Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
232 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#92566 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C5×D52 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#92560 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D52⋊D5 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 5 | C2×C5×C5⋊D5 | — | 2 | 1 | no | — |
| 6 | C2×C5×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C52×D5 | — | 2 | 1 | no | — |
| 8 | #92566 | — | 2 | 1 | no | — |
| 9 | #92566 | C22 | 1 | 2 | yes | complemented |
| 10 | C5×D52 | — | 2 | 1 | no | — |
| 11 | C5×D52 | C22 | 1 | 2 | yes | complemented |
| 12 | #92560 | C22 | 1 | 2 | yes | complemented |
| 13 | C2×D52⋊C2 | — | 5 | 1 | no | maximal |
| 14 | C2×C53 | D4 | 1 | 1 | yes | characteristic, complemented |
| 15 | C5×C5⋊D5 | — | 2 | 2 | no | — |
| 16 | C5×C5⋊D5 | D4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C5×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 18 | C52×D5 | — | 2 | 2 | no | — |
| 19 | C22×C5×D5 | — | 10 | 1 | no | — |
| 20 | C2×D52 | — | 5 | 1 | no | — |
| 21 | C2×D52 | — | 10 | 1 | no | — |
| 22 | C2×D52 | D5 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×#1044 | — | 5 | 1 | no | — |
| 24 | C2×#1042 | — | 10 | 1 | no | — |
| 25 | D52⋊C2 | — | 5 | 4 | no | — |
| 26 | C53 | C2×D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 27 | C22×C52 | — | 10 | 1 | no | — |
| 28 | C2×C5⋊D5 | — | 2 | 1 | no | — |
| 29 | C2×C5⋊D5 | C2×D5 | 1 | 1 | yes | characteristic |
| 30 | C2×C5×D5 | — | 2 | 1 | no | — |
| 31 | C2×C5×D5 | — | 2 | 1 | no | — |
| 32 | C2×C5×D5 | — | 4 | 2 | no | — |
| 33 | C2×C5×D5 | — | 10 | 1 | no | — |
| 34 | C2×C5×D5 | — | 10 | 1 | no | — |
| 35 | C2×C5×D5 | — | 10 | 1 | no | — |
| 36 | C2×C5×D5 | — | 10 | 1 | no | — |
| 37 | C2×C5×D5 | — | 10 | 2 | no | — |
| 38 | C2×C5×D5 | — | 10 | 2 | no | — |
| 39 | C2×C5×D5 | — | 10 | 1 | no | — |
| 40 | C2×C5×D5 | — | 20 | 2 | no | — |
| 41 | D52 | — | 2 | 1 | no | — |
| 42 | D52 | — | 5 | 2 | no | — |
| 43 | D52 | — | 10 | 2 | no | — |
| 44 | D52 | — | 10 | 2 | no | — |
| 45 | D52 | — | 10 | 1 | no | — |
| 46 | D52 | C2×D5 | 1 | 2 | yes | complemented |
| 47 | #1044 | — | 5 | 2 | no | — |
| 48 | #1042 | — | 10 | 2 | no | — |
| 49 | C2×C22⋊D5 | — | 25 | 1 | no | maximal |
| 50 | C2×C52 | — | 2 | 1 | no | — |
| 51 | C2×C52 | — | 2 | 1 | no | — |
| 52 | C2×C52 | — | 2 | 1 | no | — |
| 53 | C2×C52 | — | 4 | 2 | no | — |
| 54 | C2×C52 | — | 4 | 2 | no | — |
| 55 | C2×C52 | — | 8 | 1 | no | — |
| 56 | C2×C52 | — | 10 | 2 | no | — |
| 57 | C2×C52 | C22⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 58 | C5⋊D5 | — | 2 | 2 | no | — |
| 59 | C5⋊D5 | C22⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 60 | C5⋊D5 | C22×D5 | 1 | 1 | yes | characteristic |
| 61 | C5×D5 | — | 2 | 2 | no | — |
| 62 | C5×D5 | — | 2 | 2 | no | — |
| 63 | C5×D5 | — | 4 | 4 | no | — |
| 64 | C5×D5 | — | 10 | 1 | no | — |
| 65 | C5×D5 | — | 10 | 1 | no | — |
| 66 | C5×D5 | — | 10 | 1 | no | — |
| 67 | C5×D5 | — | 10 | 1 | no | — |
| 68 | C5×D5 | — | 10 | 1 | no | — |
| 69 | C5×D5 | — | 10 | 1 | no | — |
| 70 | C5×D5 | — | 10 | 2 | no | — |
| 71 | C5×D5 | — | 10 | 2 | no | — |
| 72 | C5×D5 | — | 20 | 4 | no | — |
| 73 | C23×C5 | — | 25 | 1 | no | — |
| 74 | C22×D5 | — | 10 | 1 | no | — |
| 75 | C22×D5 | — | 25 | 1 | no | — |
| 76 | C22×D5 | — | 50 | 1 | no | — |
| 77 | C2×AGL(1,𝔽5) | — | 50 | 1 | no | — |
| 78 | C22⋊D5 | — | 25 | 4 | no | — |
| 79 | C2×Dic5 | — | 25 | 1 | no | — |
| 80 | C52 | — | 2 | 1 | no | — |
| 81 | C52 | — | 2 | 1 | no | — |
| 82 | C52 | — | 2 | 1 | no | — |
| 83 | C52 | — | 4 | 2 | no | — |
| 84 | C52 | — | 4 | 2 | no | — |
| 85 | C52 | — | 8 | 1 | no | — |
| 86 | C52 | C2×C22⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 87 | C22×C5 | — | 10 | 1 | no | — |
| 88 | C22×C5 | — | 10 | 1 | no | — |
| 89 | C22×C5 | — | 20 | 2 | no | — |
| 90 | C22×C5 | — | 25 | 2 | no | — |
| 91 | C22×C5 | — | 25 | 1 | no | — |
| 92 | C22×C5 | — | 50 | 1 | no | — |
| 93 | C22×C5 | — | 50 | 1 | no | — |
| 94 | C2×D5 | — | 2 | 1 | no | — |
| 95 | C2×D5 | — | 10 | 2 | no | — |
| 96 | C2×D5 | — | 10 | 1 | no | — |
| 97 | C2×D5 | — | 10 | 1 | no | — |
| 98 | C2×D5 | — | 10 | 1 | no | — |
| 99 | C2×D5 | — | 10 | 2 | no | — |
| 100 | C2×D5 | — | 10 | 1 | no | — |
| 101 | C2×D5 | — | 10 | 1 | no | — |
| 102 | C2×D5 | — | 20 | 2 | no | — |
| 103 | C2×D5 | — | 25 | 2 | no | — |
| 104 | C2×D5 | — | 50 | 2 | no | — |
| 105 | C2×D5 | — | 50 | 1 | no | — |
| 106 | C2×D5 | — | 50 | 2 | no | — |
| 107 | AGL(1,𝔽5) | — | 50 | 2 | no | — |
| 108 | Dic5 | — | 25 | 2 | no | — |
| 109 | C2×D4 | — | 125 | 1 | no | Sylow |
| 110 | C2×C5 | — | 2 | 1 | no | — |
| 111 | C2×C5 | — | 2 | 1 | no | — |
| 112 | C2×C5 | — | 2 | 1 | no | — |
| 113 | C2×C5 | — | 4 | 2 | no | — |
| 114 | C2×C5 | — | 4 | 2 | no | — |
| 115 | C2×C5 | — | 8 | 1 | no | — |
| 116 | C2×C5 | — | 10 | 2 | no | — |
| 117 | C2×C5 | — | 10 | 2 | no | — |
| 118 | C2×C5 | — | 20 | 4 | no | — |
| 119 | C2×C5 | — | 25 | 1 | no | — |
| 120 | C2×C5 | — | 25 | 1 | no | — |
| 121 | C2×C5 | — | 50 | 2 | no | — |
| 122 | C2×C5 | D52⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 123 | D5 | — | 2 | 2 | no | — |
| 124 | D5 | — | 10 | 1 | no | — |
| 125 | D5 | — | 10 | 1 | no | — |
| 126 | D5 | — | 10 | 1 | no | — |
| 127 | D5 | — | 10 | 1 | no | — |
| 128 | D5 | — | 10 | 1 | no | — |
| 129 | D5 | — | 10 | 1 | no | — |
| 130 | D5 | — | 10 | 2 | no | — |
| 131 | D5 | — | 10 | 2 | no | — |
| 132 | D5 | — | 20 | 4 | no | — |
| 133 | C23 | — | 25 | 1 | no | — |
| 134 | C23 | — | 125 | 1 | no | — |
| 135 | D4 | — | 125 | 4 | no | — |
| 136 | C2×C4 | — | 125 | 1 | no | — |
| 137 | C5 | — | 2 | 1 | no | — |
| 138 | C5 | — | 2 | 1 | no | — |
| 139 | C5 | — | 2 | 1 | no | — |
| 140 | C5 | — | 4 | 2 | no | — |
| 141 | C5 | — | 4 | 2 | no | — |
| 142 | C5 | — | 8 | 1 | no | — |
| 143 | C5 | C2×D52⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 144 | C22 | — | 10 | 1 | no | — |
| 145 | C22 | — | 25 | 2 | no | — |
| 146 | C22 | — | 25 | 1 | no | — |
| 147 | C22 | — | 50 | 1 | no | — |
| 148 | C22 | — | 50 | 1 | no | — |
| 149 | C22 | — | 125 | 2 | no | — |
| 150 | C22 | — | 250 | 1 | no | — |
| 151 | C4 | — | 125 | 2 | no | — |
| 152 | C2 | — | 10 | 2 | no | — |
| 153 | C2 | — | 25 | 1 | no | — |
| 154 | C2 | — | 25 | 1 | no | — |
| 155 | C2 | — | 50 | 2 | no | — |
| 156 | C2 | D52⋊D5 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D52⋊D5 |
| commutator, cosocle | C5×C5⋊D5 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C53 | D4 |
| hypocenter | C53 | C2×D4 |
| LCS | G ▷ C5×C5⋊D5 ▷ C53 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#92566 | C2 | semidirect |
| 2. C2×C5×D52 | C2 | semidirect |
| 3. C2×#92560 | C2 | semidirect |
| 4. D52⋊D5 | C2 | direct (N × H) |
| 9. #92566 | C22 | semidirect |
| 11. C5×D52 | C22 | semidirect |
| 12. #92560 | C22 | semidirect |
| 14. C2×C53 | D4 | semidirect |
| 16. C5×C5⋊D5 | D4 | semidirect |
| 22. C2×D52 | D5 | semidirect |
| 26. C53 | C2×D4 | semidirect |
| 46. D52 | C2×D5 | semidirect |
| 57. C2×C52 | C22⋊D5 | semidirect |
| 59. C5⋊D5 | C22⋊D5 | semidirect |
| 86. C52 | C2×C22⋊D5 | semidirect |
| 122. C2×C5 | D52⋊C2 | semidirect |
| 143. C5 | C2×D52⋊C2 | semidirect |
| 156. C2 | D52⋊D5 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#92566 | 2 |
| C2×C5×D52 | 2 |
| C2×#92560 | 2 |
| D52⋊D5 | 2 |
| C2×D52⋊C2 | 5 |
| C2×C22⋊D5 | 25 |