Finite Group Databasebeta

← C2×#11717250 #13107318

Subgroups — C2×#11717250 #13107318

Order 1600.

1704 conjugacy classes of subgroups; only the 419 normal subgroups are shown.

normal subgroups only

This group has 1704 conjugacy classes of subgroups — too many to draw. Only its 419 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×C4×C52C211yesmaximal, characteristic, complemented
2C22×C22⋊(C5⋊D5)C212yesmaximal, complemented
3C22×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
4#11717250C218yesmaximal, direct product factor
5C2×#87808C212yesmaximal, complemented
6C2×C4⋊#1039C211yesmaximal, characteristic, complemented
7C24×C52C2211yescharacteristic
8C23×C5⋊D5C2212yes—
9C22×C4×C52—24no—
10C22×C4×C52C2211yescharacteristic
11C22×C4×C52C2211yescharacteristic
12C22×C4×C52C2214yescomplemented
13C2×C22⋊(C5⋊D5)—28no—
14C2×C22⋊(C5⋊D5)C2218yescomplemented
15C22×#1039C2212yes—
16C2×C4⋊(C5⋊D5)—24no—
17C2×C4⋊(C5⋊D5)C2214yescomplemented
18#87808C2218yescomplemented
19C4⋊#1039C2214yescomplemented
20C2×#4076—56nomaximal
21C23×C52—22no—
22C23×C52—22no—
23C23×C52D414yescomplemented
24C23×C52C2312yes—
25C23×C52C2311yescharacteristic
26C22×C5⋊D5—24no—
27C22×C5⋊D5—24no—
28C22×C5⋊D5—24no—
29C22×C5⋊D5C2314yes—
30C2×C4×C52—22no—
31C2×C4×C52—28no—
32C2×C4×C52D414yescomplemented
33C2×C4×C52C2312yes—
34C2×C4×C52C2312yes—
35C22⋊(C5⋊D5)—216no—
36C2×#1039—24no—
37C2×#1039C2314yes—
38C4⋊(C5⋊D5)—28no—
39C23×C4×C5D516yescomplemented
40C22×C22⋊D5—512no—
41C22×D20—56no—
42#4076—548no—
43C2×#685—512no—
44C2×C4⋊Dic5—56no—
45C22×C52—28no—
46C22×C52—22no—
47C22×C52—22no—
48C22×C52C2×D412yes—
49C22×C52C2×D412yes—
50C22×C52C2×D414yescomplemented
51C22×C52#4112yes—
52C22×C52C2411yescharacteristic
53C2×C5⋊D5—24no—
54C2×C5⋊D5—24no—
55C2×C5⋊D5—24no—
56C2×C5⋊D5—48no—
57C4×C52—22no—
58C4×C52C2×D414yescomplemented
59#1039—24no—
60C24×C5C2×D516yes—
61C23×D5—512no—
62C22×C4×C5—224no—
63C22×C4×C5C2×D516yes—
64C22×C4×C5C2×D516yes—
65C22×C4×C5C2×D5124yescomplemented
66C2×C22⋊D5—548no—
67C2×C22⋊D5—1048no—
68C22×Dic5—512no—
69C2×D20—524no—
70C2×D20—1024no—
71#685—548no—
72C4⋊Dic5—524no—
73C2×#121—251noSylow
74C2×C52—24no—
75C2×C52#12114yescomplemented
76C2×C52C22×D411yescharacteristic
77C2×C52C22×D411yescharacteristic
78C2×C52C2×#4111yescharacteristic
79C5⋊D5—44no—
80C23×C5—212no—
81C23×C5—212no—
82C23×C5D20124yescomplemented
83C23×C5C22×D5112yes—
84C23×C5C22×D516yes—
85C22×D5—524no—
86C22×D5—1024no—
87C22×D5—1024no—
88C22×D5—1024no—
89C2×C4×C5—212no—
90C2×C4×C5—248no—
91C2×C4×C5C22⋊D5124yescomplemented
92C2×C4×C5C22×D5112yes—
93C2×C4×C5C22×D5112yes—
94C22⋊D5—1096no—
95C2×Dic5—524no—
96C2×Dic5—1024no—
97D20—1048no—
98C22×D4—252no—
99C22×D4—251no—
100C23×C4C5⋊D511yescharacteristic, complemented
101#121—258no—
102C2×C4⋊C4—251no—
103C2×C22⋊C4—252no—
104C52C2×#12111yescharacteristic, complemented, Sylow
105C22×C5—212no—
106C22×C5—212no—
107C22×C5—248no—
108C22×C5C2×D20112yes—
109C22×C5C2×D20124yescomplemented
110C22×C5#709112yes—
111C22×C5C2×C22⋊D5112yes—
112C22×C5C23×D516yes—
113C2×D5—1024no—
114C2×D5—1024no—
115C2×D5—1024no—
116C2×D5—2048no—
117C4×C5—212no—
118C4×C5C2×C22⋊D5124yescomplemented
119Dic5—1024no—
120C24—252no—
121C24C2×C5⋊D511yescharacteristic
122C2×D4—258no—
123C2×D4—254no—
124C2×D4—508no—
125C2×D4—504no—
126C22×C4—24no—
127C22×C4—252no—
128C22×C4C2×C5⋊D511yescharacteristic
129C22×C4C2×C5⋊D511yescharacteristic
130C22×C4C2×C5⋊D514yescomplemented
131C4⋊C4—254no—
132C22⋊C4—258no—
133C2×C5—224no—
134C2×C5#4076124yescomplemented
135C2×C5C22×D2016yes—
136C2×C5C2×#70916yes—
137C2×C5C22×C22⋊D516yes—
138D5—2024no—
139C23—22no—
140C23—22no—
141C23—254no—
142C23—504no—
143C23—504no—
144C23—504no—
145C23C4⋊(C5⋊D5)14yescomplemented
146C23C22×C5⋊D512yes—
147C23C22×C5⋊D511yescharacteristic
148D4—5016no—
149D4—508no—
150C2×C4—22no—
151C2×C4—28no—
152C2×C4—254no—
153C2×C4—504no—
154C2×C4C22⋊(C5⋊D5)14yescomplemented
155C2×C4C22×C5⋊D512yes—
156C2×C4C22×C5⋊D512yes—
157C5C2×#407616yescomplemented
158C22—28no—
159C22—22no—
160C22—22no—
161C22—504no—
162C22—504no—
163C22—504no—
164C22—1008no—
165C22C2×C4⋊(C5⋊D5)12yes—
166C22C2×C4⋊(C5⋊D5)14yescomplemented
167C22#8790012yes—
168C22C2×C22⋊(C5⋊D5)12yes—
169C22C23×C5⋊D511yescharacteristic
170C4—22no—
171C4—504no—
172C4C2×C22⋊(C5⋊D5)14yescomplemented
173C2—24no—
174C2—1004no—
175C2#1171725014yesdirect product factor
176C2C22×C4⋊(C5⋊D5)11yescharacteristic
177C2C2×#8790011yescharacteristic
178C2C22×C22⋊(C5⋊D5)11yescharacteristic
subgroupG/H
centerC23C22×C5⋊D5
commutator, cosocleC22×C52C24
FrattiniC22C23×C5⋊D5
FittingC23×C4×C52C2
socleC23×C52C23
hypercenterC23×C4C5⋊D5
hypocenterC52C2×#121
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×C52C2semidirect
2. C22×C22⋊(C5⋊D5)C2semidirect
3. C22×C4⋊(C5⋊D5)C2semidirect
4. #11717250C2direct (N × H)
5. C2×#87808C2semidirect
6. C2×C4⋊#1039C2semidirect
12. C22×C4×C52C22semidirect
14. C2×C22⋊(C5⋊D5)C22semidirect
17. C2×C4⋊(C5⋊D5)C22semidirect
18. #87808C22semidirect
19. C4⋊#1039C22semidirect
23. C23×C52D4semidirect
32. C2×C4×C52D4semidirect
39. C23×C4×C5D5semidirect
50. C22×C52C2×D4semidirect
58. C4×C52C2×D4semidirect
65. C22×C4×C5C2×D5semidirect
75. C2×C52#121semidirect
82. C23×C5D20semidirect
91. C2×C4×C5C22⋊D5semidirect
100. C23×C4C5⋊D5semidirect
104. C52C2×#121semidirect
109. C22×C5C2×D20semidirect
118. C4×C5C2×C22⋊D5semidirect
130. C22×C4C2×C5⋊D5semidirect
134. C2×C5#4076semidirect
145. C23C4⋊(C5⋊D5)semidirect
154. C2×C4C22⋊(C5⋊D5)semidirect
157. C5C2×#4076semidirect
166. C22C2×C4⋊(C5⋊D5)semidirect
172. C4C2×C22⋊(C5⋊D5)semidirect
175. C2#11717250direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×C522
C22×C22⋊(C5⋊D5)2
C22×C4⋊(C5⋊D5)2
#117172502
C2×#878082
C2×C4⋊#10392
C2×#40765