← C24×#550 #72291239
Subgroups — C24×#550 #72291239
Order 1024.
914277 conjugacy classes of subgroups.
The subgroup lattice has 195 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C25×#41 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C23×#550 | C2 | 1 | 240 | yes | maximal, direct product factor |
| 3 | C24×#127 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C24×#121 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C24×C4×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C24×C4×D4 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C24×#41 | — | 2 | 128 | no | — |
| 8 | C24×#41 | C22 | 1 | 240 | yes | complemented |
| 9 | C25×Q8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C25×D4 | C22 | 1 | 3 | yes | — |
| 11 | C25×D4 | C22 | 1 | 6 | yes | — |
| 12 | C25×D4 | C22 | 1 | 6 | yes | — |
| 13 | C26×C4 | C22 | 1 | 6 | yes | — |
| 14 | C22×#550 | C22 | 1 | 8960 | yes | direct product factor |
| 15 | C23×#127 | C22 | 1 | 360 | yes | complemented |
| 16 | C23×#121 | C22 | 1 | 720 | yes | complemented |
| 17 | C23×C4×Q8 | C22 | 1 | 120 | yes | complemented |
| 18 | C23×C4×D4 | C22 | 1 | 360 | yes | complemented |
| 19 | C24×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 20 | C24×C4⋊C4 | C22 | 1 | 3 | yes | — |
| 21 | C24×C22⋊C4 | C22 | 1 | 6 | yes | — |
| 22 | C24×C42 | C22 | 1 | 3 | yes | — |
| 23 | C27 | C23 | 1 | 6 | yes | — |
| 24 | C23×#41 | — | 2 | 7680 | no | — |
| 25 | C23×#41 | C23 | 1 | 4480 | yes | complemented |
| 26 | C24×Q8 | D4 | 1 | 64 | yes | complemented |
| 27 | C24×Q8 | C23 | 1 | 60 | yes | — |
| 28 | C24×D4 | — | 2 | 192 | no | — |
| 29 | C24×D4 | — | 2 | 96 | no | — |
| 30 | C24×D4 | — | 4 | 96 | no | — |
| 31 | C24×D4 | C23 | 1 | 360 | yes | — |
| 32 | C24×D4 | C23 | 1 | 180 | yes | — |
| 33 | C24×D4 | C23 | 1 | 360 | yes | — |
| 34 | C25×C4 | — | 2 | 192 | no | — |
| 35 | C25×C4 | C23 | 1 | 360 | yes | — |
| 36 | C25×C4 | C23 | 1 | 2 | yes | — |
| 37 | C25×C4 | C23 | 1 | 3 | yes | — |
| 38 | C25×C4 | C23 | 1 | 3 | yes | — |
| 39 | C25×C4 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×#550 | C23 | 1 | 61440 | yes | direct product factor |
| 41 | C22×#127 | C23 | 1 | 6720 | yes | complemented |
| 42 | C22×#121 | C23 | 1 | 13440 | yes | complemented |
| 43 | C22×C4×Q8 | C23 | 1 | 2240 | yes | complemented |
| 44 | C22×C4×D4 | C23 | 1 | 6720 | yes | complemented |
| 45 | C23×C4⋊C4 | C23 | 1 | 60 | yes | — |
| 46 | C23×C4⋊C4 | C23 | 1 | 180 | yes | — |
| 47 | C23×C22⋊C4 | C23 | 1 | 360 | yes | — |
| 48 | C23×C42 | C23 | 1 | 180 | yes | — |
| 49 | C26 | — | 2 | 96 | no | — |
| 50 | C26 | — | 2 | 96 | no | — |
| 51 | C26 | — | 2 | 96 | no | — |
| 52 | C26 | C24 | 1 | 180 | yes | — |
| 53 | C26 | C24 | 1 | 1 | yes | characteristic |
| 54 | C22×#41 | — | 2 | 71680 | no | — |
| 55 | C22×#41 | C24 | 1 | 15360 | yes | complemented |
| 56 | C23×Q8 | C2×D4 | 1 | 1920 | yes | complemented |
| 57 | C23×Q8 | C24 | 1 | 560 | yes | — |
| 58 | C23×D4 | — | 2 | 5760 | no | — |
| 59 | C23×D4 | — | 2 | 2880 | no | — |
| 60 | C23×D4 | — | 4 | 2880 | no | — |
| 61 | C23×D4 | C24 | 1 | 3360 | yes | — |
| 62 | C23×D4 | C24 | 1 | 1680 | yes | — |
| 63 | C23×D4 | C24 | 1 | 3360 | yes | — |
| 64 | C24×C4 | — | 2 | 32 | no | — |
| 65 | C24×C4 | — | 2 | 48 | no | — |
| 66 | C24×C4 | — | 2 | 5760 | no | — |
| 67 | C24×C4 | C2×D4 | 1 | 96 | yes | — |
| 68 | C24×C4 | #41 | 1 | 32 | yes | complemented |
| 69 | C24×C4 | C24 | 1 | 60 | yes | — |
| 70 | C24×C4 | C24 | 1 | 3360 | yes | — |
| 71 | C24×C4 | C24 | 1 | 90 | yes | — |
| 72 | C24×C4 | C24 | 1 | 90 | yes | — |
| 73 | C24×C4 | C24 | 1 | 30 | yes | — |
| 74 | #550 | C24 | 1 | 65536 | yes | direct product factor |
| 75 | C2×#127 | C24 | 1 | 23040 | yes | complemented |
| 76 | C2×#121 | C24 | 1 | 46080 | yes | complemented |
| 77 | C2×C4×Q8 | C24 | 1 | 7680 | yes | complemented |
| 78 | C2×C4×D4 | C24 | 1 | 23040 | yes | complemented |
| 79 | C22×C4⋊C4 | C24 | 1 | 560 | yes | — |
| 80 | C22×C4⋊C4 | C24 | 1 | 1680 | yes | — |
| 81 | C22×C22⋊C4 | C24 | 1 | 3360 | yes | — |
| 82 | C22×C42 | C24 | 1 | 1680 | yes | — |
| 83 | C25 | — | 2 | 1440 | no | — |
| 84 | C25 | — | 2 | 1440 | no | — |
| 85 | C25 | — | 2 | 1440 | no | — |
| 86 | C25 | — | 4 | 1536 | no | — |
| 87 | C25 | C22×D4 | 1 | 16 | yes | — |
| 88 | C25 | C2×#41 | 1 | 16 | yes | — |
| 89 | C25 | H(5,𝔽2) | 1 | 16 | yes | — |
| 90 | C25 | C25 | 1 | 840 | yes | — |
| 91 | C25 | C25 | 1 | 15 | yes | — |
| 92 | C2×#41 | — | 2 | 122880 | no | — |
| 93 | C2×#41 | C25 | 1 | 8192 | yes | complemented |
| 94 | C22×Q8 | C22×D4 | 1 | 8960 | yes | complemented |
| 95 | C22×Q8 | C25 | 1 | 960 | yes | — |
| 96 | C22×D4 | — | 2 | 26880 | no | — |
| 97 | C22×D4 | — | 2 | 13440 | no | — |
| 98 | C22×D4 | — | 4 | 13440 | no | — |
| 99 | C22×D4 | C25 | 1 | 2880 | yes | — |
| 100 | C22×D4 | C25 | 1 | 5760 | yes | — |
| 101 | C22×D4 | C25 | 1 | 5760 | yes | — |
| 102 | C23×C4 | — | 2 | 480 | no | — |
| 103 | C23×C4 | — | 2 | 720 | no | — |
| 104 | C23×C4 | — | 2 | 26880 | no | — |
| 105 | C23×C4 | C22×D4 | 1 | 1440 | yes | — |
| 106 | C23×C4 | C2×#41 | 1 | 480 | yes | complemented |
| 107 | C23×C4 | C25 | 1 | 280 | yes | — |
| 108 | C23×C4 | C25 | 1 | 420 | yes | — |
| 109 | C23×C4 | C25 | 1 | 5760 | yes | — |
| 110 | C23×C4 | C25 | 1 | 420 | yes | — |
| 111 | C23×C4 | C25 | 1 | 140 | yes | — |
| 112 | #127 | C25 | 1 | 12288 | yes | complemented |
| 113 | #121 | C25 | 1 | 24576 | yes | complemented |
| 114 | C4×Q8 | C25 | 1 | 4096 | yes | complemented |
| 115 | C4×D4 | C25 | 1 | 12288 | yes | complemented |
| 116 | C2×C4⋊C4 | C25 | 1 | 960 | yes | — |
| 117 | C2×C4⋊C4 | C25 | 1 | 2880 | yes | — |
| 118 | C2×C22⋊C4 | C25 | 1 | 5760 | yes | — |
| 119 | C2×C42 | C25 | 1 | 2880 | yes | — |
| 120 | C24 | — | 2 | 3360 | no | — |
| 121 | C24 | — | 2 | 3360 | no | — |
| 122 | C24 | — | 2 | 3360 | no | — |
| 123 | C24 | — | 4 | 11520 | no | — |
| 124 | C24 | #550 | 1 | 256 | yes | direct product factor |
| 125 | C24 | C23×D4 | 1 | 120 | yes | — |
| 126 | C24 | C22×#41 | 1 | 120 | yes | — |
| 127 | C24 | C2×H(5,𝔽2) | 1 | 120 | yes | — |
| 128 | C24 | C26 | 1 | 720 | yes | — |
| 129 | C24 | C26 | 1 | 35 | yes | — |
| 130 | #41 | — | 2 | 32768 | no | — |
| 131 | C2×Q8 | C23×D4 | 1 | 7680 | yes | complemented |
| 132 | C2×Q8 | C26 | 1 | 256 | yes | — |
| 133 | C2×D4 | — | 2 | 23040 | no | — |
| 134 | C2×D4 | — | 2 | 11520 | no | — |
| 135 | C2×D4 | — | 4 | 11520 | no | — |
| 136 | C2×D4 | C26 | 1 | 768 | yes | — |
| 137 | C2×D4 | C26 | 1 | 1536 | yes | — |
| 138 | C2×D4 | C26 | 1 | 1536 | yes | — |
| 139 | C22×C4 | — | 2 | 1120 | no | — |
| 140 | C22×C4 | — | 2 | 1680 | no | — |
| 141 | C22×C4 | — | 2 | 23040 | no | — |
| 142 | C22×C4 | C23×D4 | 1 | 3360 | yes | — |
| 143 | C22×C4 | C22×#41 | 1 | 1120 | yes | complemented |
| 144 | C22×C4 | C26 | 1 | 360 | yes | — |
| 145 | C22×C4 | C26 | 1 | 240 | yes | — |
| 146 | C22×C4 | C26 | 1 | 1536 | yes | — |
| 147 | C22×C4 | C26 | 1 | 360 | yes | — |
| 148 | C22×C4 | C26 | 1 | 120 | yes | — |
| 149 | C4⋊C4 | C26 | 1 | 256 | yes | — |
| 150 | C4⋊C4 | C26 | 1 | 768 | yes | — |
| 151 | C22⋊C4 | C26 | 1 | 1536 | yes | — |
| 152 | C42 | C26 | 1 | 768 | yes | — |
| 153 | C23 | — | 2 | 1440 | no | — |
| 154 | C23 | — | 2 | 1440 | no | — |
| 155 | C23 | — | 2 | 1440 | no | — |
| 156 | C23 | — | 4 | 13440 | no | — |
| 157 | C23 | C2×#550 | 1 | 960 | yes | direct product factor |
| 158 | C23 | C24×D4 | 1 | 140 | yes | — |
| 159 | C23 | C23×#41 | 1 | 140 | yes | — |
| 160 | C23 | C22×H(5,𝔽2) | 1 | 140 | yes | — |
| 161 | C23 | C27 | 1 | 96 | yes | — |
| 162 | C23 | C27 | 1 | 15 | yes | — |
| 163 | Q8 | C24×D4 | 1 | 1024 | yes | complemented |
| 164 | D4 | — | 2 | 3072 | no | — |
| 165 | D4 | — | 2 | 1536 | no | — |
| 166 | D4 | — | 4 | 1536 | no | — |
| 167 | C2×C4 | — | 2 | 480 | no | — |
| 168 | C2×C4 | — | 2 | 720 | no | — |
| 169 | C2×C4 | — | 2 | 3072 | no | — |
| 170 | C2×C4 | C24×D4 | 1 | 1440 | yes | — |
| 171 | C2×C4 | C23×#41 | 1 | 480 | yes | complemented |
| 172 | C2×C4 | C27 | 1 | 48 | yes | — |
| 173 | C2×C4 | C27 | 1 | 32 | yes | — |
| 174 | C2×C4 | C27 | 1 | 48 | yes | — |
| 175 | C2×C4 | C27 | 1 | 16 | yes | — |
| 176 | C22 | — | 2 | 96 | no | — |
| 177 | C22 | — | 2 | 96 | no | — |
| 178 | C22 | — | 2 | 96 | no | — |
| 179 | C22 | — | 4 | 2880 | no | — |
| 180 | C22 | C22×#550 | 1 | 560 | yes | direct product factor |
| 181 | C22 | C25×D4 | 1 | 30 | yes | — |
| 182 | C22 | C24×#41 | 1 | 30 | yes | — |
| 183 | C22 | C23×H(5,𝔽2) | 1 | 30 | yes | — |
| 184 | C22 | C28 | 1 | 1 | yes | characteristic |
| 185 | C4 | — | 2 | 32 | no | — |
| 186 | C4 | — | 2 | 48 | no | — |
| 187 | C4 | C25×D4 | 1 | 96 | yes | — |
| 188 | C4 | C24×#41 | 1 | 32 | yes | complemented |
| 189 | C2 | — | 4 | 96 | no | — |
| 190 | C2 | C23×#550 | 1 | 60 | yes | direct product factor |
| 191 | C2 | C26×D4 | 1 | 1 | yes | characteristic |
| 192 | C2 | C25×#41 | 1 | 1 | yes | characteristic |
| 193 | C2 | C24×H(5,𝔽2) | 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