Finite Group Databasebeta

← C2×C32×C22⋊(C5⋊D5) #72230635

Subgroups — C2×C32×C22⋊(C5⋊D5) #72230635

Order 3600.

1296 conjugacy classes of subgroups; only the 450 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 450 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
1C23×C32×C52C211yesmaximal, characteristic, complemented
2C22×C32×C5⋊D5C211yesmaximal, characteristic, complemented
3C32×C22⋊(C5⋊D5)C214yesmaximal, direct product factor
4C2×C32×#1039C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊(C5⋊D5)C314yesmaximal, direct product factor
6C22×C32×C52—22no—
7C22×C32×C52C2211yescharacteristic
8C22×C32×C52C2212yescomplemented
9C2×C32×C5⋊D5—22no—
10C2×C32×C5⋊D5C2212yescomplemented
11C32×#1039C2212yescomplemented
12C2×C32×C22⋊D5—56nomaximal
13C23×C3×C52C2×C314yescomplemented
14C22×C3×C5⋊D5C2×C314yescomplemented
15C3×C22⋊(C5⋊D5)C2×C3116yesdirect product factor
16C2×C3×#1039C2×C314yescomplemented
17C2×C32×C52—22no—
18C2×C32×C52D412yescomplemented
19C2×C32×C52C2311yescharacteristic
20C32×C5⋊D5—22no—
21C2×C22⋊(C5⋊D5)C3211yescharacteristic, direct product factor, Hall
22C23×C32×C5D516yescomplemented
23C22×C32×D5—56no—
24C32×C22⋊D5—524no—
25C2×C32×Dic5—56no—
26C22×C3×C52—28no—
27C22×C3×C52C22×C314yes—
28C22×C3×C52C22×C318yescomplemented
29C2×C3×C5⋊D5—28no—
30C2×C3×C5⋊D5C22×C318yescomplemented
31C3×#1039C22×C318yescomplemented
32C2×C3×C22⋊D5—524no—
33C32×C52C2×D411yescharacteristic, complemented, Hall
34C23×C52C2×C3211yescharacteristic, complemented
35C22×C5⋊D5C2×C3211yescharacteristic, complemented
36C22⋊(C5⋊D5)C2×C3214yesdirect product factor
37C2×#1039C2×C3211yescharacteristic, complemented
38C22×C32×C5—212no—
39C22×C32×C5C2×D516yes—
40C22×C32×C5C2×D5112yescomplemented
41C2×C32×D5—512no—
42C2×C32×D5—1012no—
43C32×Dic5—512no—
44C2×C3×C52—28no—
45C2×C3×C52C3×D418yescomplemented
46C2×C3×C52C23×C314yes—
47C3×C5⋊D5—28no—
48C2×C32×D4—251noHall
49C23×C3×C5C3×D5124yescomplemented
50C22×C3×D5—524no—
51C3×C22⋊D5—596no—
52C2×C3×Dic5—524no—
53C22×C52—22no—
54C22×C52C22×C3211yescharacteristic
55C22×C52C22×C3212yescomplemented
56C2×C5⋊D5—22no—
57C2×C5⋊D5C22×C3212yescomplemented
58#1039C22×C3212yescomplemented
59C2×C32×C5—212no—
60C2×C32×C5C22⋊D5112yescomplemented
61C2×C32×C5C22×D516yes—
62C32×D5—1012no—
63C2×C22⋊D5—56no—
64C3×C52C2×C3×D414yescomplemented
65C23×C32—251no—
66C23×C32C5⋊D511yescharacteristic, complemented
67C32×D4—254no—
68C2×C32×C4—251no—
69C22×C3×C5—248no—
70C22×C3×C5C2×C3×D5124yes—
71C22×C3×C5C2×C3×D5148yescomplemented
72C2×C3×D5—548no—
73C2×C3×D5—1048no—
74C3×Dic5—548no—
75C2×C52—22no—
76C2×C52C32×D412yescomplemented
77C2×C52C23×C3211yescharacteristic
78C5⋊D5—22no—
79C2×C3×D4—254no—
80C32×C5C2×C22⋊D516yescomplemented
81C23×C5C32×D516yescomplemented
82C22×D5—56no—
83C22⋊D5—524no—
84C2×Dic5—56no—
85C22×C32—22no—
86C22×C32—252no—
87C22×C32—502no—
88C22×C32C2×C5⋊D511yescharacteristic
89C22×C32C2×C5⋊D512yescomplemented
90C32×C4—252no—
91C2×C3×C5—248no—
92C2×C3×C5C3×C22⋊D5148yescomplemented
93C2×C3×C5C22×C3×D5124yes—
94C3×D5—1048no—
95C52C2×C32×D411yescharacteristic, complemented, Sylow
96C23×C3—254no—
97C23×C3C3×C5⋊D514yescomplemented
98C3×D4—2516no—
99C2×C3×C4—254no—
100C22×C5—212no—
101C22×C5C2×C32×D516yes—
102C22×C5C2×C32×D5112yescomplemented
103C2×D5—512no—
104C2×D5—1012no—
105Dic5—512no—
106C2×C32—22no—
107C2×C32—502no—
108C2×C32C22⋊(C5⋊D5)12yesdirect product factor
109C2×C32C22×C5⋊D511yescharacteristic
110C2×D4—251noSylow
111C3×C5C2×C3×C22⋊D5124yescomplemented
112C22×C3—28no—
113C22×C3—258no—
114C22×C3—508no—
115C22×C3C2×C3×C5⋊D514yes—
116C22×C3C2×C3×C5⋊D518yescomplemented
117C3×C4—258no—
118C2×C5—212no—
119C2×C5C32×C22⋊D5112yescomplemented
120C2×C5C22×C32×D516yes—
121D5—1012no—
122C32C2×C22⋊(C5⋊D5)11yescharacteristic, direct product factor, Sylow
123C23—251no—
124C23C32×C5⋊D511yescharacteristic, complemented
125D4—254no—
126C2×C4—251no—
127C2×C3—28no—
128C2×C3—508no—
129C2×C3C3×C22⋊(C5⋊D5)18yesdirect product factor
130C2×C3C22×C3×C5⋊D514yes—
131C5C2×C32×C22⋊D516yescomplemented
132C22—22no—
133C22—252no—
134C22—502no—
135C22C2×C32×C5⋊D511yescharacteristic
136C22C2×C32×C5⋊D512yescomplemented
137C4—252no—
138C3C2×C3×C22⋊(C5⋊D5)14yesdirect product factor
139C2—22no—
140C2—502no—
141C2C32×C22⋊(C5⋊D5)12yesdirect product factor
142C2C22×C32×C5⋊D511yescharacteristic
subgroupG/H
centerC22×C32C2×C5⋊D5
commutator, cosocleC2×C52C23×C32
FrattiniC2C22×C32×C5⋊D5
FittingC23×C32×C52C2
socleC22×C32×C52C22
hypercenterC23×C32C5⋊D5
hypocenterC52C2×C32×D4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22×C32 ◁ C23×C32

Complements

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

NH ≅ G/Ntype
1. C23×C32×C52C2semidirect
2. C22×C32×C5⋊D5C2semidirect
3. C32×C22⋊(C5⋊D5)C2direct (N × H)
4. C2×C32×#1039C2semidirect
5. C2×C3×C22⋊(C5⋊D5)C3direct (N × H)
8. C22×C32×C52C22semidirect
10. C2×C32×C5⋊D5C22semidirect
11. C32×#1039C22semidirect
13. C23×C3×C52C2×C3semidirect
14. C22×C3×C5⋊D5C2×C3semidirect
15. C3×C22⋊(C5⋊D5)C2×C3direct (N × H)
16. C2×C3×#1039C2×C3semidirect
18. C2×C32×C52D4semidirect
21. C2×C22⋊(C5⋊D5)C32direct (N × H)
22. C23×C32×C5D5semidirect
28. C22×C3×C52C22×C3semidirect
30. C2×C3×C5⋊D5C22×C3semidirect
31. C3×#1039C22×C3semidirect
33. C32×C52C2×D4semidirect
34. C23×C52C2×C32semidirect
35. C22×C5⋊D5C2×C32semidirect
36. C22⋊(C5⋊D5)C2×C32direct (N × H)
37. C2×#1039C2×C32semidirect
40. C22×C32×C5C2×D5semidirect
45. C2×C3×C52C3×D4semidirect
49. C23×C3×C5C3×D5semidirect
55. C22×C52C22×C32semidirect
57. C2×C5⋊D5C22×C32semidirect
58. #1039C22×C32semidirect
60. C2×C32×C5C22⋊D5semidirect
64. C3×C52C2×C3×D4semidirect
66. C23×C32C5⋊D5semidirect
71. C22×C3×C5C2×C3×D5semidirect
76. C2×C52C32×D4semidirect
80. C32×C5C2×C22⋊D5semidirect
81. C23×C5C32×D5semidirect
89. C22×C32C2×C5⋊D5semidirect
92. C2×C3×C5C3×C22⋊D5semidirect
95. C52C2×C32×D4semidirect
97. C23×C3C3×C5⋊D5semidirect
102. C22×C5C2×C32×D5semidirect
108. C2×C32C22⋊(C5⋊D5)direct (N × H)
111. C3×C5C2×C3×C22⋊D5semidirect
116. C22×C3C2×C3×C5⋊D5semidirect
119. C2×C5C32×C22⋊D5semidirect
122. C32C2×C22⋊(C5⋊D5)direct (N × H)
124. C23C32×C5⋊D5semidirect
129. C2×C3C3×C22⋊(C5⋊D5)direct (N × H)
131. C5C2×C32×C22⋊D5semidirect
136. C22C2×C32×C5⋊D5semidirect
138. C3C2×C3×C22⋊(C5⋊D5)direct (N × H)
141. C2C32×C22⋊(C5⋊D5)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×C522
C22×C32×C5⋊D52
C32×C22⋊(C5⋊D5)2
C2×C32×#10392
C2×C3×C22⋊(C5⋊D5)3
C2×C32×C22⋊D55