Finite Group Databasebeta

← #26012125

Subgroups — #26012125

Order 11072.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×D692C211yesmaximal, characteristic, complemented
2C2×D4×D173C212yesmaximal, complemented
3C2×D4×D173C211yesmaximal, characteristic, complemented
4#19149675C211yesmaximal, characteristic, complemented
5#19149654C211yesmaximal, characteristic, complemented
6#19149652C212yesmaximal, complemented
7#19149649C212yesmaximal, complemented
8#19149647C212yesmaximal, complemented
9#19149641C211yesmaximal, characteristic, complemented
10C4×C22⋊D173C211yesmaximal, characteristic, complemented
11#121×C173C211yesmaximal, characteristic, complemented
12C23×D173C2212yescomplemented
13C23×D173C2212yescomplemented
14C2×D692—24no—
15C2×D692C2211yescharacteristic
16C2×D692C2212yescomplemented
17C2×D692C2212yescomplemented
18C2×D692C2211yescharacteristic, complemented
19C2×C22⋊D173C2211yescharacteristic
20C2×C22⋊D173C2212yescomplemented
21C2×C22⋊D173C2212yescomplemented
22C2×C22⋊D173C2212yescomplemented
23D4×D173—24no—
24D4×D173—28no—
25C2×C4×D173C2211yescharacteristic, complemented
26C2×C4×D173C2212yescomplemented
27#14583805C2211yescharacteristic, complemented
28C22⋊Dic173C2211yescharacteristic, complemented
29#14583803C2211yescharacteristic, complemented
30#14583803C2212yescomplemented
31#14583803C2212yescomplemented
32C4×Dic173C2211yescharacteristic, complemented
33C2×D4×C173C2212yescomplemented
34C2×D4×C173C2211yescharacteristic, complemented
35C22×C4×C173C2211yescharacteristic, complemented
36C4⋊C4×C173C2211yescharacteristic, complemented
37C22⋊C4×C173C2212yescomplemented
38C22×D173—24no—
39C22×D173—24no—
40C22×D173—24no—
41C22×D173—24no—
42C22×D173—42no—
43C22×D173—42no—
44C22×D173C2312yes—
45C22×D173C2311yescharacteristic
46C22×D173C2312yes—
47D692—24no—
48D692—24no—
49D692—22no—
50D692D414yescomplemented
51C22⋊D173—24no—
52C22⋊D173—24no—
53C22⋊D173—42no—
54C22⋊D173D414yescomplemented
55C2×Dic173C2311yescharacteristic
56C2×Dic173C2311yescharacteristic
57C2×Dic173C2311yescharacteristic
58C4×D173—22no—
59C4×D173—24no—
60C23×C173C2312yes—
61C23×C173C2311yescharacteristic
62D4×C173—24no—
63D4×C173—22no—
64C2×C4×C173—22no—
65C2×C4×C173C2312yes—
66C2×C4×C173C2311yescharacteristic
67C2×C4×C173C2311yescharacteristic
68C22×C173—21no—
69C22×C173—21no—
70C22×C173—22no—
71C22×C173—22no—
72C22×C173—22no—
73C22×C173C2×D412yescomplemented
74C22×C173C2411yescharacteristic
75C2×D173—21no—
76C2×D173—22no—
77C2×D173—21no—
78C2×D173—22no—
79C2×D173—22no—
80C2×D173—22no—
81C2×D173—22no—
82C2×D173—48no—
83C2×D173—48no—
84C2×D173C2×D414yescomplemented
85C2×D173C2×D412yescomplemented
86C4×C173—21no—
87C4×C173—22no—
88C4×C173C2×D412yescomplemented
89Dic173—21no—
90Dic173—21no—
91Dic173C2×D412yescomplemented
92C2×C173—22no—
93C2×C173—42no—
94C2×C173C22×D411yescharacteristic
95C2×C173C22×D411yescharacteristic
96C2×C173H(5,𝔽2)11yescharacteristic
97D173—22no—
98D173—24no—
99D173—42no—
100C173D4211yescharacteristic, complemented, Sylow
101D42—1731nomaximal, Sylow
102C22×D4—1731no—
103C22×D4—1732no—
104C22×D4—1731no—
105#127—1731no—
106#121—1731no—
107#121—1732no—
108#121D17311yescharacteristic, complemented
109#120—1732no—
110#120—1732no—
111C4×D4—1731no—
112C4×D4—1731no—
113C24—1732no—
114C24—1732no—
115C2×D4—3464no—
116C2×D4—3468no—
117C2×D4—3464no—
118C2×D4—1732no—
119C2×D4—1732no—
120C2×D4—1731no—
121C2×D4—1732no—
122C2×D4—1732no—
123C2×D4—1732no—
124C2×D4—1731no—
125C2×D4—1731no—
126C2×D4C2×D17312yescomplemented
127C2×D4C2×D17311yescharacteristic, complemented
128C22×C4—1732no—
129C22×C4—1731no—
130C22×C4C2×D17311yescharacteristic, complemented
131C4⋊C4—1731no—
132C4⋊C4C2×D17311yescharacteristic, complemented
133C22⋊C4—1731no—
134C22⋊C4—1732no—
135C22⋊C4—1731no—
136C22⋊C4—1732no—
137C22⋊C4C2×D17312yescomplemented
138C42—1731no—
139C23—3464no—
140C23—3464no—
141C23—3464no—
142C23—3464no—
143C23—1732no—
144C23—1731no—
145C23—1732no—
146C23—6922no—
147C23—6922no—
148C23C22×D17312yes—
149C23C22×D17311yescharacteristic
150D4—24no—
151D4—22no—
152D4—3464no—
153D4—3464no—
154D4—3464no—
155D4—3464no—
156D4—3462no—
157D4—1734no—
158D4—1734no—
159D4—6922no—
160C2×C4—22no—
161C2×C4—3464no—
162C2×C4—3462no—
163C2×C4—1731no—
164C2×C4—1731no—
165C2×C4—1731no—
166C2×C4C22×D17312yes—
167C2×C4C22×D17311yescharacteristic
168C2×C4C22×D17311yescharacteristic
169C22—21no—
170C22—21no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—3462no—
175C22—3461no—
176C22—3462no—
177C22—3461no—
178C22—3462no—
179C22—3462no—
180C22—3462no—
181C22—1732no—
182C22—1734no—
183C22—6928no—
184C22—6928no—
185C22D4×D17312yescomplemented
186C22C23×D17311yescharacteristic
187C4—21no—
188C4—22no—
189C4—3461no—
190C4—3461no—
191C4—1732no—
192C4D4×D17312yescomplemented
193C2—22no—
194C2—42no—
195C2—3464no—
196C2—3462no—
197C2—6922no—
198C2C2×D4×D17311yescharacteristic
199C2C2×D4×D17311yescharacteristic
200C2#1914972511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#121×C173C2
hypercenter#121D173
hypocenterC173D42
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D692C2semidirect
2. C2×D4×D173C2semidirect
3. C2×D4×D173C2semidirect
4. #19149675C2semidirect
5. #19149654C2semidirect
6. #19149652C2semidirect
7. #19149649C2semidirect
8. #19149647C2semidirect
9. #19149641C2semidirect
10. C4×C22⋊D173C2semidirect
11. #121×C173C2semidirect
12. C23×D173C22semidirect
13. C23×D173C22semidirect
16. C2×D692C22semidirect
17. C2×D692C22semidirect
18. C2×D692C22semidirect
20. C2×C22⋊D173C22semidirect
21. C2×C22⋊D173C22semidirect
22. C2×C22⋊D173C22semidirect
25. C2×C4×D173C22semidirect
26. C2×C4×D173C22semidirect
27. #14583805C22semidirect
28. C22⋊Dic173C22semidirect
29. #14583803C22semidirect
30. #14583803C22semidirect
31. #14583803C22semidirect
32. C4×Dic173C22semidirect
33. C2×D4×C173C22semidirect
34. C2×D4×C173C22semidirect
35. C22×C4×C173C22semidirect
36. C4⋊C4×C173C22semidirect
37. C22⋊C4×C173C22semidirect
50. D692D4semidirect
54. C22⋊D173D4semidirect
73. C22×C173C2×D4semidirect
84. C2×D173C2×D4semidirect
85. C2×D173C2×D4semidirect
88. C4×C173C2×D4semidirect
91. Dic173C2×D4semidirect
100. C173D42semidirect
108. #121D173semidirect
126. C2×D4C2×D173semidirect
127. C2×D4C2×D173semidirect
130. C22×C4C2×D173semidirect
132. C4⋊C4C2×D173semidirect
137. C22⋊C4C2×D173semidirect
185. C22D4×D173semidirect
192. C4D4×D173semidirect

Maximal subgroups

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

subgroupindex
C22×D6922
C2×D4×D1732
#191496752
#191496542
#191496522
#191496492
#191496472
#191496412
C4×C22⋊D1732
#121×C1732
D42173