← #10617729
Subgroups — #10617729
Order 640.
1060 conjugacy classes of subgroups.
The subgroup lattice has 198 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×C5×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #66161 | C2 | 1 | 2 | yes | maximal |
| 3 | #65904 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | Q8×Dic10 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #65877 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | D4×Dic10 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #65853 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #65804 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | C2×#4015 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | C2×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×Dic10 | C22 | 1 | 6 | yes | — |
| 12 | C4×C5×Q8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C4×C5×Q8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×C5×D4 | C22 | 1 | 3 | yes | — |
| 15 | C4×C5×D4 | C22 | 1 | 3 | yes | — |
| 16 | C5×#117 | C22 | 1 | 3 | yes | — |
| 17 | C2×C42×C5 | C22 | 1 | 3 | yes | — |
| 18 | Q8×Dic5 | C22 | 1 | 4 | yes | — |
| 19 | D4×Dic5 | C22 | 1 | 12 | yes | — |
| 20 | #4072 | C22 | 1 | 6 | yes | — |
| 21 | C2×C4⋊Dic5 | C22 | 1 | 6 | yes | — |
| 22 | C2×C4⋊Dic5 | C22 | 1 | 6 | yes | — |
| 23 | #4070 | C22 | 1 | 6 | yes | — |
| 24 | #4070 | C22 | 1 | 6 | yes | — |
| 25 | #4036 | C22 | 1 | 12 | yes | — |
| 26 | #4034 | C22 | 1 | 12 | yes | — |
| 27 | #4024 | C22 | 1 | 24 | yes | — |
| 28 | #4016 | C22 | 1 | 6 | yes | — |
| 29 | #4015 | C22 | 1 | 3 | yes | — |
| 30 | #4015 | C22 | 1 | 6 | yes | — |
| 31 | #4015 | C22 | 1 | 6 | yes | — |
| 32 | #4015 | C22 | 1 | 6 | yes | — |
| 33 | #4015 | C22 | 1 | 1 | yes | characteristic |
| 34 | C4×Dic10 | C22 | 1 | 6 | yes | — |
| 35 | C4×Dic10 | C22 | 1 | 6 | yes | — |
| 36 | #3503 | — | 5 | 1 | no | maximal, Sylow |
| 37 | C5×#41 | Q8 | 1 | 8 | yes | complemented |
| 38 | C2×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×C5×D4 | C23 | 1 | 3 | yes | — |
| 40 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 41 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 42 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 43 | C22×Dic5 | C23 | 1 | 12 | yes | — |
| 44 | C2×Dic10 | — | 4 | 12 | no | — |
| 45 | C2×Dic10 | C23 | 1 | 6 | yes | — |
| 46 | C2×Dic10 | C23 | 1 | 2 | yes | — |
| 47 | C2×Dic10 | C23 | 1 | 12 | yes | — |
| 48 | C5×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 49 | C5×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 50 | C5×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 51 | C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 52 | C42×C5 | C23 | 1 | 3 | yes | — |
| 53 | C42×C5 | C23 | 1 | 3 | yes | — |
| 54 | C42×C5 | C23 | 1 | 3 | yes | — |
| 55 | C22⋊Dic5 | C23 | 1 | 12 | yes | — |
| 56 | C4⋊Dic5 | C23 | 1 | 2 | yes | — |
| 57 | C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 58 | C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 59 | C4⋊Dic5 | C23 | 1 | 12 | yes | — |
| 60 | C4⋊Dic5 | C23 | 1 | 12 | yes | — |
| 61 | C4⋊Dic5 | C23 | 1 | 2 | yes | — |
| 62 | #683 | C23 | 1 | 12 | yes | — |
| 63 | #683 | C23 | 1 | 12 | yes | — |
| 64 | C4×Dic5 | C23 | 1 | 12 | yes | — |
| 65 | Q82 | — | 5 | 2 | no | — |
| 66 | #557 | — | 5 | 2 | no | — |
| 67 | #554 | — | 5 | 6 | no | — |
| 68 | D4×Q8 | — | 5 | 6 | no | — |
| 69 | #544 | — | 5 | 6 | no | — |
| 70 | #533 | — | 5 | 3 | no | — |
| 71 | C2×C4⋊Q8 | — | 5 | 3 | no | — |
| 72 | #520 | — | 5 | 2 | no | — |
| 73 | C4×#41 | D5 | 1 | 1 | yes | characteristic |
| 74 | C23×C5 | C24 | 1 | 3 | yes | — |
| 75 | C5×Q8 | C2×Q8 | 1 | 4 | yes | complemented |
| 76 | C5×D4 | C2×Q8 | 1 | 12 | yes | complemented |
| 77 | C2×C4×C5 | — | 2 | 6 | no | — |
| 78 | C2×C4×C5 | — | 2 | 6 | no | — |
| 79 | C2×C4×C5 | C2×Q8 | 1 | 12 | yes | complemented |
| 80 | C2×C4×C5 | C24 | 1 | 3 | yes | — |
| 81 | C2×C4×C5 | C24 | 1 | 3 | yes | — |
| 82 | C2×C4×C5 | C24 | 1 | 3 | yes | — |
| 83 | C2×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 86 | C2×Dic5 | — | 4 | 12 | no | — |
| 87 | C2×Dic5 | C24 | 1 | 4 | yes | — |
| 88 | C2×Dic5 | C24 | 1 | 12 | yes | — |
| 89 | Dic10 | — | 2 | 12 | no | — |
| 90 | Dic10 | — | 2 | 4 | no | — |
| 91 | Dic10 | — | 4 | 12 | no | — |
| 92 | C2×#41 | C2×D5 | 1 | 1 | yes | characteristic |
| 93 | C22×Q8 | — | 5 | 6 | no | — |
| 94 | C4⋊Q8 | — | 5 | 12 | no | — |
| 95 | C4⋊Q8 | — | 5 | 3 | no | — |
| 96 | C4⋊Q8 | — | 5 | 6 | no | — |
| 97 | C4⋊Q8 | — | 5 | 6 | no | — |
| 98 | C4⋊Q8 | — | 5 | 6 | no | — |
| 99 | C4⋊Q8 | — | 5 | 1 | no | — |
| 100 | #125 | — | 5 | 12 | no | — |
| 101 | #125 | — | 5 | 6 | no | — |
| 102 | C22⋊Q8 | — | 5 | 6 | no | — |
| 103 | C22⋊Q8 | — | 5 | 6 | no | — |
| 104 | C22⋊Q8 | — | 5 | 24 | no | — |
| 105 | C4×Q8 | — | 5 | 4 | no | — |
| 106 | C4×Q8 | — | 5 | 6 | no | — |
| 107 | C4×Q8 | — | 5 | 6 | no | — |
| 108 | C4×Q8 | C2×D5 | 1 | 1 | yes | characteristic |
| 109 | C4×Q8 | C2×D5 | 1 | 1 | yes | characteristic |
| 110 | C4×D4 | — | 5 | 12 | no | — |
| 111 | C4×D4 | C2×D5 | 1 | 3 | yes | — |
| 112 | C4×D4 | C2×D5 | 1 | 3 | yes | — |
| 113 | #117 | — | 5 | 6 | no | — |
| 114 | #117 | C2×D5 | 1 | 3 | yes | — |
| 115 | C2×C4⋊C4 | — | 5 | 6 | no | — |
| 116 | C2×C4⋊C4 | — | 5 | 6 | no | — |
| 117 | C2×C42 | C2×D5 | 1 | 3 | yes | — |
| 118 | C22×C5 | — | 2 | 3 | no | — |
| 119 | C22×C5 | — | 2 | 3 | no | — |
| 120 | C22×C5 | C22×Q8 | 1 | 6 | yes | — |
| 121 | C22×C5 | C25 | 1 | 1 | yes | characteristic |
| 122 | C4×C5 | — | 2 | 3 | no | — |
| 123 | C4×C5 | — | 2 | 3 | no | — |
| 124 | C4×C5 | C22×Q8 | 1 | 2 | yes | — |
| 125 | C4×C5 | C22×Q8 | 1 | 6 | yes | — |
| 126 | C4×C5 | H(5,𝔽2) | 1 | 2 | yes | — |
| 127 | C4×C5 | 2−1+4 | 1 | 2 | yes | — |
| 128 | Dic5 | — | 2 | 12 | no | — |
| 129 | Dic5 | — | 2 | 4 | no | — |
| 130 | #41 | Dic10 | 1 | 8 | yes | complemented |
| 131 | C2×Q8 | — | 5 | 2 | no | — |
| 132 | C2×Q8 | — | 5 | 6 | no | — |
| 133 | C2×Q8 | — | 5 | 12 | no | — |
| 134 | C2×Q8 | — | 20 | 12 | no | — |
| 135 | C2×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 136 | C2×D4 | C22×D5 | 1 | 3 | yes | — |
| 137 | C22×C4 | — | 5 | 12 | no | — |
| 138 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 139 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 140 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 141 | C4⋊C4 | — | 5 | 6 | no | — |
| 142 | C4⋊C4 | — | 5 | 6 | no | — |
| 143 | C4⋊C4 | — | 5 | 12 | no | — |
| 144 | C4⋊C4 | — | 5 | 12 | no | — |
| 145 | C4⋊C4 | — | 5 | 2 | no | — |
| 146 | C4⋊C4 | — | 5 | 2 | no | — |
| 147 | C4⋊C4 | — | 5 | 12 | no | — |
| 148 | C4⋊C4 | — | 5 | 12 | no | — |
| 149 | C4⋊C4 | C22×D5 | 1 | 3 | yes | — |
| 150 | C4⋊C4 | C22×D5 | 1 | 3 | yes | — |
| 151 | C22⋊C4 | — | 5 | 12 | no | — |
| 152 | C22⋊C4 | C22×D5 | 1 | 6 | yes | — |
| 153 | C42 | — | 5 | 12 | no | — |
| 154 | C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 155 | C42 | C22×D5 | 1 | 3 | yes | — |
| 156 | C42 | C22×D5 | 1 | 3 | yes | — |
| 157 | C42 | C22×D5 | 1 | 3 | yes | — |
| 158 | C2×C5 | — | 2 | 6 | no | — |
| 159 | C2×C5 | C23×Q8 | 1 | 1 | yes | characteristic |
| 160 | C2×C5 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 161 | C2×C5 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 162 | C23 | C23×D5 | 1 | 3 | yes | — |
| 163 | Q8 | — | 10 | 12 | no | — |
| 164 | Q8 | — | 10 | 4 | no | — |
| 165 | Q8 | — | 20 | 12 | no | — |
| 166 | Q8 | C2×Dic10 | 1 | 4 | yes | complemented |
| 167 | D4 | C2×Dic10 | 1 | 12 | yes | complemented |
| 168 | C2×C4 | — | 2 | 6 | no | — |
| 169 | C2×C4 | — | 2 | 6 | no | — |
| 170 | C2×C4 | — | 5 | 4 | no | — |
| 171 | C2×C4 | — | 5 | 12 | no | — |
| 172 | C2×C4 | — | 20 | 12 | no | — |
| 173 | C2×C4 | C2×Dic10 | 1 | 12 | yes | complemented |
| 174 | C2×C4 | C23×D5 | 1 | 3 | yes | — |
| 175 | C2×C4 | C23×D5 | 1 | 3 | yes | — |
| 176 | C2×C4 | C23×D5 | 1 | 3 | yes | — |
| 177 | C2×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 178 | C2×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 179 | C2×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 180 | C5 | #3503 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 181 | C22 | — | 2 | 3 | no | — |
| 182 | C22 | — | 2 | 3 | no | — |
| 183 | C22 | C22×Dic10 | 1 | 6 | yes | — |
| 184 | C22 | C24×D5 | 1 | 1 | yes | characteristic |
| 185 | C4 | — | 2 | 3 | no | — |
| 186 | C4 | — | 2 | 3 | no | — |
| 187 | C4 | — | 10 | 12 | no | — |
| 188 | C4 | — | 10 | 4 | no | — |
| 189 | C4 | C22×Dic10 | 1 | 2 | yes | — |
| 190 | C4 | C22×Dic10 | 1 | 6 | yes | — |
| 191 | C4 | #4149 | 1 | 2 | yes | — |
| 192 | C4 | #4150 | 1 | 2 | yes | — |
| 193 | C2 | — | 2 | 6 | no | — |
| 194 | C2 | C23×Dic10 | 1 | 1 | yes | characteristic |
| 195 | C2 | C2×#4149 | 1 | 1 | yes | characteristic |
| 196 | C2 | C2×#4150 | 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