Finite Group DatabasebetaFinite Group DatabasebetaOrder 2880.
276 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 35 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 | C5×Ω+(4,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5⋊#5950 | C3 | 1 | 2 | yes | maximal, complemented |
| 3 | #11756944 | C3 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×C5⋊(#41⋊C3) | — | 4 | 2 | no | maximal |
| 5 | (#41⋊C3)⋊A4 | — | 5 | 1 | no | maximal, Hall |
| 6 | C5×#997 | C2×C3 | 1 | 2 | yes | complemented |
| 7 | C5×#994 | C2×C3 | 1 | 2 | yes | complemented |
| 8 | #4150⋊C3 | — | 3 | 2 | no | — |
| 9 | D5×#41⋊C3 | — | 3 | 2 | no | — |
| 10 | C3×C5×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 11 | #66285 | C32 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | Ω+(4,𝔽3) | D5 | 1 | 1 | yes | characteristic, complemented |
| 13 | C3×#713 | — | 4 | 2 | no | — |
| 14 | C5×#41⋊C3 | — | 3 | 2 | no | — |
| 15 | A4×Dic5 | — | 12 | 2 | no | — |
| 16 | D5×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 17 | C5⋊(#41⋊C3) | — | 4 | 4 | no | — |
| 18 | C5⋊(#41⋊C3) | A4 | 1 | 2 | yes | complemented |
| 19 | #5950 | — | 5 | 2 | no | — |
| 20 | #5945 | — | 5 | 2 | no | — |
| 21 | C32×Dic5 | — | 16 | 1 | no | — |
| 22 | C5×H(5,𝔽2) | C2×C32 | 1 | 1 | yes | characteristic |
| 23 | #4150 | — | 3 | 2 | no | — |
| 24 | D5×#41 | — | 3 | 2 | no | — |
| 25 | #4144 | — | 9 | 1 | no | — |
| 26 | C2×#711 | — | 9 | 1 | no | — |
| 27 | C3×#41⋊C3 | — | 20 | 2 | no | — |
| 28 | C2×C5×A4 | — | 12 | 2 | no | — |
| 29 | C3×C5×Q8 | — | 4 | 2 | no | — |
| 30 | C3×D20 | — | 12 | 2 | no | — |
| 31 | C3×C4×D5 | — | 12 | 2 | no | — |
| 32 | C5×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 33 | C5×SL(2,𝔽3) | C2×A4 | 1 | 2 | yes | — |
| 34 | #997 | C3×D5 | 1 | 2 | yes | complemented |
| 35 | 2−1+4⋊C3 | — | 15 | 2 | no | — |
| 36 | #994 | C3×D5 | 1 | 2 | yes | complemented |
| 37 | C2×#41⋊C3 | — | 15 | 2 | no | — |
| 38 | C2×C32×C5 | — | 16 | 1 | no | — |
| 39 | C5×#41 | — | 3 | 2 | no | — |
| 40 | C2×C5×D4 | — | 9 | 1 | no | — |
| 41 | C2×C22⋊D5 | — | 9 | 2 | no | — |
| 42 | C22×Dic5 | — | 3 | 2 | no | — |
| 43 | #713 | C3×A4 | 1 | 2 | yes | — |
| 44 | Q8×D5 | — | 3 | 2 | no | — |
| 45 | #711 | — | 9 | 2 | no | — |
| 46 | #711 | — | 9 | 4 | no | — |
| 47 | D4×D5 | — | 9 | 2 | no | — |
| 48 | #709 | — | 9 | 2 | no | — |
| 49 | C2×C4×D5 | — | 9 | 1 | no | — |
| 50 | C2×Dic10 | — | 9 | 1 | no | — |
| 51 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 52 | #585 | — | 5 | 1 | no | Sylow |
| 53 | C2×C3×D5 | — | 12 | 2 | no | — |
| 54 | C5×A4 | — | 12 | 2 | no | — |
| 55 | C3×C4×C5 | — | 12 | 2 | no | — |
| 56 | C3×Dic5 | — | 4 | 2 | no | — |
| 57 | C3×Dic5 | — | 16 | 2 | no | — |
| 58 | C3×#41 | — | 20 | 2 | no | — |
| 59 | #41⋊C3 | — | 3 | 2 | no | — |
| 60 | #41⋊C3 | — | 5 | 2 | no | — |
| 61 | #41⋊C3 | — | 20 | 4 | no | — |
| 62 | C2×SL(2,𝔽3) | — | 15 | 2 | no | — |
| 63 | C4×A4 | — | 60 | 2 | no | — |
| 64 | C32×C5 | — | 16 | 1 | no | Hall |
| 65 | C23×C5 | — | 3 | 2 | no | — |
| 66 | C22×D5 | — | 9 | 1 | no | — |
| 67 | C5×Q8 | C2×C3×A4 | 1 | 2 | yes | — |
| 68 | C5×D4 | — | 9 | 2 | no | — |
| 69 | C2×C4×C5 | — | 9 | 1 | no | — |
| 70 | C22⋊D5 | — | 9 | 4 | no | — |
| 71 | C2×Dic5 | — | 9 | 1 | no | — |
| 72 | C2×Dic5 | — | 9 | 2 | no | — |
| 73 | D20 | — | 3 | 2 | no | — |
| 74 | C4×D5 | — | 3 | 2 | no | — |
| 75 | C4×D5 | — | 9 | 2 | no | — |
| 76 | Dic10 | — | 9 | 2 | no | — |
| 77 | C32×C4 | — | 80 | 1 | no | — |
| 78 | 2−1+4 | — | 15 | 2 | no | — |
| 79 | H(5,𝔽2) | — | 45 | 1 | no | — |
| 80 | H(5,𝔽2) | C32×D5 | 1 | 1 | yes | characteristic |
| 81 | C2×#41 | — | 15 | 2 | no | — |
| 82 | C2×#41 | — | 45 | 1 | no | — |
| 83 | C2×C3×C5 | — | 4 | 2 | no | — |
| 84 | C2×C3×C5 | — | 16 | 2 | no | — |
| 85 | C3×D5 | — | 24 | 2 | no | — |
| 86 | C2×A4 | — | 12 | 2 | no | — |
| 87 | C3×Q8 | — | 4 | 2 | no | — |
| 88 | C3×D4 | — | 60 | 2 | no | — |
| 89 | C2×C3×C4 | — | 60 | 2 | no | — |
| 90 | SL(2,𝔽3) | — | 4 | 4 | no | — |
| 91 | SL(2,𝔽3) | D5×A4 | 1 | 2 | yes | — |
| 92 | C22×C5 | — | 9 | 1 | no | — |
| 93 | C22×C5 | — | 12 | 2 | no | — |
| 94 | C2×D5 | — | 3 | 2 | no | — |
| 95 | C2×D5 | — | 36 | 1 | no | — |
| 96 | C4×C5 | — | 3 | 2 | no | — |
| 97 | Dic5 | — | 9 | 1 | no | — |
| 98 | Dic5 | A42 | 1 | 1 | yes | characteristic |
| 99 | C2×C32 | — | 16 | 1 | no | — |
| 100 | #41 | — | 3 | 2 | no | — |
| 101 | #41 | — | 5 | 2 | no | — |
| 102 | #41 | — | 45 | 2 | no | — |
| 103 | #41 | — | 45 | 2 | no | — |
| 104 | #41 | — | 45 | 4 | no | — |
| 105 | C2×Q8 | — | 15 | 2 | no | — |
| 106 | C2×Q8 | — | 45 | 1 | no | — |
| 107 | C2×D4 | — | 9 | 1 | no | — |
| 108 | C2×D4 | — | 45 | 2 | no | — |
| 109 | C2×D4 | — | 45 | 2 | no | — |
| 110 | C22×C4 | — | 15 | 2 | no | — |
| 111 | C22×C4 | — | 45 | 1 | no | — |
| 112 | C3×C5 | — | 4 | 2 | no | — |
| 113 | C3×C5 | — | 16 | 2 | no | — |
| 114 | C22×C3 | — | 60 | 2 | no | — |
| 115 | A4 | — | 12 | 2 | no | — |
| 116 | C3×C4 | — | 12 | 2 | no | — |
| 117 | C3×C4 | — | 20 | 2 | no | — |
| 118 | C3×C4 | — | 80 | 2 | no | — |
| 119 | C2×C5 | — | 18 | 1 | no | — |
| 120 | C2×C5 | C2×A42 | 1 | 1 | yes | characteristic |
| 121 | D5 | — | 6 | 2 | no | — |
| 122 | C32 | — | 16 | 1 | no | Sylow |
| 123 | C23 | — | 3 | 2 | no | — |
| 124 | C23 | — | 45 | 1 | no | — |
| 125 | Q8 | — | 45 | 2 | no | — |
| 126 | Q8 | C3×D5×A4 | 1 | 2 | yes | — |
| 127 | D4 | — | 9 | 2 | no | — |
| 128 | D4 | — | 15 | 2 | no | — |
| 129 | D4 | — | 45 | 4 | no | — |
| 130 | C2×C4 | — | 9 | 1 | no | — |
| 131 | C2×C4 | — | 15 | 2 | no | — |
| 132 | C2×C4 | — | 45 | 2 | no | — |
| 133 | C2×C4 | — | 45 | 1 | no | — |
| 134 | C2×C4 | — | 45 | 2 | no | — |
| 135 | C2×C3 | — | 4 | 2 | no | — |
| 136 | C2×C3 | — | 16 | 2 | no | — |
| 137 | C2×C3 | — | 120 | 2 | no | — |
| 138 | C5 | (#41⋊C3)⋊A4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 139 | C22 | — | 9 | 1 | no | — |
| 140 | C22 | — | 12 | 2 | no | — |
| 141 | C22 | — | 15 | 2 | no | — |
| 142 | C22 | — | 180 | 1 | no | — |
| 143 | C4 | — | 3 | 2 | no | — |
| 144 | C4 | — | 5 | 1 | no | — |
| 145 | C4 | — | 45 | 1 | no | — |
| 146 | C3 | — | 4 | 2 | no | — |
| 147 | C3 | — | 16 | 2 | no | — |
| 148 | C2 | — | 18 | 1 | no | — |
| 149 | C2 | — | 30 | 2 | no | — |
| 150 | C2 | D5×A42 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×A42 |
| commutator, Fitting, cosocle, hypocenter | C5×H(5,𝔽2) | C2×C32 |
| socle | C2×C5 | C2×A42 |
| LCS | G ▷ C5×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×Ω+(4,𝔽3) | C2 | semidirect |
| 2. C5⋊#5950 | C3 | semidirect |
| 3. #11756944 | C3 | semidirect |
| 6. C5×#997 | C2×C3 | semidirect |
| 7. C5×#994 | C2×C3 | semidirect |
| 11. #66285 | C32 | semidirect |
| 12. Ω+(4,𝔽3) | D5 | semidirect |
| 18. C5⋊(#41⋊C3) | A4 | semidirect |
| 34. #997 | C3×D5 | semidirect |
| 36. #994 | C3×D5 | semidirect |
| 138. C5 | (#41⋊C3)⋊A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×Ω+(4,𝔽3) | 2 |
| C5⋊#5950 | 3 |
| #11756944 | 3 |
| C3×C5⋊(#41⋊C3) | 4 |
| (#41⋊C3)⋊A4 | 5 |