×C4⋊C4 | C2×D5 | 1 | 9 | yes | — |
| 102 | #376 | — | 5 | 9 | no | — |
| 103 | #376 | — | 5 | 9 | no | — |
| 104 | #376 | — | 5 | 3 | no | — |
| 105 | #376 | C2×D5 | 1 | 3 | yes | — |
| 106 | C2×#95 | — | 5 | 1 | no | — |
| 107 | C2×#95 | — | 5 | 9 | no | — |
| 108 | C23×C5 | C25 | 1 | 1 | yes | characteristic |
| 109 | C22×D5 | C23×C4 | 1 | 6 | yes | — |
| 110 | C22×D5 | H(5,𝔽2) | 1 | 2 | yes | — |
| 111 | C22×D5 | 2−1+4 | 1 | 6 | yes | — |
| 112 | C5×Q8 | C2×C42 | 1 | 16 | yes | complemented |
| 113 | C2×C4×C5 | — | 2 | 9 | no | — |
| 114 | C2×C4×C5 | — | 2 | 6 | no | — |
| 115 | C2×C4×C5 | — | 2 | 9 | no | — |
| 116 | C2×C4×C5 | C23×C4 | 1 | 6 | yes | — |
| 117 | C2×C4×C5 | C23×C4 | 1 | 18 | yes | — |
| 118 | C2×C4×C5 | C23×C4 | 1 | 18 | yes | — |
| 119 | C2×Dic5 | — | 2 | 9 | no | — |
| 120 | C2×Dic5 | — | 2 | 6 | no | — |
| 121 | C2×Dic5 | — | 2 | 9 | no | — |
| 122 | C2×Dic5 | C23×C4 | 1 | 6 | yes | — |
| 123 | C2×Dic5 | C23×C4 | 1 | 18 | yes | — |
| 124 | C2×Dic5 | C23×C4 | 1 | 18 | yes | — |
| 125 | C4×D5 | — | 2 | 24 | no | — |
| 126 | C4×D5 | — | 2 | 72 | no | — |
| 127 | C4×D5 | C2×C42 | 1 | 48 | yes | — |
| 128 | Dic10 | C2×C42 | 1 | 48 | yes | complemented |
| 129 | C22×Q8 | — | 5 | 24 | no | — |
| 130 | C22×Q8 | — | 5 | 3 | no | — |
| 131 | C22×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 132 | C23×C4 | — | 5 | 3 | no | — |
| 133 | C23×C4 | — | 5 | 9 | no | — |
| 134 | C23×C4 | — | 5 | 3 | no | — |
| 135 | C4×Q8 | — | 5 | 24 | no | — |
| 136 | C4×Q8 | — | 5 | 72 | no | — |
| 137 | C4×Q8 | — | 5 | 72 | no | — |
| 138 | C4×Q8 | C4×D5 | 1 | 24 | yes | complemented |
| 139 | C2×C4⋊C4 | — | 5 | 18 | no | — |
| 140 | C2×C4⋊C4 | — | 5 | 9 | no | — |
| 141 | C2×C4⋊C4 | — | 5 | 72 | no | — |
| 142 | C2×C4⋊C4 | C22×D5 | 1 | 9 | yes | — |
| 143 | C2×C42 | — | 5 | 9 | no | — |
| 144 | C2×C42 | — | 5 | 9 | no | — |
| 145 | C2×C42 | — | 5 | 9 | no | — |
| 146 | C2×C42 | — | 5 | 72 | no | — |
| 147 | C2×C42 | — | 5 | 18 | no | — |
| 148 | C2×C42 | — | 20 | 12 | no | — |
| 149 | C2×C42 | C22×D5 | 1 | 6 | yes | — |
| 150 | C2×C42 | C22×D5 | 1 | 9 | yes | — |
| 151 | #95 | — | 5 | 3 | no | — |
| 152 | #95 | — | 5 | 18 | no | — |
| 153 | #95 | — | 5 | 9 | no | — |
| 154 | #95 | C22×D5 | 1 | 1 | yes | characteristic |
| 155 | #95 | C22×D5 | 1 | 9 | yes | — |
| 156 | C22×C5 | C24×C4 | 1 | 3 | yes | — |
| 157 | C22×C5 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 158 | C22×C5 | C2×2−1+4 | 1 | 3 | yes | — |
| 159 | C2×D5 | C22×C42 | 1 | 4 | yes | — |
| 160 | C2×D5 | #519 | 1 | 12 | yes | — |
| 161 | C2×D5 | #520 | 1 | 12 | yes | — |
| 162 | C4×C5 | — | 2 | 6 | no | — |
| 163 | C4×C5 | — | 2 | 18 | no | — |
| 164 | C4×C5 | C22×C42 | 1 | 12 | yes | — |
| 165 | Dic5 | — | 2 | 6 | no | — |
| 166 | Dic5 | — | 2 | 18 | no | — |
| 167 | Dic5 | C22×C42 | 1 | 12 | yes | — |
| 168 | C24 | — | 5 | 1 | no | — |
| 169 | C2×Q8 | — | 5 | 64 | no | — |
| 170 | C2×Q8 | — | 5 | 36 | no | — |
| 171 | C2×Q8 | C2×C4×D5 | 1 | 12 | yes | — |
| 172 | C22×C4 | — | 5 | 3 | no | — |
| 173 | C22×C4 | — | 5 | 36 | no | — |
| 174 | C22×C4 | — | 5 | 3 | no | — |
| 175 | C22×C4 | — | 5 | 9 | no | — |
| 176 | C22×C4 | — | 5 | 12 | no | — |
| 177 | C22×C4 | — | 5 | 36 | no | — |
| 178 | C22×C4 | — | 10 | 18 | no | — |
| 179 | C22×C4 | — | 10 | 18 | no | — |
| 180 | C22×C4 | — | 10 | 12 | no | — |
| 181 | C22×C4 | C23×D5 | 1 | 3 | yes | — |
| 182 | C22×C4 | C23×D5 | 1 | 9 | yes | — |
| 183 | C22×C4 | C23×D5 | 1 | 3 | yes | — |
| 184 | C4⋊C4 | — | 5 | 72 | no | — |
| 185 | C4⋊C4 | — | 5 | 36 | no | — |
| 186 | C4⋊C4 | C2×C4×D5 | 1 | 36 | yes | — |
| 187 | C42 | — | 4 | 6 | no | — |
| 188 | C42 | — | 5 | 36 | no | — |
| 189 | C42 | — | 5 | 36 | no | — |
| 190 | C42 | — | 5 | 36 | no | — |
| 191 | C42 | — | 20 | 18 | no | — |
| 192 | C42 | C2×C4×D5 | 1 | 36 | yes | — |
| 193 | C2×C5 | C23×C42 | 1 | 1 | yes | characteristic |
| 194 | C2×C5 | C2×#519 | 1 | 3 | yes | — |
| 195 | C2×C5 | C2×#520 | 1 | 3 | yes | — |
| 196 | D5 | #2276 | 1 | 8 | yes | direct product factor |
| 197 | C23 | — | 5 | 6 | no | — |
| 198 | C23 | — | 5 | 2 | no | — |
| 199 | C23 | — | 5 | 6 | no | — |
| 200 | C23 | C24×D5 | 1 | 1 | yes | characteristic |
| 201 | Q8 | — | 5 | 48 | no | — |
| 202 | Q8 | C42×D5 | 1 | 16 | yes | complemented |
| 203 | C2×C4 | — | 2 | 9 | no | — |
| 204 | C2×C4 | — | 2 | 6 | no | — |
| 205 | C2×C4 | — | 2 | 9 | no | — |
| 206 | C2×C4 | — | 5 | 18 | no | — |
| 207 | C2×C4 | — | 5 | 48 | no | — |
| 208 | C2×C4 | — | 5 | 6 | no | — |
| 209 | C2×C4 | — | 5 | 18 | no | — |
| 210 | C2×C4 | — | 10 | 24 | no | — |
| 211 | C2×C4 | — | 10 | 72 | no | — |
| 212 | C2×C4 | — | 10 | 9 | no | — |
| 213 | C2×C4 | — | 10 | 6 | no | — |
| 214 | C2×C4 | — | 10 | 9 | no | — |
| 215 | C2×C4 | C22×C4×D5 | 1 | 6 | yes | — |
| 216 | C2×C4 | C22×C4×D5 | 1 | 18 | yes | — |
| 217 | C2×C4 | C22×C4×D5 | 1 | 18 | yes | — |
| 218 | C5 | C2×#2276 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 219 | C22 | — | 5 | 12 | no | — |
| 220 | C22 | — | 5 | 12 | no | — |
| 221 | C22 | — | 5 | 4 | no | — |
| 222 | C22 | C23×C4×D5 | 1 | 3 | yes | — |
| 223 | C22 | D5×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 224 | C22 | D5×2−1+4 | 1 | 3 | yes | — |
| 225 | C4 | — | 2 | 6 | no | — |
| 226 | C4 | — | 2 | 18 | no | — |
| 227 | C4 | — | 5 | 12 | no | — |
| 228 | C4 | — | 10 | 6 | no | — |
| 229 | C4 | — | 10 | 18 | no | — |
| 230 | C4 | C2×C42×D5 | 1 | 12 | yes | — |
| 231 | C2 | — | 5 | 8 | no | — |
| 232 | C2 | C22×C42×D5 | 1 | 1 | yes | characteristic |
| 233 | C2 | D5×#519 | 1 | 3 | yes | — |
| 234 | C2 | D5×#520 | 1 | 3 | yes | — |
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