← C22×#16864231 #31769723
Subgroups — C22×#16864231 #31769723
Order 17152.
60366 conjugacy classes of subgroups.
The subgroup lattice has 246 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16864231 | C2 | 1 | 96 | yes | maximal, direct product factor |
| 2 | C22×#14524090 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | C22×#14524088 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | C23×#11761627 | C2 | 1 | 9 | yes | maximal, complemented |
| 5 | C22×#41×D67 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C22×H(5,𝔽2)×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #16864231 | C22 | 1 | 1024 | yes | direct product factor |
| 8 | C2×#14524090 | C22 | 1 | 288 | yes | complemented |
| 9 | C2×#14524088 | C22 | 1 | 432 | yes | complemented |
| 10 | C22×#11761629 | C22 | 1 | 2 | yes | complemented |
| 11 | C22×#11761628 | C22 | 1 | 18 | yes | complemented |
| 12 | C22×#11761627 | C22 | 1 | 18 | yes | complemented |
| 13 | C22×#11761627 | C22 | 1 | 36 | yes | complemented |
| 14 | C22×#11761627 | C22 | 1 | 432 | yes | complemented |
| 15 | C2×#41×D67 | C22 | 1 | 288 | yes | complemented |
| 16 | C23×Dic134 | C22 | 1 | 9 | yes | complemented |
| 17 | C22×Q8×D67 | C22 | 1 | 6 | yes | complemented |
| 18 | C23×C22⋊D67 | C22 | 1 | 18 | yes | complemented |
| 19 | C22×D4×D67 | C22 | 1 | 18 | yes | complemented |
| 20 | C24×Dic67 | C22 | 1 | 6 | yes | complemented |
| 21 | C23×C4×D67 | C22 | 1 | 9 | yes | complemented |
| 22 | C2×H(5,𝔽2)×C67 | C22 | 1 | 48 | yes | complemented |
| 23 | C22×#41×C67 | C22 | 1 | 6 | yes | complemented |
| 24 | C23×D4×C67 | C22 | 1 | 9 | yes | complemented |
| 25 | C24×D67 | C23 | 1 | 9 | yes | — |
| 26 | #14524090 | C23 | 1 | 1536 | yes | complemented |
| 27 | #14524088 | C23 | 1 | 2304 | yes | complemented |
| 28 | C2×#11761629 | C23 | 1 | 48 | yes | complemented |
| 29 | C2×#11761628 | C23 | 1 | 432 | yes | complemented |
| 30 | C2×#11761627 | C23 | 1 | 432 | yes | complemented |
| 31 | C2×#11761627 | C23 | 1 | 864 | yes | complemented |
| 32 | C2×#11761627 | C23 | 1 | 2304 | yes | complemented |
| 33 | #41×D67 | C23 | 1 | 1536 | yes | complemented |
| 34 | C22×Dic134 | C23 | 1 | 18 | yes | — |
| 35 | C22×Dic134 | C23 | 1 | 216 | yes | complemented |
| 36 | C2×Q8×D67 | C23 | 1 | 144 | yes | complemented |
| 37 | C22×D268 | C23 | 1 | 6 | yes | — |
| 38 | C22×C22⋊D67 | C23 | 1 | 36 | yes | — |
| 39 | C22×C22⋊D67 | C23 | 1 | 432 | yes | complemented |
| 40 | C2×D4×D67 | C23 | 1 | 432 | yes | complemented |
| 41 | C23×Dic67 | C23 | 1 | 9 | yes | — |
| 42 | C23×Dic67 | C23 | 1 | 18 | yes | — |
| 43 | C23×Dic67 | C23 | 1 | 144 | yes | complemented |
| 44 | C22×C4×D67 | C23 | 1 | 6 | yes | — |
| 45 | C22×C4×D67 | C23 | 1 | 18 | yes | — |
| 46 | C22×C4×D67 | C23 | 1 | 216 | yes | complemented |
| 47 | C25×C67 | C23 | 1 | 6 | yes | — |
| 48 | H(5,𝔽2)×C67 | C23 | 1 | 256 | yes | complemented |
| 49 | C2×#41×C67 | C23 | 1 | 144 | yes | complemented |
| 50 | C22×Q8×C67 | C23 | 1 | 2 | yes | — |
| 51 | C22×D4×C67 | C23 | 1 | 18 | yes | — |
| 52 | C22×D4×C67 | C23 | 1 | 216 | yes | complemented |
| 53 | C23×C4×C67 | C23 | 1 | 9 | yes | — |
| 54 | C23×D67 | — | 4 | 36 | no | — |
| 55 | C23×D67 | C24 | 1 | 6 | yes | — |
| 56 | C23×D67 | C24 | 1 | 108 | yes | — |
| 57 | #11761629 | C24 | 1 | 128 | yes | complemented |
| 58 | #11761628 | C24 | 1 | 1152 | yes | complemented |
| 59 | #11761627 | C24 | 1 | 1152 | yes | complemented |
| 60 | #11761627 | C24 | 1 | 2304 | yes | complemented |
| 61 | C2×Dic134 | C24 | 1 | 216 | yes | — |
| 62 | C2×Dic134 | C24 | 1 | 576 | yes | complemented |
| 63 | Q8×D67 | C24 | 1 | 384 | yes | complemented |
| 64 | C2×D268 | C24 | 1 | 72 | yes | — |
| 65 | C2×C22⋊D67 | C24 | 1 | 432 | yes | — |
| 66 | C2×C22⋊D67 | C24 | 1 | 1152 | yes | complemented |
| 67 | D4×D67 | C24 | 1 | 1152 | yes | complemented |
| 68 | C22×Dic67 | C24 | 1 | 1 | yes | characteristic |
| 69 | C22×Dic67 | C24 | 1 | 9 | yes | — |
| 70 | C22×Dic67 | C24 | 1 | 108 | yes | — |
| 71 | C22×Dic67 | C24 | 1 | 216 | yes | — |
| 72 | C22×Dic67 | C24 | 1 | 384 | yes | complemented |
| 73 | C2×C4×D67 | C24 | 1 | 72 | yes | — |
| 74 | C2×C4×D67 | C24 | 1 | 216 | yes | — |
| 75 | C2×C4×D67 | C24 | 1 | 576 | yes | complemented |
| 76 | C24×C67 | — | 4 | 24 | no | — |
| 77 | C24×C67 | C24 | 1 | 9 | yes | — |
| 78 | C24×C67 | C24 | 1 | 72 | yes | — |
| 79 | #41×C67 | C24 | 1 | 384 | yes | complemented |
| 80 | C2×Q8×C67 | C24 | 1 | 24 | yes | — |
| 81 | C2×D4×C67 | C24 | 1 | 216 | yes | — |
| 82 | C2×D4×C67 | C24 | 1 | 576 | yes | complemented |
| 83 | C22×C4×C67 | C24 | 1 | 6 | yes | — |
| 84 | C22×C4×C67 | C24 | 1 | 108 | yes | — |
| 85 | C23×C67 | — | 2 | 36 | no | — |
| 86 | C23×C67 | — | 4 | 144 | no | — |
| 87 | C23×C67 | C25 | 1 | 54 | yes | — |
| 88 | C23×C67 | C25 | 1 | 96 | yes | — |
| 89 | C23×C67 | C25 | 1 | 1 | yes | characteristic |
| 90 | C22×D67 | — | 2 | 24 | no | — |
| 91 | C22×D67 | — | 4 | 216 | no | — |
| 92 | C22×D67 | C25 | 1 | 36 | yes | — |
| 93 | C22×D67 | C25 | 1 | 144 | yes | — |
| 94 | Q8×C67 | C25 | 1 | 32 | yes | — |
| 95 | D4×C67 | C25 | 1 | 288 | yes | — |
| 96 | C2×C4×C67 | C25 | 1 | 36 | yes | — |
| 97 | C2×C4×C67 | C25 | 1 | 144 | yes | — |
| 98 | C22⋊D67 | C25 | 1 | 576 | yes | — |
| 99 | C2×Dic67 | C25 | 1 | 6 | yes | — |
| 100 | C2×Dic67 | C25 | 1 | 54 | yes | — |
| 101 | C2×Dic67 | C25 | 1 | 144 | yes | — |
| 102 | C2×Dic67 | C25 | 1 | 288 | yes | — |
| 103 | D268 | C25 | 1 | 96 | yes | — |
| 104 | C4×D67 | C25 | 1 | 96 | yes | — |
| 105 | C4×D67 | C25 | 1 | 288 | yes | — |
| 106 | Dic134 | C25 | 1 | 288 | yes | — |
| 107 | C22×C67 | — | 2 | 108 | no | — |
| 108 | C22×C67 | — | 4 | 96 | no | — |
| 109 | C22×C67 | #585 | 1 | 4 | yes | complemented |
| 110 | C22×C67 | C26 | 1 | 36 | yes | — |
| 111 | C22×C67 | C26 | 1 | 3 | yes | — |
| 112 | C2×D67 | — | 2 | 72 | no | — |
| 113 | C2×D67 | — | 4 | 144 | no | — |
| 114 | C2×D67 | C26 | 1 | 24 | yes | — |
| 115 | C4×C67 | C26 | 1 | 24 | yes | — |
| 116 | Dic67 | C26 | 1 | 36 | yes | — |
| 117 | Dic67 | C26 | 1 | 4 | yes | — |
| 118 | C22×#585 | — | 67 | 1 | no | maximal, Sylow |
| 119 | C2×C67 | — | 2 | 36 | no | — |
| 120 | C2×C67 | C2×#585 | 1 | 6 | yes | complemented |
| 121 | C2×C67 | C27 | 1 | 1 | yes | characteristic |
| 122 | D67 | — | 2 | 24 | no | — |
| 123 | C2×#585 | — | 67 | 96 | no | — |
| 124 | C22×2−1+4 | — | 67 | 6 | no | — |
| 125 | C22×H(5,𝔽2) | — | 67 | 9 | no | — |
| 126 | C22×H(5,𝔽2) | D67 | 1 | 1 | yes | characteristic, complemented |
| 127 | C23×#41 | — | 67 | 6 | no | — |
| 128 | C23×#41 | — | 67 | 9 | no | — |
| 129 | C67 | C22×#585 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 130 | #585 | — | 67 | 1024 | no | — |
| 131 | C2×2−1+4 | — | 67 | 288 | no | — |
| 132 | C2×H(5,𝔽2) | — | 67 | 432 | no | — |
| 133 | C2×H(5,𝔽2) | C2×D67 | 1 | 48 | yes | complemented |
| 134 | C22×#41 | — | 67 | 288 | no | — |
| 135 | C22×#41 | — | 67 | 2 | no | — |
| 136 | C22×#41 | — | 67 | 18 | no | — |
| 137 | C22×#41 | — | 67 | 36 | no | — |
| 138 | C22×#41 | — | 67 | 432 | no | — |
| 139 | C22×#41 | — | 67 | 18 | no | — |
| 140 | C22×#41 | C2×D67 | 1 | 6 | yes | complemented |
| 141 | C23×Q8 | — | 67 | 6 | no | — |
| 142 | C23×Q8 | — | 67 | 9 | no | — |
| 143 | C23×D4 | — | 67 | 18 | no | — |
| 144 | C23×D4 | — | 67 | 18 | no | — |
| 145 | C23×D4 | C2×D67 | 1 | 9 | yes | complemented |
| 146 | C24×C4 | — | 67 | 6 | no | — |
| 147 | C24×C4 | — | 67 | 9 | no | — |
| 148 | C25 | — | 67 | 9 | no | — |
| 149 | C25 | C22×D67 | 1 | 6 | yes | — |
| 150 | 2−1+4 | — | 67 | 1536 | no | — |
| 151 | H(5,𝔽2) | — | 67 | 2304 | no | — |
| 152 | H(5,𝔽2) | C22×D67 | 1 | 256 | yes | complemented |
| 153 | C2×#41 | — | 67 | 1536 | no | — |
| 154 | C2×#41 | — | 67 | 48 | no | — |
| 155 | C2×#41 | — | 67 | 432 | no | — |
| 156 | C2×#41 | — | 67 | 864 | no | — |
| 157 | C2×#41 | — | 67 | 2304 | no | — |
| 158 | C2×#41 | — | 67 | 432 | no | — |
| 159 | C2×#41 | C22×D67 | 1 | 144 | yes | complemented |
| 160 | C22×Q8 | — | 67 | 144 | no | — |
| 161 | C22×Q8 | — | 67 | 18 | no | — |
| 162 | C22×Q8 | — | 67 | 216 | no | — |
| 163 | C22×Q8 | C22×D67 | 1 | 2 | yes | — |
| 164 | C22×D4 | — | 67 | 36 | no | — |
| 165 | C22×D4 | — | 67 | 432 | no | — |
| 166 | C22×D4 | — | 67 | 432 | no | — |
| 167 | C22×D4 | — | 67 | 6 | no | — |
| 168 | C22×D4 | C22×D67 | 1 | 18 | yes | — |
| 169 | C22×D4 | C22×D67 | 1 | 216 | yes | complemented |
| 170 | C23×C4 | — | 67 | 9 | no | — |
| 171 | C23×C4 | — | 67 | 18 | no | — |
| 172 | C23×C4 | — | 67 | 144 | no | — |
| 173 | C23×C4 | — | 67 | 6 | no | — |
| 174 | C23×C4 | — | 67 | 18 | no | — |
| 175 | C23×C4 | — | 67 | 216 | no | — |
| 176 | C23×C4 | C22×D67 | 1 | 9 | yes | — |
| 177 | C24 | — | 4 | 24 | no | — |
| 178 | C24 | — | 268 | 36 | no | — |
| 179 | C24 | — | 67 | 108 | no | — |
| 180 | C24 | — | 67 | 6 | no | — |
| 181 | C24 | C23×D67 | 1 | 9 | yes | — |
| 182 | C24 | C23×D67 | 1 | 72 | yes | — |
| 183 | #41 | — | 67 | 128 | no | — |
| 184 | #41 | — | 67 | 1152 | no | — |
| 185 | #41 | — | 67 | 2304 | no | — |
| 186 | #41 | — | 67 | 1152 | no | — |
| 187 | #41 | C23×D67 | 1 | 384 | yes | complemented |
| 188 | C2×Q8 | — | 67 | 384 | no | — |
| 189 | C2×Q8 | — | 67 | 216 | no | — |
| 190 | C2×Q8 | — | 67 | 576 | no | — |
| 191 | C2×Q8 | C23×D67 | 1 | 24 | yes | — |
| 192 | C2×D4 | — | 67 | 432 | no | — |
| 193 | C2×D4 | — | 67 | 1152 | no | — |
| 194 | C2×D4 | — | 67 | 1152 | no | — |
| 195 | C2×D4 | — | 67 | 72 | no | — |
| 196 | C2×D4 | C23×D67 | 1 | 216 | yes | — |
| 197 | C2×D4 | C23×D67 | 1 | 576 | yes | complemented |
| 198 | C22×C4 | — | 67 | 108 | no | — |
| 199 | C22×C4 | — | 67 | 216 | no | — |
| 200 | C22×C4 | — | 67 | 384 | no | — |
| 201 | C22×C4 | — | 67 | 1 | no | — |
| 202 | C22×C4 | — | 67 | 9 | no | — |
| 203 | C22×C4 | — | 67 | 72 | no | — |
| 204 | C22×C4 | — | 67 | 216 | no | — |
| 205 | C22×C4 | — | 67 | 576 | no | — |
| 206 | C22×C4 | C23×D67 | 1 | 108 | yes | — |
| 207 | C22×C4 | C23×D67 | 1 | 6 | yes | — |
| 208 | C23 | — | 2 | 36 | no | — |
| 209 | C23 | — | 4 | 144 | no | — |
| 210 | C23 | — | 268 | 216 | no | — |
| 211 | C23 | — | 67 | 144 | no | — |
| 212 | C23 | — | 67 | 36 | no | — |
| 213 | C23 | — | 134 | 24 | no | — |
| 214 | C23 | C24×D67 | 1 | 54 | yes | — |
| 215 | C23 | C24×D67 | 1 | 96 | yes | — |
| 216 | C23 | C24×D67 | 1 | 1 | yes | characteristic |
| 217 | Q8 | — | 67 | 288 | no | — |
| 218 | Q8 | C24×D67 | 1 | 32 | yes | — |
| 219 | D4 | — | 67 | 576 | no | — |
| 220 | D4 | — | 67 | 96 | no | — |
| 221 | D4 | C24×D67 | 1 | 288 | yes | — |
| 222 | C2×C4 | — | 67 | 6 | no | — |
| 223 | C2×C4 | — | 67 | 54 | no | — |
| 224 | C2×C4 | — | 67 | 144 | no | — |
| 225 | C2×C4 | — | 67 | 288 | no | — |
| 226 | C2×C4 | — | 67 | 96 | no | — |
| 227 | C2×C4 | — | 67 | 288 | no | — |
| 228 | C2×C4 | C24×D67 | 1 | 36 | yes | — |
| 229 | C2×C4 | C24×D67 | 1 | 144 | yes | — |
| 230 | C22 | — | 2 | 108 | no | — |
| 231 | C22 | — | 4 | 96 | no | — |
| 232 | C22 | — | 268 | 144 | no | — |
| 233 | C22 | — | 67 | 24 | no | — |
| 234 | C22 | — | 134 | 72 | no | — |
| 235 | C22 | #16864231 | 1 | 4 | yes | direct product factor |
| 236 | C22 | C25×D67 | 1 | 36 | yes | — |
| 237 | C22 | C25×D67 | 1 | 3 | yes | — |
| 238 | C4 | — | 67 | 4 | no | — |
| 239 | C4 | — | 67 | 36 | no | — |
| 240 | C4 | C25×D67 | 1 | 24 | yes | — |
| 241 | C2 | — | 2 | 36 | no | — |
| 242 | C2 | — | 134 | 24 | no | — |
| 243 | C2 | C2×#16864231 | 1 | 6 | yes | direct product factor |
| 244 | C2 | C26×D67 | 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