← C67×#33385 #30712636
Subgroups — C67×#33385 #30712636
Order 17152.
1114 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 | C67×#3584 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#504×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C67×#2916 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C67×#2143 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#359×C67 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #585×C67 | — | 2 | 1 | no | — |
| 7 | #578×C67 | C4 | 1 | 4 | yes | — |
| 8 | #577×C67 | — | 2 | 4 | no | — |
| 9 | #576×C67 | C4 | 1 | 4 | yes | — |
| 10 | #575×C67 | — | 2 | 3 | no | — |
| 11 | C2×#135×C67 | C22 | 1 | 6 | yes | — |
| 12 | #568×C67 | — | 2 | 1 | no | — |
| 13 | C2×#131×C67 | C22 | 1 | 1 | yes | characteristic |
| 14 | #504×C67 | — | 2 | 4 | no | — |
| 15 | C2×M32×C67 | C22 | 1 | 3 | yes | complemented |
| 16 | C22×C16×C67 | C22 | 1 | 3 | yes | complemented |
| 17 | #430×C67 | — | 2 | 3 | no | — |
| 18 | C2×#108×C67 | C22 | 1 | 6 | yes | complemented |
| 19 | #359×C67 | C22 | 1 | 4 | yes | complemented |
| 20 | #359×C67 | C22 | 1 | 12 | yes | complemented |
| 21 | 2−1+4×C67 | — | 2 | 4 | no | — |
| 22 | 2−1+4×C67 | — | 4 | 1 | no | — |
| 23 | H(5,𝔽2)×C67 | — | 2 | 4 | no | — |
| 24 | H(5,𝔽2)×C67 | — | 4 | 3 | no | — |
| 25 | C2×#41×C67 | — | 2 | 6 | no | — |
| 26 | C2×#41×C67 | — | 4 | 4 | no | — |
| 27 | C2×#41×C67 | D4 | 1 | 1 | yes | characteristic |
| 28 | #137×C67 | — | 2 | 12 | no | — |
| 29 | AGL(1,ℤ/8)×C67 | — | 2 | 12 | no | — |
| 30 | #135×C67 | — | 2 | 6 | no | — |
| 31 | #135×C67 | — | 2 | 6 | no | — |
| 32 | #135×C67 | C2×C4 | 1 | 12 | yes | — |
| 33 | #135×C67 | C23 | 1 | 4 | yes | — |
| 34 | C2×Q16×C67 | C2×C4 | 1 | 6 | yes | — |
| 35 | C2×SD16×C67 | — | 2 | 6 | no | — |
| 36 | C2×D8×C67 | C2×C4 | 1 | 6 | yes | — |
| 37 | #131×C67 | — | 4 | 4 | no | — |
| 38 | #131×C67 | C2×C4 | 1 | 4 | yes | — |
| 39 | #131×C67 | D4 | 1 | 4 | yes | — |
| 40 | C2×M16×C67 | — | 2 | 6 | no | — |
| 41 | C2×M16×C67 | D4 | 1 | 3 | yes | — |
| 42 | C22×C8×C67 | C23 | 1 | 3 | yes | — |
| 43 | M32×C67 | — | 2 | 6 | no | — |
| 44 | C2×C16×C67 | — | 2 | 6 | no | — |
| 45 | C2×C16×C67 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C16×C67 | C23 | 1 | 3 | yes | — |
| 47 | #108×C67 | — | 2 | 6 | no | — |
| 48 | #108×C67 | C23 | 1 | 4 | yes | — |
| 49 | #41×C67 | — | 2 | 12 | no | — |
| 50 | #41×C67 | — | 2 | 4 | no | — |
| 51 | #41×C67 | — | 4 | 4 | no | — |
| 52 | #41×C67 | — | 4 | 12 | no | — |
| 53 | #41×C67 | — | 4 | 6 | no | — |
| 54 | #41×C67 | — | 4 | 6 | no | — |
| 55 | #41×C67 | C22⋊C4 | 1 | 4 | yes | — |
| 56 | #41×C67 | D8 | 1 | 4 | yes | — |
| 57 | C2×Q8×C67 | — | 2 | 6 | no | — |
| 58 | C2×Q8×C67 | — | 4 | 4 | no | — |
| 59 | C2×Q8×C67 | SD16 | 1 | 1 | yes | characteristic |
| 60 | C2×D4×C67 | — | 2 | 6 | no | — |
| 61 | C2×D4×C67 | — | 4 | 6 | no | — |
| 62 | C2×D4×C67 | — | 4 | 12 | no | — |
| 63 | C2×D4×C67 | SD16 | 1 | 3 | yes | — |
| 64 | C22×C4×C67 | — | 4 | 6 | no | — |
| 65 | C22×C4×C67 | C2×D4 | 1 | 3 | yes | — |
| 66 | Q16×C67 | — | 2 | 6 | no | — |
| 67 | Q16×C67 | C22×C4 | 1 | 4 | yes | — |
| 68 | SD16×C67 | — | 2 | 12 | no | — |
| 69 | SD16×C67 | — | 2 | 4 | no | — |
| 70 | D8×C67 | — | 2 | 6 | no | — |
| 71 | D8×C67 | C22×C4 | 1 | 4 | yes | — |
| 72 | M16×C67 | — | 2 | 4 | no | — |
| 73 | M16×C67 | — | 4 | 6 | no | — |
| 74 | M16×C67 | C22⋊C4 | 1 | 12 | yes | — |
| 75 | C2×C8×C67 | — | 4 | 6 | no | — |
| 76 | C2×C8×C67 | C22×C4 | 1 | 6 | yes | — |
| 77 | C2×C8×C67 | C2×D4 | 1 | 3 | yes | — |
| 78 | C2×C8×C67 | C2×D4 | 1 | 6 | yes | — |
| 79 | C2×C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 80 | C16×C67 | — | 2 | 1 | no | — |
| 81 | C16×C67 | — | 2 | 3 | no | — |
| 82 | C23×C67 | — | 4 | 6 | no | — |
| 83 | C23×C67 | C2×SD16 | 1 | 3 | yes | — |
| 84 | Q8×C67 | — | 2 | 4 | no | — |
| 85 | Q8×C67 | — | 4 | 6 | no | — |
| 86 | Q8×C67 | #102 | 1 | 4 | yes | — |
| 87 | D4×C67 | — | 2 | 4 | no | — |
| 88 | D4×C67 | — | 4 | 4 | no | — |
| 89 | D4×C67 | — | 4 | 12 | no | — |
| 90 | D4×C67 | — | 4 | 6 | no | — |
| 91 | D4×C67 | #102 | 1 | 12 | yes | — |
| 92 | C2×C4×C67 | — | 4 | 6 | no | — |
| 93 | C2×C4×C67 | — | 4 | 4 | no | — |
| 94 | C2×C4×C67 | — | 4 | 12 | no | — |
| 95 | C2×C4×C67 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 96 | C2×C4×C67 | C2×D8 | 1 | 6 | yes | — |
| 97 | C2×C4×C67 | C2×SD16 | 1 | 3 | yes | — |
| 98 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 99 | C8×C67 | — | 4 | 4 | no | — |
| 100 | C8×C67 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 101 | C8×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 102 | C8×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×C67 | — | 4 | 3 | no | — |
| 104 | C22×C67 | — | 4 | 4 | no | — |
| 105 | C22×C67 | — | 16 | 6 | no | — |
| 106 | C22×C67 | C2×#102 | 1 | 6 | yes | — |
| 107 | C22×C67 | C22×SD16 | 1 | 1 | yes | characteristic |
| 108 | C4×C67 | — | 4 | 4 | no | — |
| 109 | C4×C67 | C2×#102 | 1 | 6 | yes | — |
| 110 | C4×C67 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C4×C67 | C22×D8 | 1 | 1 | yes | characteristic |
| 112 | #33385 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 113 | C2×C67 | — | 2 | 1 | no | — |
| 114 | C2×C67 | — | 2 | 6 | no | — |
| 115 | C2×C67 | — | 8 | 4 | no | — |
| 116 | C2×C67 | C22×#102 | 1 | 1 | yes | characteristic |
| 117 | #3584 | C2×C67 | 1 | 1 | yes | characteristic |
| 118 | C2×#504 | C2×C67 | 1 | 1 | yes | characteristic, complemented |
| 119 | #2916 | C2×C67 | 1 | 1 | yes | characteristic, complemented |
| 120 | #2143 | C2×C67 | 1 | 6 | yes | complemented |
| 121 | C2×#359 | C2×C67 | 1 | 6 | yes | complemented |
| 122 | C67 | #33385 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 123 | #585 | — | 2 | 1 | no | — |
| 124 | #578 | C4×C67 | 1 | 4 | yes | — |
| 125 | #577 | — | 2 | 4 | no | — |
| 126 | #576 | C4×C67 | 1 | 4 | yes | — |
| 127 | #575 | — | 2 | 3 | no | — |
| 128 | C2×#135 | C22×C67 | 1 | 6 | yes | — |
| 129 | #568 | — | 2 | 1 | no | — |
| 130 | C2×#131 | C22×C67 | 1 | 1 | yes | characteristic |
| 131 | #504 | — | 2 | 4 | no | — |
| 132 | C2×M32 | C22×C67 | 1 | 3 | yes | complemented |
| 133 | C22×C16 | C22×C67 | 1 | 3 | yes | complemented |
| 134 | #430 | — | 2 | 3 | no | — |
| 135 | C2×#108 | C22×C67 | 1 | 6 | yes | complemented |
| 136 | #359 | C22×C67 | 1 | 4 | yes | complemented |
| 137 | #359 | C22×C67 | 1 | 12 | yes | complemented |
| 138 | 2−1+4 | — | 2 | 4 | no | — |
| 139 | 2−1+4 | — | 4 | 1 | no | — |
| 140 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 141 | H(5,𝔽2) | — | 4 | 3 | no | — |
| 142 | C2×#41 | — | 2 | 6 | no | — |
| 143 | C2×#41 | — | 4 | 4 | no | — |
| 144 | C2×#41 | D4×C67 | 1 | 1 | yes | characteristic |
| 145 | #137 | — | 2 | 12 | no | — |
| 146 | AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 147 | #135 | — | 2 | 6 | no | — |
| 148 | #135 | — | 2 | 6 | no | — |
| 149 | #135 | C2×C4×C67 | 1 | 12 | yes | — |
| 150 | #135 | C23×C67 | 1 | 4 | yes | — |
| 151 | C2×Q16 | C2×C4×C67 | 1 | 6 | yes | — |
| 152 | C2×SD16 | — | 2 | 6 | no | — |
| 153 | C2×D8 | C2×C4×C67 | 1 | 6 | yes | — |
| 154 | #131 | — | 4 | 4 | no | — |
| 155 | #131 | C2×C4×C67 | 1 | 4 | yes | — |
| 156 | #131 | D4×C67 | 1 | 4 | yes | — |
| 157 | C2×M16 | — | 2 | 6 | no | — |
| 158 | C2×M16 | D4×C67 | 1 | 3 | yes | — |
| 159 | C22×C8 | C23×C67 | 1 | 3 | yes | — |
| 160 | M32 | — | 2 | 6 | no | — |
| 161 | C2×C16 | — | 2 | 6 | no | — |
| 162 | C2×C16 | C23×C67 | 1 | 1 | yes | characteristic |
| 163 | C2×C16 | C23×C67 | 1 | 3 | yes | — |
| 164 | #108 | — | 2 | 6 | no | — |
| 165 | #108 | C23×C67 | 1 | 4 | yes | — |
| 166 | #41 | — | 2 | 12 | no | — |
| 167 | #41 | — | 2 | 4 | no | — |
| 168 | #41 | — | 4 | 4 | no | — |
| 169 | #41 | — | 4 | 12 | no | — |
| 170 | #41 | — | 4 | 6 | no | — |
| 171 | #41 | — | 4 | 6 | no | — |
| 172 | #41 | C22⋊C4×C67 | 1 | 4 | yes | — |
| 173 | #41 | D8×C67 | 1 | 4 | yes | — |
| 174 | C2×Q8 | — | 2 | 6 | no | — |
| 175 | C2×Q8 | — | 4 | 4 | no | — |
| 176 | C2×Q8 | SD16×C67 | 1 | 1 | yes | characteristic |
| 177 | C2×D4 | — | 2 | 6 | no | — |
| 178 | C2×D4 | — | 4 | 6 | no | — |
| 179 | C2×D4 | — | 4 | 12 | no | — |
| 180 | C2×D4 | SD16×C67 | 1 | 3 | yes | — |
| 181 | C22×C4 | — | 4 | 6 | no | — |
| 182 | C22×C4 | C2×D4×C67 | 1 | 3 | yes | — |
| 183 | Q16 | — | 2 | 6 | no | — |
| 184 | Q16 | C22×C4×C67 | 1 | 4 | yes | — |
| 185 | SD16 | — | 2 | 12 | no | — |
| 186 | SD16 | — | 2 | 4 | no | — |
| 187 | D8 | — | 2 | 6 | no | — |
| 188 | D8 | C22×C4×C67 | 1 | 4 | yes | — |
| 189 | M16 | — | 2 | 4 | no | — |
| 190 | M16 | — | 4 | 6 | no | — |
| 191 | M16 | C22⋊C4×C67 | 1 | 12 | yes | — |
| 192 | C2×C8 | — | 4 | 6 | no | — |
| 193 | C2×C8 | C22×C4×C67 | 1 | 6 | yes | — |
| 194 | C2×C8 | C2×D4×C67 | 1 | 3 | yes | — |
| 195 | C2×C8 | C2×D4×C67 | 1 | 6 | yes | — |
| 196 | C2×C8 | C24×C67 | 1 | 1 | yes | characteristic |
| 197 | C16 | — | 2 | 1 | no | — |
| 198 | C16 | — | 2 | 3 | no | — |
| 199 | C23 | — | 4 | 6 | no | — |
| 200 | C23 | C2×SD16×C67 | 1 | 3 | yes | — |
| 201 | Q8 | — | 2 | 4 | no | — |
| 202 | Q8 | — | 4 | 6 | no | — |
| 203 | Q8 | #102×C67 | 1 | 4 | yes | — |
| 204 | D4 | — | 2 | 4 | no | — |
| 205 | D4 | — | 4 | 4 | no | — |
| 206 | D4 | — | 4 | 12 | no | — |
| 207 | D4 | — | 4 | 6 | no | — |
| 208 | D4 | #102×C67 | 1 | 12 | yes | — |
| 209 | C2×C4 | — | 4 | 6 | no | — |
| 210 | C2×C4 | — | 4 | 4 | no | — |
| 211 | C2×C4 | — | 4 | 12 | no | — |
| 212 | C2×C4 | C2×C22⋊C4×C67 | 1 | 6 | yes | — |
| 213 | C2×C4 | C2×D8×C67 | 1 | 6 | yes | — |
| 214 | C2×C4 | C2×SD16×C67 | 1 | 3 | yes | — |
| 215 | C2×C4 | C22×D4×C67 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 4 | 4 | no | — |
| 217 | C8 | C2×C22⋊C4×C67 | 1 | 6 | yes | — |
| 218 | C8 | C23×C4×C67 | 1 | 1 | yes | characteristic |
| 219 | C8 | C22×D4×C67 | 1 | 1 | yes | characteristic |
| 220 | C22 | — | 4 | 3 | no | — |
| 221 | C22 | — | 4 | 4 | no | — |
| 222 | C22 | — | 16 | 6 | no | — |
| 223 | C22 | C2×#102×C67 | 1 | 6 | yes | — |
| 224 | C22 | C22×SD16×C67 | 1 | 1 | yes | characteristic |
| 225 | C4 | — | 4 | 4 | no | — |
| 226 | C4 | C2×#102×C67 | 1 | 6 | yes | — |
| 227 | C4 | C22×C22⋊C4×C67 | 1 | 1 | yes | characteristic |
| 228 | C4 | C22×D8×C67 | 1 | 1 | yes | characteristic |
| 229 | C2 | — | 2 | 1 | no | — |
| 230 | C2 | — | 2 | 6 | no | — |
| 231 | C2 | — | 8 | 4 | no | — |
| 232 | C2 | C22×#102×C67 | 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