← #10617331
Subgroups — #10617331
Order 640.
356 conjugacy classes of subgroups.
The subgroup lattice has 234 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #66022 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #65709 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #65708 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #65708 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #65707 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #65707 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #65703 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #65701 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #65701 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | D5⋊(C4⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C4×#700 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#700 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#700 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×D5⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×D5⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 16 | C5×#125 | C4 | 1 | 1 | yes | characteristic |
| 17 | #4042 | C4 | 1 | 2 | yes | — |
| 18 | #4042 | C4 | 1 | 2 | yes | — |
| 19 | #4038 | C22 | 1 | 1 | yes | characteristic |
| 20 | #4038 | C22 | 1 | 1 | yes | characteristic |
| 21 | #4034 | C4 | 1 | 1 | yes | characteristic |
| 22 | #4034 | C4 | 1 | 1 | yes | characteristic |
| 23 | #4033 | C22 | 1 | 2 | yes | — |
| 24 | #4033 | C22 | 1 | 2 | yes | — |
| 25 | C42×D5 | C22 | 1 | 1 | yes | characteristic |
| 26 | #4016 | C4 | 1 | 1 | yes | characteristic |
| 27 | #4004 | C22 | 1 | 1 | yes | characteristic |
| 28 | #4004 | C22 | 1 | 1 | yes | characteristic |
| 29 | #4004 | C22 | 1 | 2 | yes | — |
| 30 | #4004 | C22 | 1 | 2 | yes | — |
| 31 | #4002 | C22 | 1 | 2 | yes | — |
| 32 | #4002 | C22 | 1 | 2 | yes | — |
| 33 | #4002 | C22 | 1 | 2 | yes | — |
| 34 | C4⋊#170 | C22 | 1 | 1 | yes | characteristic |
| 35 | C4⋊#170 | C22 | 1 | 1 | yes | characteristic |
| 36 | C4⋊#170 | C22 | 1 | 2 | yes | — |
| 37 | C4⋊#170 | C22 | 1 | 2 | yes | — |
| 38 | C4×#170 | C22 | 1 | 2 | yes | — |
| 39 | C4×#170 | C22 | 1 | 2 | yes | — |
| 40 | C4×#170 | C22 | 1 | 2 | yes | — |
| 41 | #2962 | — | 5 | 1 | no | maximal, Sylow |
| 42 | C2×C4×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C4×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C2×C4×D5 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×Dic10 | C2×C4 | 1 | 2 | yes | — |
| 46 | C2×Dic10 | C2×C4 | 1 | 2 | yes | — |
| 47 | C2×#170 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×#170 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×#170 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×#170 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×#170 | C23 | 1 | 2 | yes | — |
| 52 | C2×#170 | C23 | 1 | 2 | yes | — |
| 53 | #700 | — | 2 | 2 | no | — |
| 54 | #700 | Q8 | 1 | 4 | yes | complemented |
| 55 | D5⋊C8 | — | 2 | 2 | no | — |
| 56 | D5⋊C8 | — | 2 | 2 | no | — |
| 57 | C5×C4⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 58 | C5×C4⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 59 | C5×C4⋊C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C5×C4⋊C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | #685 | C2×C4 | 1 | 2 | yes | — |
| 63 | #685 | C2×C4 | 1 | 2 | yes | — |
| 64 | C4⋊Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 65 | C4⋊Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | #683 | C2×C4 | 1 | 2 | yes | — |
| 67 | #683 | C2×C4 | 1 | 2 | yes | — |
| 68 | #683 | C2×C4 | 1 | 2 | yes | — |
| 69 | #683 | C2×C4 | 1 | 2 | yes | — |
| 70 | C4×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 72 | C4×Dic5 | C23 | 1 | 2 | yes | — |
| 73 | C4×Dic5 | C23 | 1 | 2 | yes | — |
| 74 | C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 75 | #533 | — | 5 | 1 | no | — |
| 76 | #446 | — | 5 | 2 | no | — |
| 77 | #446 | — | 5 | 2 | no | — |
| 78 | #446 | — | 5 | 2 | no | — |
| 79 | C8×Q8 | — | 5 | 2 | no | — |
| 80 | #424 | — | 5 | 1 | no | — |
| 81 | #424 | — | 5 | 1 | no | — |
| 82 | C2×C4⋊C8 | — | 5 | 1 | no | — |
| 83 | #405 | — | 5 | 1 | no | — |
| 84 | #405 | — | 5 | 1 | no | — |
| 85 | C4×M16 | — | 5 | 1 | no | — |
| 86 | C22×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C5 | C22×C4 | 1 | 2 | yes | — |
| 90 | C2×C4×C5 | C22×C4 | 1 | 2 | yes | — |
| 91 | C2×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C2×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C2×Dic5 | C22×C4 | 1 | 2 | yes | — |
| 94 | C2×Dic5 | C22×C4 | 1 | 2 | yes | — |
| 95 | C2×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 97 | C4×D5 | — | 2 | 2 | no | — |
| 98 | C4×D5 | — | 4 | 1 | no | — |
| 99 | C4×D5 | C2×Q8 | 1 | 2 | yes | — |
| 100 | C4×D5 | #41 | 1 | 2 | yes | — |
| 101 | Dic10 | — | 2 | 4 | no | — |
| 102 | Dic10 | — | 4 | 2 | no | — |
| 103 | #170 | — | 2 | 1 | no | — |
| 104 | #170 | — | 2 | 1 | no | — |
| 105 | #170 | — | 2 | 1 | no | — |
| 106 | #170 | — | 2 | 1 | no | — |
| 107 | #170 | — | 2 | 2 | no | — |
| 108 | #170 | C2×Q8 | 1 | 4 | yes | — |
| 109 | C2×M16 | — | 5 | 1 | no | — |
| 110 | C2×M16 | — | 5 | 1 | no | — |
| 111 | C22×C8 | — | 5 | 1 | no | — |
| 112 | C22×C8 | — | 5 | 1 | no | — |
| 113 | C4⋊Q8 | — | 5 | 1 | no | — |
| 114 | C4⋊Q8 | — | 5 | 1 | no | — |
| 115 | #125 | — | 5 | 1 | no | — |
| 116 | #125 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 117 | C22⋊Q8 | — | 5 | 2 | no | — |
| 118 | C22⋊Q8 | — | 5 | 2 | no | — |
| 119 | C4×Q8 | — | 5 | 2 | no | — |
| 120 | C4×Q8 | — | 5 | 2 | no | — |
| 121 | #117 | — | 5 | 1 | no | — |
| 122 | #117 | — | 5 | 1 | no | — |
| 123 | C2×C42 | — | 5 | 1 | no | — |
| 124 | C4⋊C8 | — | 5 | 1 | no | — |
| 125 | C4⋊C8 | — | 5 | 1 | no | — |
| 126 | C4⋊C8 | — | 5 | 2 | no | — |
| 127 | C4⋊C8 | — | 5 | 2 | no | — |
| 128 | C4⋊C8 | — | 5 | 1 | no | — |
| 129 | C4⋊C8 | — | 5 | 1 | no | — |
| 130 | C4⋊C8 | — | 5 | 2 | no | — |
| 131 | C4⋊C8 | — | 5 | 2 | no | — |
| 132 | #97 | — | 5 | 2 | no | — |
| 133 | #97 | — | 5 | 2 | no | — |
| 134 | #97 | — | 5 | 2 | no | — |
| 135 | C4×C8 | — | 5 | 2 | no | — |
| 136 | C4×C8 | — | 5 | 2 | no | — |
| 137 | C4×C8 | — | 5 | 2 | no | — |
| 138 | C22×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 139 | C2×D5 | — | 2 | 1 | no | — |
| 140 | C2×D5 | — | 2 | 1 | no | — |
| 141 | C2×D5 | C4×Q8 | 1 | 2 | yes | — |
| 142 | C4×C5 | — | 2 | 2 | no | — |
| 143 | C4×C5 | — | 2 | 1 | no | — |
| 144 | C4×C5 | — | 2 | 2 | no | — |
| 145 | C4×C5 | — | 2 | 1 | no | — |
| 146 | C4×C5 | C4×Q8 | 1 | 2 | yes | — |
| 147 | Dic5 | — | 2 | 2 | no | — |
| 148 | Dic5 | — | 2 | 1 | no | — |
| 149 | Dic5 | — | 2 | 2 | no | — |
| 150 | Dic5 | #131 | 1 | 2 | yes | — |
| 151 | Dic5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 152 | Dic5 | C2×#41 | 1 | 1 | yes | characteristic |
| 153 | C2×Q8 | — | 5 | 2 | no | — |
| 154 | C2×Q8 | — | 5 | 2 | no | — |
| 155 | C22×C4 | — | 5 | 1 | no | — |
| 156 | C22×C4 | — | 5 | 1 | no | — |
| 157 | C22×C4 | — | 5 | 1 | no | — |
| 158 | M16 | — | 5 | 4 | no | — |
| 159 | M16 | — | 10 | 2 | no | — |
| 160 | C2×C8 | — | 5 | 2 | no | — |
| 161 | C2×C8 | — | 5 | 1 | no | — |
| 162 | C2×C8 | — | 5 | 1 | no | — |
| 163 | C2×C8 | — | 5 | 2 | no | — |
| 164 | C2×C8 | — | 5 | 1 | no | — |
| 165 | C2×C8 | — | 5 | 1 | no | — |
| 166 | C2×C8 | — | 10 | 2 | no | — |
| 167 | C2×C8 | — | 10 | 2 | no | — |
| 168 | C4⋊C4 | — | 5 | 1 | no | — |
| 169 | C4⋊C4 | — | 5 | 1 | no | — |
| 170 | C4⋊C4 | — | 5 | 2 | no | — |
| 171 | C4⋊C4 | — | 5 | 2 | no | — |
| 172 | C4⋊C4 | — | 5 | 2 | no | — |
| 173 | C4⋊C4 | — | 5 | 2 | no | — |
| 174 | C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 175 | C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 176 | C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 177 | C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 178 | C22⋊C4 | — | 5 | 2 | no | — |
| 179 | C22⋊C4 | — | 5 | 2 | no | — |
| 180 | C42 | — | 5 | 1 | no | — |
| 181 | C42 | — | 5 | 2 | no | — |
| 182 | C42 | — | 5 | 2 | no | — |
| 183 | C42 | — | 5 | 1 | no | — |
| 184 | C42 | — | 5 | 1 | no | — |
| 185 | C42 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 186 | C2×C5 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 187 | C2×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 188 | C2×C5 | #568 | 1 | 1 | yes | characteristic |
| 189 | D5 | — | 2 | 2 | no | — |
| 190 | C23 | — | 5 | 1 | no | — |
| 191 | Q8 | — | 10 | 4 | no | — |
| 192 | Q8 | — | 20 | 2 | no | — |
| 193 | C2×C4 | — | 5 | 1 | no | — |
| 194 | C2×C4 | — | 5 | 1 | no | — |
| 195 | C2×C4 | — | 5 | 2 | no | — |
| 196 | C2×C4 | — | 5 | 2 | no | — |
| 197 | C2×C4 | — | 5 | 1 | no | — |
| 198 | C2×C4 | — | 5 | 2 | no | — |
| 199 | C2×C4 | — | 5 | 2 | no | — |
| 200 | C2×C4 | — | 10 | 2 | no | — |
| 201 | C2×C4 | — | 20 | 1 | no | — |
| 202 | C2×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 205 | C2×C4 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 206 | C2×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 207 | C8 | — | 5 | 4 | no | — |
| 208 | C8 | — | 10 | 1 | no | — |
| 209 | C8 | — | 10 | 1 | no | — |
| 210 | C8 | — | 10 | 1 | no | — |
| 211 | C8 | — | 10 | 1 | no | — |
| 212 | C8 | — | 10 | 2 | no | — |
| 213 | C5 | #2962 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 214 | C22 | — | 5 | 2 | no | — |
| 215 | C22 | — | 10 | 1 | no | — |
| 216 | C22 | — | 10 | 1 | no | — |
| 217 | C22 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 218 | C4 | — | 2 | 1 | no | — |
| 219 | C4 | — | 2 | 2 | no | — |
| 220 | C4 | — | 2 | 1 | no | — |
| 221 | C4 | — | 2 | 2 | no | — |
| 222 | C4 | — | 5 | 1 | no | — |
| 223 | C4 | — | 5 | 1 | no | — |
| 224 | C4 | — | 5 | 2 | no | — |
| 225 | C4 | — | 10 | 2 | no | — |
| 226 | C4 | — | 10 | 1 | no | — |
| 227 | C4 | — | 10 | 2 | no | — |
| 228 | C4 | Q8×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 229 | C2 | — | 10 | 2 | no | — |
| 230 | C2 | C2×Q8×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 231 | C2 | D5⋊#131 | 1 | 1 | yes | characteristic |
| 232 | C2 | #66264 | 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