Finite Group Databasebeta

← #26012051

Subgroups — #26012051

Order 11072.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×C22⋊D173C211yesmaximal, characteristic, complemented
2C2×D4×D173C211yesmaximal, characteristic, complemented
3#19149663C211yesmaximal, characteristic, complemented
4#19149661C212yesmaximal, complemented
5#19149660C211yesmaximal, characteristic, complemented
6#19149652C212yesmaximal, complemented
7#19149650C212yesmaximal, complemented
8#19149648C211yesmaximal, characteristic, complemented
9#19149647C212yesmaximal, complemented
10C22⋊C4×D173C211yesmaximal, characteristic, complemented
11#120×C173C211yesmaximal, characteristic, complemented
12C23×D173C2211yescharacteristic
13C23×D173C2211yescharacteristic, complemented
14C2×D692C2211yescharacteristic, complemented
15C2×C22⋊D173—24no—
16C2×C22⋊D173C2212yes—
17C2×C22⋊D173C2211yescharacteristic
18C2×C22⋊D173C2211yescharacteristic, complemented
19C2×C22⋊D173C2212yescomplemented
20C2×C22⋊D173C2212yescomplemented
21D4×D173—24no—
22C22×Dic173C2211yescharacteristic, complemented
23C2×C4×D173C2211yescharacteristic, complemented
24C2×C4×D173C2212yescomplemented
25C4⋊Dic173C2212yescomplemented
26#14583805C2212yescomplemented
27C22⋊Dic173C2211yescharacteristic, complemented
28C22⋊Dic173C2212yescomplemented
29C22⋊Dic173C2212yescomplemented
30#14583803C2212yescomplemented
31#14583803C2212yescomplemented
32C24×C173C2211yescharacteristic, complemented
33C2×D4×C173C2211yescharacteristic
34C2×D4×C173C2212yescomplemented
35C22⋊C4×C173C2211yescharacteristic, complemented
36C22⋊C4×C173C2212yescomplemented
37C22×D173—22no—
38C22×D173—42no—
39C22×D173—42no—
40C22×D173D414yescomplemented
41C22×D173C2311yescharacteristic
42C22×D173C2311yescharacteristic
43C22×D173C2311yescharacteristic
44D692—22no—
45C22⋊D173—22no—
46C22⋊D173—22no—
47C22⋊D173—24no—
48C22⋊D173—24no—
49C22⋊D173—42no—
50C2×Dic173—22no—
51C2×Dic173C2311yescharacteristic
52C2×Dic173C2312yes—
53C2×Dic173C2312yes—
54C4×D173—22no—
55C4×D173—42no—
56C23×C173—22no—
57C23×C173—42no—
58C23×C173C2312yes—
59C23×C173C2311yescharacteristic
60C23×C173C2311yescharacteristic
61D4×C173—22no—
62D4×C173—42no—
63C2×C4×C173C2311yescharacteristic
64C2×C4×C173C2312yes—
65C22×C173—22no—
66C22×C173—21no—
67C22×C173—22no—
68C22×C173—22no—
69C22×C173—22no—
70C22×C173—22no—
71C22×C173—44no—
72C22×C173C2×D412yes—
73C22×C173C2411yescharacteristic
74C2×D173—22no—
75C2×D173—22no—
76C2×D173—21no—
77C2×D173—22no—
78C2×D173—44no—
79C2×D173—44no—
80C2×D173C2×D412yes—
81C2×D173C2×D412yes—
82C4×C173—21no—
83C4×C173—22no—
84Dic173—21no—
85Dic173—22no—
86Dic173—22no—
87C2×C173—22no—
88C2×C173—41no—
89C2×C173—42no—
90C2×C173C22×D411yescharacteristic
91C2×C173H(5,𝔽2)12yes—
92D173—22no—
93D173—22no—
94D173—41no—
95C173#53411yescharacteristic, complemented, Sylow
96#534—1731nomaximal, Sylow
97C22×D4—1731no—
98C22×D4—1731no—
99#123—1731no—
100#123—1732no—
101#123—1731no—
102#121—1732no—
103#121—1732no—
104#120—1731no—
105#120—1732no—
106#120D17311yescharacteristic, complemented
107C2×C22⋊C4—1731no—
108C24—1731no—
109C24—1731no—
110C24C2×D17311yescharacteristic, complemented
111C2×D4—3464no—
112C2×D4—3464no—
113C2×D4—1732no—
114C2×D4—1731no—
115C2×D4—1731no—
116C2×D4—1732no—
117C2×D4—1732no—
118C2×D4—1731no—
119C2×D4C2×D17311yescharacteristic
120C2×D4C2×D17312yescomplemented
121C22×C4—1731no—
122C22×C4—1731no—
123C22×C4—1732no—
124C4⋊C4—1732no—
125C4⋊C4—1732no—
126C22⋊C4—1731no—
127C22⋊C4—1732no—
128C22⋊C4—1732no—
129C22⋊C4—1732no—
130C22⋊C4—1732no—
131C22⋊C4C2×D17311yescharacteristic, complemented
132C22⋊C4C2×D17312yescomplemented
133C23—22no—
134C23—42no—
135C23—3462no—
136C23—1731no—
137C23—1731no—
138C23—1731no—
139C23—1734no—
140C23—6922no—
141C23—6922no—
142C23C22×D17311yescharacteristic
143C23C22×D17312yes—
144C23C22×D17311yescharacteristic
145D4—22no—
146D4—42no—
147D4—3462no—
148D4—3462no—
149D4—3464no—
150D4—3464no—
151D4—3462no—
152D4—6922no—
153C2×C4—3462no—
154C2×C4—3462no—
155C2×C4—1732no—
156C2×C4—1731no—
157C2×C4—1732no—
158C2×C4—6922no—
159C2×C4C22×D17312yes—
160C2×C4C22×D17311yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—3462no—
169C22—3462no—
170C22—3461no—
171C22—3462no—
172C22—1732no—
173C22—1732no—
174C22—6924no—
175C22—6924no—
176C22D4×D17312yes—
177C22C23×D17311yescharacteristic
178C4—21no—
179C4—22no—
180C4—3461no—
181C4—3462no—
182C4—3462no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—3462no—
187C2—3462no—
188C2—6921no—
189C2C2×D4×D17311yescharacteristic
190C2#1914972412yes—
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#120×C173C2
hypercenter#120D173
hypocenterC173#534
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D173C2semidirect
2. C2×D4×D173C2semidirect
3. #19149663C2semidirect
4. #19149661C2semidirect
5. #19149660C2semidirect
6. #19149652C2semidirect
7. #19149650C2semidirect
8. #19149648C2semidirect
9. #19149647C2semidirect
10. C22⋊C4×D173C2semidirect
11. #120×C173C2semidirect
13. C23×D173C22semidirect
14. C2×D692C22semidirect
18. C2×C22⋊D173C22semidirect
19. C2×C22⋊D173C22semidirect
20. C2×C22⋊D173C22semidirect
22. C22×Dic173C22semidirect
23. C2×C4×D173C22semidirect
24. C2×C4×D173C22semidirect
25. C4⋊Dic173C22semidirect
26. #14583805C22semidirect
27. C22⋊Dic173C22semidirect
28. C22⋊Dic173C22semidirect
29. C22⋊Dic173C22semidirect
30. #14583803C22semidirect
31. #14583803C22semidirect
32. C24×C173C22semidirect
34. C2×D4×C173C22semidirect
35. C22⋊C4×C173C22semidirect
36. C22⋊C4×C173C22semidirect
40. C22×D173D4semidirect
95. C173#534semidirect
106. #120D173semidirect
110. C24C2×D173semidirect
120. C2×D4C2×D173semidirect
131. C22⋊C4C2×D173semidirect
132. C22⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D1732
C2×D4×D1732
#191496632
#191496612
#191496602
#191496522
#191496502
#191496482
#191496472
C22⋊C4×D1732
#120×C1732
#534173