← #29495161
Subgroups — #29495161
Order 15104.
2038 conjugacy classes of subgroups.
The subgroup lattice has 260 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #21438807 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #21437198 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #21437197 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | C2×#16837820 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#16837819 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #504×D59 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | C59×#3407 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #16838238 | C4 | 1 | 1 | yes | characteristic |
| 9 | #16838169 | C4 | 1 | 1 | yes | characteristic |
| 10 | #16838168 | C4 | 1 | 6 | yes | — |
| 11 | #16838167 | C4 | 1 | 8 | yes | — |
| 12 | C2×#14273733 | C22 | 1 | 6 | yes | — |
| 13 | #131×D59 | C22 | 1 | 8 | yes | — |
| 14 | #16837820 | C22 | 1 | 12 | yes | complemented |
| 15 | #16837820 | C22 | 1 | 4 | yes | complemented |
| 16 | #16837820 | C22 | 1 | 12 | yes | complemented |
| 17 | #16837820 | C22 | 1 | 12 | yes | complemented |
| 18 | #16837819 | C22 | 1 | 4 | yes | complemented |
| 19 | #16837819 | C22 | 1 | 4 | yes | complemented |
| 20 | #16837818 | C22 | 1 | 24 | yes | complemented |
| 21 | C2×#14273667 | C22 | 1 | 3 | yes | complemented |
| 22 | C2×#14273666 | C22 | 1 | 12 | yes | complemented |
| 23 | M32×D59 | C22 | 1 | 24 | yes | complemented |
| 24 | C22×C59⋊C16 | C22 | 1 | 3 | yes | complemented |
| 25 | C2×C16×D59 | C22 | 1 | 12 | yes | complemented |
| 26 | C2×#131×C59 | C22 | 1 | 1 | yes | characteristic |
| 27 | C59×#504 | C22 | 1 | 8 | yes | complemented |
| 28 | C2×M32×C59 | C22 | 1 | 6 | yes | complemented |
| 29 | #14273775 | C8 | 1 | 2 | yes | — |
| 30 | #14273774 | C8 | 1 | 4 | yes | — |
| 31 | #14273773 | C8 | 1 | 4 | yes | — |
| 32 | #14273772 | C8 | 1 | 6 | yes | — |
| 33 | C2×#11745685 | C2×C4 | 1 | 6 | yes | — |
| 34 | #41×D59 | C2×C4 | 1 | 8 | yes | — |
| 35 | #14273735 | C2×C4 | 1 | 24 | yes | — |
| 36 | #14273734 | C2×C4 | 1 | 8 | yes | — |
| 37 | #14273733 | C2×C4 | 1 | 12 | yes | — |
| 38 | #14273733 | C2×C4 | 1 | 12 | yes | — |
| 39 | #14273733 | C23 | 1 | 12 | yes | — |
| 40 | #14273733 | C23 | 1 | 4 | yes | — |
| 41 | C2×#11745664 | C2×C4 | 1 | 6 | yes | — |
| 42 | C2×#11745659 | C2×C4 | 1 | 12 | yes | — |
| 43 | M16×D59 | C2×C4 | 1 | 24 | yes | — |
| 44 | C2×C8×D59 | C23 | 1 | 12 | yes | — |
| 45 | #14273667 | C23 | 1 | 12 | yes | — |
| 46 | #14273666 | C23 | 1 | 24 | yes | — |
| 47 | #14273666 | C23 | 1 | 8 | yes | — |
| 48 | C2×C59⋊C16 | C23 | 1 | 6 | yes | — |
| 49 | C2×C59⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×C59⋊C16 | C23 | 1 | 6 | yes | — |
| 51 | C2×C59⋊C16 | C23 | 1 | 3 | yes | — |
| 52 | C16×D59 | C23 | 1 | 8 | yes | — |
| 53 | C16×D59 | C23 | 1 | 24 | yes | — |
| 54 | C2×#41×C59 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #131×C59 | C2×C4 | 1 | 4 | yes | — |
| 56 | #131×C59 | C23 | 1 | 4 | yes | — |
| 57 | C2×M16×C59 | C2×C4 | 1 | 3 | yes | — |
| 58 | C22×C8×C59 | C23 | 1 | 3 | yes | — |
| 59 | M32×C59 | C23 | 1 | 4 | yes | — |
| 60 | M32×C59 | C23 | 1 | 12 | yes | — |
| 61 | C2×C16×C59 | C23 | 1 | 12 | yes | — |
| 62 | #41×C59 | C2×C8 | 1 | 4 | yes | — |
| 63 | #41×C59 | C22×C4 | 1 | 4 | yes | — |
| 64 | C2×Q8×C59 | C2×C8 | 1 | 1 | yes | characteristic |
| 65 | C2×D4×C59 | C2×C8 | 1 | 3 | yes | — |
| 66 | C22×C4×C59 | C22×C4 | 1 | 3 | yes | — |
| 67 | C2×C22⋊D59 | C2×C8 | 1 | 12 | yes | — |
| 68 | #11745689 | C2×C8 | 1 | 8 | yes | — |
| 69 | Q8×D59 | C2×C8 | 1 | 8 | yes | — |
| 70 | #11745687 | C2×C8 | 1 | 24 | yes | — |
| 71 | D4×D59 | C2×C8 | 1 | 24 | yes | — |
| 72 | #11745685 | C2×C8 | 1 | 12 | yes | — |
| 73 | #11745685 | C2×C8 | 1 | 12 | yes | — |
| 74 | #11745685 | C22×C4 | 1 | 4 | yes | — |
| 75 | #11745685 | C22×C4 | 1 | 12 | yes | — |
| 76 | C2×D236 | C2×C8 | 1 | 6 | yes | — |
| 77 | C2×C4×D59 | C22×C4 | 1 | 12 | yes | — |
| 78 | C2×Dic118 | C2×C8 | 1 | 6 | yes | — |
| 79 | M16×C59 | C22×C4 | 1 | 12 | yes | — |
| 80 | C2×C8×C59 | C22×C4 | 1 | 3 | yes | — |
| 81 | C2×C8×C59 | C22×C4 | 1 | 6 | yes | — |
| 82 | C2×C8×C59 | C24 | 1 | 6 | yes | — |
| 83 | C2×C8×C59 | C24 | 1 | 1 | yes | characteristic |
| 84 | #11745664 | C22×C4 | 1 | 12 | yes | — |
| 85 | #11745664 | C22×C4 | 1 | 4 | yes | — |
| 86 | C2×C59⋊C8 | C22×C4 | 1 | 12 | yes | — |
| 87 | #11745659 | C22×C4 | 1 | 24 | yes | — |
| 88 | #11745659 | C22×C4 | 1 | 8 | yes | — |
| 89 | C8×D59 | C22×C4 | 1 | 24 | yes | — |
| 90 | C8×D59 | C24 | 1 | 8 | yes | — |
| 91 | C16×C59 | C24 | 1 | 8 | yes | — |
| 92 | C59⋊C16 | C24 | 1 | 1 | yes | characteristic |
| 93 | C59⋊C16 | C24 | 1 | 6 | yes | — |
| 94 | C59⋊C16 | C24 | 1 | 1 | yes | characteristic |
| 95 | C23×C59 | C22×C8 | 1 | 3 | yes | — |
| 96 | C22×D59 | C22×C8 | 1 | 12 | yes | — |
| 97 | Q8×C59 | C22×C8 | 1 | 4 | yes | — |
| 98 | D4×C59 | C22×C8 | 1 | 12 | yes | — |
| 99 | C2×C4×C59 | C22×C8 | 1 | 6 | yes | — |
| 100 | C2×C4×C59 | C22×C8 | 1 | 3 | yes | — |
| 101 | C2×C4×C59 | C23×C4 | 1 | 6 | yes | — |
| 102 | C2×C4×C59 | C23×C4 | 1 | 1 | yes | characteristic |
| 103 | C22⋊D59 | C22×C8 | 1 | 8 | yes | — |
| 104 | C22⋊D59 | C22×C8 | 1 | 24 | yes | — |
| 105 | C2×Dic59 | C22×C8 | 1 | 12 | yes | — |
| 106 | D236 | C22×C8 | 1 | 4 | yes | — |
| 107 | D236 | C22×C8 | 1 | 12 | yes | — |
| 108 | C4×D59 | C22×C8 | 1 | 24 | yes | — |
| 109 | C4×D59 | C23×C4 | 1 | 8 | yes | — |
| 110 | Dic118 | C22×C8 | 1 | 4 | yes | — |
| 111 | Dic118 | C22×C8 | 1 | 12 | yes | — |
| 112 | C8×C59 | C23×C4 | 1 | 6 | yes | — |
| 113 | C8×C59 | C23×C4 | 1 | 1 | yes | characteristic |
| 114 | C8×C59 | C25 | 1 | 1 | yes | characteristic |
| 115 | C59⋊C8 | C23×C4 | 1 | 8 | yes | — |
| 116 | #62624 | — | 59 | 1 | no | maximal, Sylow |
| 117 | C22×C59 | — | 4 | 3 | no | — |
| 118 | C22×C59 | C23×C8 | 1 | 6 | yes | — |
| 119 | C22×C59 | C23×C8 | 1 | 1 | yes | characteristic |
| 120 | C2×D59 | — | 4 | 12 | no | — |
| 121 | C2×D59 | C23×C8 | 1 | 8 | yes | — |
| 122 | C4×C59 | C23×C8 | 1 | 1 | yes | characteristic |
| 123 | C4×C59 | C23×C8 | 1 | 6 | yes | — |
| 124 | C4×C59 | C24×C4 | 1 | 1 | yes | characteristic |
| 125 | Dic59 | C23×C8 | 1 | 8 | yes | — |
| 126 | #3573 | — | 59 | 1 | no | — |
| 127 | #3407 | — | 59 | 8 | no | — |
| 128 | #3407 | — | 59 | 6 | no | — |
| 129 | #3407 | D59 | 1 | 1 | yes | characteristic, complemented |
| 130 | C2×#504 | — | 59 | 8 | no | — |
| 131 | C2×#504 | — | 59 | 6 | no | — |
| 132 | C2×#504 | — | 59 | 1 | no | — |
| 133 | C2×C59 | — | 2 | 6 | no | — |
| 134 | C2×C59 | — | 2 | 1 | no | — |
| 135 | C2×C59 | C24×C8 | 1 | 1 | yes | characteristic |
| 136 | D59 | — | 2 | 8 | no | — |
| 137 | #585 | — | 59 | 1 | no | — |
| 138 | #568 | — | 59 | 1 | no | — |
| 139 | #568 | — | 59 | 8 | no | — |
| 140 | #568 | — | 59 | 6 | no | — |
| 141 | C2×#131 | — | 59 | 8 | no | — |
| 142 | C2×#131 | — | 59 | 6 | no | — |
| 143 | C2×#131 | C2×D59 | 1 | 1 | yes | characteristic |
| 144 | #504 | — | 59 | 4 | no | — |
| 145 | #504 | — | 59 | 4 | no | — |
| 146 | #504 | — | 59 | 12 | no | — |
| 147 | #504 | — | 59 | 4 | no | — |
| 148 | #504 | — | 59 | 12 | no | — |
| 149 | #504 | — | 59 | 12 | no | — |
| 150 | #504 | — | 59 | 24 | no | — |
| 151 | #504 | C2×D59 | 1 | 8 | yes | complemented |
| 152 | C2×M32 | — | 59 | 3 | no | — |
| 153 | C2×M32 | — | 59 | 24 | no | — |
| 154 | C2×M32 | — | 59 | 12 | no | — |
| 155 | C2×M32 | C2×D59 | 1 | 6 | yes | complemented |
| 156 | C22×C16 | — | 59 | 3 | no | — |
| 157 | C22×C16 | — | 59 | 12 | no | — |
| 158 | C59 | #62624 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 159 | 2−1+4 | — | 59 | 4 | no | — |
| 160 | 2−1+4 | — | 59 | 2 | no | — |
| 161 | H(5,𝔽2) | — | 59 | 4 | no | — |
| 162 | H(5,𝔽2) | — | 59 | 6 | no | — |
| 163 | C2×#41 | — | 59 | 8 | no | — |
| 164 | C2×#41 | — | 59 | 6 | no | — |
| 165 | C2×#41 | C4×D59 | 1 | 1 | yes | characteristic |
| 166 | #131 | — | 59 | 8 | no | — |
| 167 | #131 | — | 59 | 24 | no | — |
| 168 | #131 | — | 59 | 12 | no | — |
| 169 | #131 | — | 59 | 12 | no | — |
| 170 | #131 | — | 59 | 4 | no | — |
| 171 | #131 | — | 59 | 12 | no | — |
| 172 | #131 | C4×D59 | 1 | 4 | yes | — |
| 173 | #131 | C22×D59 | 1 | 4 | yes | — |
| 174 | C2×M16 | — | 59 | 6 | no | — |
| 175 | C2×M16 | — | 59 | 24 | no | — |
| 176 | C2×M16 | — | 59 | 12 | no | — |
| 177 | C2×M16 | C4×D59 | 1 | 3 | yes | — |
| 178 | C22×C8 | — | 59 | 12 | no | — |
| 179 | C22×C8 | C22×D59 | 1 | 3 | yes | — |
| 180 | M32 | — | 59 | 12 | no | — |
| 181 | M32 | — | 59 | 24 | no | — |
| 182 | M32 | — | 59 | 8 | no | — |
| 183 | M32 | C22×D59 | 1 | 12 | yes | — |
| 184 | M32 | C22×D59 | 1 | 4 | yes | — |
| 185 | C2×C16 | — | 59 | 1 | no | — |
| 186 | C2×C16 | — | 59 | 6 | no | — |
| 187 | C2×C16 | — | 59 | 3 | no | — |
| 188 | C2×C16 | — | 59 | 6 | no | — |
| 189 | C2×C16 | — | 59 | 8 | no | — |
| 190 | C2×C16 | — | 59 | 24 | no | — |
| 191 | C2×C16 | C22×D59 | 1 | 12 | yes | — |
| 192 | #41 | — | 59 | 8 | no | — |
| 193 | #41 | — | 59 | 24 | no | — |
| 194 | #41 | — | 59 | 12 | no | — |
| 195 | #41 | — | 59 | 12 | no | — |
| 196 | #41 | — | 59 | 12 | no | — |
| 197 | #41 | — | 59 | 4 | no | — |
| 198 | #41 | C8×D59 | 1 | 4 | yes | — |
| 199 | #41 | C2×C4×D59 | 1 | 4 | yes | — |
| 200 | C2×Q8 | — | 59 | 8 | no | — |
| 201 | C2×Q8 | — | 59 | 6 | no | — |
| 202 | C2×Q8 | C8×D59 | 1 | 1 | yes | characteristic |
| 203 | C2×D4 | — | 59 | 12 | no | — |
| 204 | C2×D4 | — | 59 | 24 | no | — |
| 205 | C2×D4 | — | 59 | 6 | no | — |
| 206 | C2×D4 | C8×D59 | 1 | 3 | yes | — |
| 207 | C22×C4 | — | 59 | 12 | no | — |
| 208 | C22×C4 | C2×C4×D59 | 1 | 3 | yes | — |
| 209 | M16 | — | 59 | 12 | no | — |
| 210 | M16 | — | 59 | 4 | no | — |
| 211 | M16 | — | 59 | 24 | no | — |
| 212 | M16 | — | 59 | 8 | no | — |
| 213 | M16 | C2×C4×D59 | 1 | 12 | yes | — |
| 214 | C2×C8 | — | 59 | 12 | no | — |
| 215 | C2×C8 | — | 59 | 24 | no | — |
| 216 | C2×C8 | — | 59 | 8 | no | — |
| 217 | C2×C8 | C2×C4×D59 | 1 | 6 | yes | — |
| 218 | C2×C8 | C2×C4×D59 | 1 | 3 | yes | — |
| 219 | C2×C8 | C23×D59 | 1 | 1 | yes | characteristic |
| 220 | C2×C8 | C23×D59 | 1 | 6 | yes | — |
| 221 | C16 | — | 59 | 1 | no | — |
| 222 | C16 | — | 59 | 1 | no | — |
| 223 | C16 | — | 59 | 6 | no | — |
| 224 | C16 | C23×D59 | 1 | 8 | yes | — |
| 225 | C23 | — | 59 | 12 | no | — |
| 226 | C23 | C2×C8×D59 | 1 | 3 | yes | — |
| 227 | Q8 | — | 59 | 12 | no | — |
| 228 | Q8 | — | 59 | 4 | no | — |
| 229 | Q8 | C2×C8×D59 | 1 | 4 | yes | — |
| 230 | D4 | — | 59 | 24 | no | — |
| 231 | D4 | — | 59 | 8 | no | — |
| 232 | D4 | — | 59 | 12 | no | — |
| 233 | D4 | — | 59 | 4 | no | — |
| 234 | D4 | C2×C8×D59 | 1 | 12 | yes | — |
| 235 | C2×C4 | — | 59 | 12 | no | — |
| 236 | C2×C4 | — | 59 | 24 | no | — |
| 237 | C2×C4 | — | 59 | 8 | no | — |
| 238 | C2×C4 | C2×C8×D59 | 1 | 6 | yes | — |
| 239 | C2×C4 | C2×C8×D59 | 1 | 3 | yes | — |
| 240 | C2×C4 | C22×C4×D59 | 1 | 6 | yes | — |
| 241 | C2×C4 | C22×C4×D59 | 1 | 1 | yes | characteristic |
| 242 | C8 | — | 59 | 8 | no | — |
| 243 | C8 | C22×C4×D59 | 1 | 6 | yes | — |
| 244 | C8 | C22×C4×D59 | 1 | 1 | yes | characteristic |
| 245 | C8 | C24×D59 | 1 | 1 | yes | characteristic |
| 246 | C22 | — | 4 | 3 | no | — |
| 247 | C22 | — | 59 | 8 | no | — |
| 248 | C22 | — | 236 | 12 | no | — |
| 249 | C22 | C22×C8×D59 | 1 | 6 | yes | — |
| 250 | C22 | C22×C8×D59 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 59 | 8 | no | — |
| 252 | C4 | C22×C8×D59 | 1 | 6 | yes | — |
| 253 | C4 | C22×C8×D59 | 1 | 1 | yes | characteristic |
| 254 | C4 | C23×C4×D59 | 1 | 1 | yes | characteristic |
| 255 | C2 | — | 2 | 1 | no | — |
| 256 | C2 | — | 2 | 6 | no | — |
| 257 | C2 | — | 118 | 8 | no | — |
| 258 | C2 | C23×C8×D59 | 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