Finite Group Databasebeta

← (C5⋊(D31⋊C5))⋊(C3⋊D3) #42124731

Subgroups — (C5⋊(D31⋊C5))⋊(C3⋊D3) #42124731

Order 27900.

288 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D15×ΓL(1,𝔽32)C211yesmaximal, characteristic, complemented
2C5×C3⋊(C3⋊(D31⋊C5))C211yesmaximal, characteristic, complemented
3C32×C5⋊(D31⋊C5)C211yesmaximal, characteristic, complemented
4D15⋊(D31⋊C5)—34nomaximal
5C32×C5×ΓL(1,𝔽32)C2211yescharacteristic, complemented, Hall
6D155⋊(C3⋊D3)C511yesmaximal, characteristic, complemented
7C3⋊D3×D31⋊C5—51nomaximal
8D15×ΓL(1,𝔽32)—34no—
9C5×C3⋊(D31⋊C5)—34no—
10C3×C5⋊(D31⋊C5)D314yescomplemented
11D5×D31⋊C5—91noHall
12C31×C3⋊D15C2×C511yescharacteristic, complemented
13C5×C3⋊D93C2×C511yescharacteristic, complemented
14C32×D155C2×C511yescharacteristic, complemented
15C3⋊D3×ΓL(1,𝔽32)—51no—
16C3⋊(C3⋊(D31⋊C5))—22no—
17C3⋊(C3⋊(D31⋊C5))D511yescharacteristic, complemented
18C32×D31⋊C5—51no—
19C3×C5×ΓL(1,𝔽32)C2×D314yescomplemented
20D15⋊D31—34no—
21D3×D31⋊C5—154no—
22C5⋊(D31⋊C5)C3⋊D311yescharacteristic, complemented
23D5×ΓL(1,𝔽32)—91no—
24C5×D31⋊C5—91no—
25C32×C5×C31C22×C511yescharacteristic, complemented
26C32×ΓL(1,𝔽32)—22no—
27C32×ΓL(1,𝔽32)C2×D511yescharacteristic, complemented
28C3⋊D3×D31—51noHall
29D15×C31—34no—
30C5×D93—34no—
31C3×D155C5×D314yescomplemented
32C3⋊(D31⋊C5)—34no—
33C3⋊(D31⋊C5)—68no—
34D3×ΓL(1,𝔽32)—154no—
35C3×D31⋊C5—54no—
36C5×D5×C3⋊D3—311nomaximal, Hall
37C5×ΓL(1,𝔽32)C2×C3⋊D311yescharacteristic, complemented, Hall
38D5×D31—91no—
39C2×D31⋊C5—451no—
40C3⋊D93C5×D511yescharacteristic, complemented
41C3⋊D3×C31—51no—
42C32×D31—51no—
43C3×C5×C31C2×C5×D314yescomplemented
44C3×ΓL(1,𝔽32)—28no—
45C3×ΓL(1,𝔽32)D3×D514yescomplemented
46C5×C3⋊D15—311no—
47C52×C3⋊D3—311no—
48C32×C5×D5—311no—
49D3×D31—154no—
50D155C5×C3⋊D311yescharacteristic, complemented
51C5×D31—91no—
52D5×C31—91no—
53C2×ΓL(1,𝔽32)—451no—
54D31⋊C5—51no—
55D31⋊C5—91no—
56D31⋊C5—182no—
57C5×D3×D5—934no—
58C32×C31C2×C5×D511yescharacteristic, complemented, Hall
59C32×C52—311noHall
60D93—34no—
61C3×D31—54no—
62D3×C31—154no—
63C2×C5×C3⋊D3—1551no—
64D5×C3⋊D3—311no—
65C5×C31C2×C5×C3⋊D311yescharacteristic, complemented
66ΓL(1,𝔽32)—22no—
67ΓL(1,𝔽32)D5×C3⋊D311yescharacteristic, complemented
68C5×D15—934no—
69C52×D3—934no—
70C3×C5×D5—314no—
71C2×D31—451noHall
72C2×C5×D5—2791noHall
73C3×C31C5×D3×D514yescomplemented
74C2×C32×C5—1551no—
75C3⋊D15D31⋊C511yescharacteristic, complemented
76C5×C3⋊D3—311no—
77C5×C3⋊D3—311no—
78C5×C3⋊D3—622no—
79C5×C3⋊D3—1551no—
80C32×D5—311no—
81C3×C52—314no—
82C2×C31—451no—
83D31—51no—
84D31—91no—
85C2×C5×D3—4654no—
86D3×D5—934no—
87C2×C52—2791no—
88C5×D5—2791no—
89C5×D5—311no—
90C32×C5—311no—
91C32×C5—622no—
92C32×C5C2×D31⋊C511yescharacteristic, complemented
93C2×C3⋊D3—1551noHall
94C31C5×D5×C3⋊D311yescharacteristic, complemented, Sylow
95C2×C3×C5—1554no—
96D15—34no—
97C3×D5—314no—
98C5×D3—934no—
99C5×D3—934no—
100C5×D3—1868no—
101C5×D3—4654no—
102C52—311noSylow
103C22×C5—13951no—
104C2×D5—2791no—
105C2×C32—1551no—
106C3⋊D3—51no—
107C3⋊D3—311no—
108C3×C5—314no—
109C3×C5—628no—
110C3×C5D3×D31⋊C514yescomplemented
111C2×D3—4654no—
112C2×C5—2791no—
113C2×C5—2791no—
114C2×C5—5582no—
115C2×C5—13951no—
116C2×C5—1551no—
117D5—91no—
118D5—311no—
119C32D5×D31⋊C511yescharacteristic, complemented, Sylow
120C2×C3—1554no—
121D3—154no—
122D3—934no—
123C5—311no—
124C5—622no—
125C5C3⋊D3×D31⋊C511yescharacteristic, complemented
126C22—13951noSylow
127C3D15⋊(D31⋊C5)14yescomplemented
128C2—2791no—
129C2—451no—
130C2—1551no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C5×C31C22×C5
LCSG ▷ C32×C5×C31
UCSC1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3⋊D15×ΓL(1,𝔽32)C2semidirect
2. C5×C3⋊(C3⋊(D31⋊C5))C2semidirect
3. C32×C5⋊(D31⋊C5)C2semidirect
5. C32×C5×ΓL(1,𝔽32)C22semidirect
6. D155⋊(C3⋊D3)C5semidirect
10. C3×C5⋊(D31⋊C5)D3semidirect
12. C31×C3⋊D15C2×C5semidirect
13. C5×C3⋊D93C2×C5semidirect
14. C32×D155C2×C5semidirect
17. C3⋊(C3⋊(D31⋊C5))D5semidirect
19. C3×C5×ΓL(1,𝔽32)C2×D3semidirect
22. C5⋊(D31⋊C5)C3⋊D3semidirect
25. C32×C5×C31C22×C5semidirect
27. C32×ΓL(1,𝔽32)C2×D5semidirect
31. C3×D155C5×D3semidirect
37. C5×ΓL(1,𝔽32)C2×C3⋊D3semidirect
40. C3⋊D93C5×D5semidirect
43. C3×C5×C31C2×C5×D3semidirect
45. C3×ΓL(1,𝔽32)D3×D5semidirect
50. D155C5×C3⋊D3semidirect
58. C32×C31C2×C5×D5semidirect
65. C5×C31C2×C5×C3⋊D3semidirect
67. ΓL(1,𝔽32)D5×C3⋊D3semidirect
73. C3×C31C5×D3×D5semidirect
75. C3⋊D15D31⋊C5semidirect
92. C32×C5C2×D31⋊C5semidirect
94. C31C5×D5×C3⋊D3semidirect
110. C3×C5D3×D31⋊C5semidirect
119. C32D5×D31⋊C5semidirect
125. C5C3⋊D3×D31⋊C5semidirect
127. C3D15⋊(D31⋊C5)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3⋊D15×ΓL(1,𝔽32)2
C5×C3⋊(C3⋊(D31⋊C5))2
C32×C5⋊(D31⋊C5)2
D15⋊(D31⋊C5)3
D155⋊(C3⋊D3)5
C3⋊D3×D31⋊C55
C5×D5×C3⋊D331