← #10617415
Subgroups — #10617415
Order 640.
556 conjugacy classes of subgroups.
The subgroup lattice has 184 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #41×AGL(1,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D5⋊#131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | D5×#131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #65728 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | M16×AGL(1,𝔽5) | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #65718 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#3992 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | D5×#41 | C22 | 1 | 1 | yes | characteristic |
| 9 | Q8×AGL(1,𝔽5) | C4 | 1 | 2 | yes | — |
| 10 | #4133 | C4 | 1 | 2 | yes | complemented |
| 11 | D4×AGL(1,𝔽5) | C4 | 1 | 6 | yes | — |
| 12 | #4131 | C4 | 1 | 6 | yes | complemented |
| 13 | #4130 | C22 | 1 | 3 | yes | — |
| 14 | C2×C4×AGL(1,𝔽5) | C22 | 1 | 3 | yes | — |
| 15 | D5⋊M16 | C22 | 1 | 3 | yes | — |
| 16 | C2×D5⋊C8 | C22 | 1 | 3 | yes | — |
| 17 | C5×#131 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | #4094 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | #4053 | C4 | 1 | 3 | yes | complemented |
| 20 | D5×M16 | C22 | 1 | 3 | yes | — |
| 21 | #4047 | C4 | 1 | 3 | yes | complemented |
| 22 | C2×C8×D5 | C22 | 1 | 3 | yes | — |
| 23 | #3992 | C22 | 1 | 1 | yes | characteristic |
| 24 | #3992 | C22 | 1 | 3 | yes | — |
| 25 | #3992 | C22 | 1 | 3 | yes | — |
| 26 | #3992 | C22 | 1 | 3 | yes | — |
| 27 | C8×AGL(1,𝔽5) | C22 | 1 | 6 | yes | — |
| 28 | #2875 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C22×AGL(1,𝔽5) | C2×C4 | 1 | 6 | yes | — |
| 30 | C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | #713 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | Q8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | #711 | C2×C4 | 1 | 3 | yes | — |
| 34 | D4×D5 | C2×C4 | 1 | 3 | yes | — |
| 35 | #709 | C2×C4 | 1 | 3 | yes | — |
| 36 | C2×C4×D5 | C23 | 1 | 3 | yes | — |
| 37 | C22⋊AGL(1,𝔽5) | C2×C4 | 1 | 6 | yes | — |
| 38 | #704 | C2×C4 | 1 | 6 | yes | complemented |
| 39 | C2×#170 | C2×C4 | 1 | 6 | yes | complemented |
| 40 | C4⋊AGL(1,𝔽5) | C2×C4 | 1 | 6 | yes | — |
| 41 | C4×AGL(1,𝔽5) | C2×C4 | 1 | 6 | yes | — |
| 42 | C4×AGL(1,𝔽5) | C23 | 1 | 1 | yes | characteristic |
| 43 | C4×AGL(1,𝔽5) | C23 | 1 | 3 | yes | — |
| 44 | #700 | C2×C4 | 1 | 6 | yes | complemented |
| 45 | D5⋊C8 | C2×C4 | 1 | 6 | yes | complemented |
| 46 | D5⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 47 | D5⋊C8 | C23 | 1 | 3 | yes | — |
| 48 | C5×M16 | C2×C4 | 1 | 3 | yes | complemented |
| 49 | C2×C5×C8 | C2×C4 | 1 | 3 | yes | complemented |
| 50 | #681 | C2×C4 | 1 | 3 | yes | complemented |
| 51 | C2×#168 | C2×C4 | 1 | 3 | yes | complemented |
| 52 | #676 | C2×C4 | 1 | 6 | yes | complemented |
| 53 | C8×D5 | C2×C4 | 1 | 3 | yes | complemented |
| 54 | C8×D5 | C2×C4 | 1 | 3 | yes | complemented |
| 55 | C8×D5 | C23 | 1 | 3 | yes | — |
| 56 | C8×D5 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×#131 | — | 5 | 1 | no | — |
| 58 | C2×#131 | — | 5 | 1 | no | — |
| 59 | C4×#41 | — | 5 | 1 | no | — |
| 60 | #405 | — | 5 | 3 | no | — |
| 61 | #405 | — | 5 | 3 | no | — |
| 62 | C4×M16 | — | 5 | 3 | no | — |
| 63 | C2×#97 | — | 5 | 3 | no | — |
| 64 | C22×D5 | C22×C4 | 1 | 3 | yes | — |
| 65 | C2×AGL(1,𝔽5) | — | 4 | 6 | no | — |
| 66 | C2×AGL(1,𝔽5) | C22×C4 | 1 | 2 | yes | — |
| 67 | C2×AGL(1,𝔽5) | C22×C4 | 1 | 6 | yes | — |
| 68 | C5×Q8 | C42 | 1 | 1 | yes | characteristic |
| 69 | C5×D4 | C42 | 1 | 3 | yes | — |
| 70 | C2×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 71 | C22⋊D5 | C42 | 1 | 6 | yes | — |
| 72 | C2×Dic5 | C22×C4 | 1 | 3 | yes | — |
| 73 | D20 | C42 | 1 | 3 | yes | — |
| 74 | C4×D5 | C22×C4 | 1 | 3 | yes | — |
| 75 | C4×D5 | C22×C4 | 1 | 3 | yes | — |
| 76 | C4×D5 | C22×C4 | 1 | 3 | yes | — |
| 77 | C4×D5 | C24 | 1 | 1 | yes | characteristic |
| 78 | Dic10 | C42 | 1 | 3 | yes | — |
| 79 | #170 | C22×C4 | 1 | 2 | yes | — |
| 80 | #170 | C22×C4 | 1 | 6 | yes | — |
| 81 | C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C5×C8 | C22×C4 | 1 | 3 | yes | — |
| 83 | #168 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | #168 | C22×C4 | 1 | 3 | yes | — |
| 85 | C2×#41 | — | 5 | 1 | no | — |
| 86 | #131 | — | 5 | 6 | no | — |
| 87 | #131 | — | 5 | 2 | no | — |
| 88 | #131 | — | 5 | 1 | no | — |
| 89 | #131 | — | 5 | 3 | no | — |
| 90 | #131 | — | 5 | 3 | no | — |
| 91 | #131 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 92 | C2×M16 | — | 5 | 3 | no | — |
| 93 | C2×M16 | — | 5 | 3 | no | — |
| 94 | C22×C8 | — | 5 | 3 | no | — |
| 95 | C22×C8 | — | 5 | 3 | no | — |
| 96 | C4×Q8 | — | 5 | 2 | no | — |
| 97 | C4×D4 | — | 5 | 6 | no | — |
| 98 | #117 | — | 5 | 3 | no | — |
| 99 | C2×C42 | — | 5 | 3 | no | — |
| 100 | #97 | — | 5 | 1 | no | — |
| 101 | #97 | — | 5 | 3 | no | — |
| 102 | #97 | — | 5 | 3 | no | — |
| 103 | #97 | — | 5 | 3 | no | — |
| 104 | C4×C8 | — | 5 | 6 | no | — |
| 105 | C22×C5 | C2×C42 | 1 | 3 | yes | — |
| 106 | C2×D5 | — | 2 | 6 | no | — |
| 107 | C2×D5 | C2×C42 | 1 | 3 | yes | — |
| 108 | C2×D5 | C23×C4 | 1 | 1 | yes | characteristic |
| 109 | AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 110 | AGL(1,𝔽5) | — | 2 | 6 | no | — |
| 111 | C4×C5 | C2×C42 | 1 | 3 | yes | — |
| 112 | C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 113 | Dic5 | C2×C42 | 1 | 3 | yes | — |
| 114 | Dic5 | C23×C4 | 1 | 1 | yes | characteristic |
| 115 | #41 | — | 5 | 1 | no | — |
| 116 | #41 | — | 5 | 3 | no | — |
| 117 | #41 | — | 5 | 3 | no | — |
| 118 | #41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 119 | C2×Q8 | — | 5 | 1 | no | — |
| 120 | C2×D4 | — | 5 | 3 | no | — |
| 121 | C22×C4 | — | 5 | 3 | no | — |
| 122 | C22×C4 | — | 5 | 6 | no | — |
| 123 | M16 | — | 5 | 6 | no | — |
| 124 | M16 | — | 5 | 6 | no | — |
| 125 | M16 | — | 5 | 3 | no | — |
| 126 | M16 | — | 5 | 6 | no | — |
| 127 | M16 | C2×AGL(1,𝔽5) | 1 | 3 | yes | complemented |
| 128 | C2×C8 | — | 5 | 3 | no | — |
| 129 | C2×C8 | — | 5 | 1 | no | — |
| 130 | C2×C8 | — | 5 | 6 | no | — |
| 131 | C2×C8 | — | 5 | 6 | no | — |
| 132 | C2×C8 | — | 5 | 3 | no | — |
| 133 | C2×C8 | — | 5 | 3 | no | — |
| 134 | C2×C8 | — | 5 | 1 | no | — |
| 135 | C2×C8 | — | 5 | 3 | no | — |
| 136 | C2×C8 | — | 5 | 3 | no | — |
| 137 | C2×C8 | C2×AGL(1,𝔽5) | 1 | 3 | yes | complemented |
| 138 | C4⋊C4 | — | 5 | 6 | no | — |
| 139 | C22⋊C4 | — | 5 | 6 | no | — |
| 140 | C42 | — | 5 | 3 | no | — |
| 141 | C42 | — | 5 | 1 | no | — |
| 142 | C42 | — | 5 | 6 | no | — |
| 143 | C2×C5 | — | 2 | 3 | no | — |
| 144 | C2×C5 | C22×C42 | 1 | 1 | yes | characteristic |
| 145 | D5 | — | 2 | 3 | no | — |
| 146 | D5 | #568 | 1 | 2 | yes | — |
| 147 | C23 | — | 5 | 3 | no | — |
| 148 | Q8 | — | 5 | 3 | no | — |
| 149 | Q8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 150 | D4 | — | 5 | 6 | no | — |
| 151 | D4 | — | 5 | 3 | no | — |
| 152 | D4 | C4×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 153 | C2×C4 | — | 5 | 3 | no | — |
| 154 | C2×C4 | — | 5 | 3 | no | — |
| 155 | C2×C4 | — | 5 | 3 | no | — |
| 156 | C2×C4 | — | 5 | 1 | no | — |
| 157 | C2×C4 | — | 5 | 3 | no | — |
| 158 | C2×C4 | — | 5 | 2 | no | — |
| 159 | C2×C4 | — | 5 | 6 | no | — |
| 160 | C2×C4 | — | 20 | 6 | no | — |
| 161 | C2×C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 162 | C8 | — | 5 | 2 | no | — |
| 163 | C8 | — | 5 | 6 | no | — |
| 164 | C8 | — | 5 | 1 | no | — |
| 165 | C8 | — | 5 | 3 | no | — |
| 166 | C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 167 | C8 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 168 | C5 | #2875 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 169 | C22 | — | 5 | 1 | no | — |
| 170 | C22 | — | 5 | 3 | no | — |
| 171 | C22 | — | 10 | 6 | no | — |
| 172 | C22 | C2×C4×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 173 | C4 | — | 5 | 1 | no | — |
| 174 | C4 | — | 5 | 3 | no | — |
| 175 | C4 | — | 10 | 2 | no | — |
| 176 | C4 | — | 10 | 6 | no | — |
| 177 | C4 | C2×C4×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 178 | C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 179 | C2 | — | 2 | 3 | no | — |
| 180 | C2 | — | 5 | 2 | no | — |
| 181 | C2 | — | 10 | 3 | no | — |
| 182 | C2 | C22×C4×AGL(1,𝔽5) | 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