Finite Group Databasebeta

← #26012057

Subgroups — #26012057

Order 11072.

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

normal subgroups only

This group has 334 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
1C2×#14583826C211yesmaximal, characteristic, complemented
2C2×D4×D173C211yesmaximal, characteristic, complemented
3#19149675C212yesmaximal, complemented
4#19149665C212yesmaximal, complemented
5#19149654C211yesmaximal, characteristic, complemented
6#19149653C211yesmaximal, characteristic, complemented
7#19149650C211yesmaximal, characteristic, complemented
8#19149649C212yesmaximal, complemented
9#19149647C212yesmaximal, complemented
10#19149637C211yesmaximal, characteristic, complemented
11#121×C173C211yesmaximal, characteristic, complemented
12C23×D173C2212yes—
13#14583826—24no—
14C2×Dic346C2211yescharacteristic, complemented
15C2×D692C2211yescharacteristic
16C2×D692C2211yescharacteristic
17C2×D692C2212yescomplemented
18C2×C22⋊D173C2211yescharacteristic
19C2×C22⋊D173C2212yes—
20C2×C22⋊D173C2211yescharacteristic, complemented
21C2×C22⋊D173C2212yescomplemented
22C2×C22⋊D173C2212yescomplemented
23D4×D173—24no—
24C2×C4×D173C2211yescharacteristic
25C2×C4×D173C2211yescharacteristic, complemented
26C4⋊Dic173C2211yescharacteristic, complemented
27C22⋊Dic173C2212yescomplemented
28#14583803C2212yes—
29#14583803C2212yescomplemented
30#14583803C2212yescomplemented
31C4×Dic173C2212yescomplemented
32C2×D4×C173C2211yescharacteristic
33C2×D4×C173C2212yescomplemented
34C22×C4×C173C2211yescharacteristic, complemented
35C4⋊C4×C173C2211yescharacteristic, complemented
36C22⋊C4×C173C2212yescomplemented
37C22×D173—24no—
38C22×D173—42no—
39C22×D173—42no—
40C22×D173C2311yescharacteristic
41C22×D173C2312yes—
42C22×D173C2311yescharacteristic
43Dic346—22no—
44D692—22no—
45D692—22no—
46D692—42no—
47C22⋊D173—22no—
48C22⋊D173—24no—
49C22⋊D173—22no—
50C22⋊D173—42no—
51C22⋊D173—42no—
52C2×Dic173C2311yescharacteristic
53C2×Dic173C2312yes—
54C2×Dic173C2311yescharacteristic
55C4×D173—22no—
56C4×D173D414yescomplemented
57C23×C173C2311yescharacteristic
58C23×C173C2312yes—
59D4×C173—22no—
60D4×C173—42no—
61C2×C4×C173—22no—
62C2×C4×C173C2311yescharacteristic
63C2×C4×C173C2312yes—
64C2×C4×C173C2311yescharacteristic
65C22×C173—21no—
66C22×C173—22no—
67C22×C173—21no—
68C22×C173—22no—
69C22×C173—21no—
70C22×C173—22no—
71C22×C173C2411yescharacteristic
72C2×D173—21no—
73C2×D173—21no—
74C2×D173—21no—
75C2×D173—22no—
76C2×D173—21no—
77C2×D173—22no—
78C2×D173—21no—
79C2×D173—22no—
80C2×D173—48no—
81C2×D173C2×D412yes—
82C4×C173—21no—
83C4×C173—22no—
84C4×C173C2×D412yes—
85Dic173—21no—
86Dic173—22no—
87Dic173C2×D412yes—
88C2×C173—42no—
89C2×C173—41no—
90C2×C173C22×D411yescharacteristic
91C2×C173H(5,𝔽2)11yescharacteristic
92C2×C173H(5,𝔽2)11yescharacteristic
93D173—22no—
94D173—41no—
95D173—42no—
96C173#53511yescharacteristic, complemented, Sylow
97#535—1731nomaximal, Sylow
98C2×#41—1731no—
99C22×D4—1731no—
100#127—1732no—
101#124—1732no—
102#121—1731no—
103#121—1731no—
104#121—1731no—
105#121D17311yescharacteristic, complemented
106#120—1732no—
107#120—1732no—
108#117—1731no—
109C24—1732no—
110#41—3464no—
111C2×Q8—1731no—
112C2×D4—3464no—
113C2×D4—1731no—
114C2×D4—1732no—
115C2×D4—1731no—
116C2×D4—1732no—
117C2×D4—1732no—
118C2×D4—1731no—
119C2×D4—1731no—
120C2×D4—1732no—
121C2×D4C2×D17311yescharacteristic
122C2×D4C2×D17312yescomplemented
123C22×C4—1731no—
124C22×C4—1731no—
125C22×C4C2×D17311yescharacteristic, complemented
126C4⋊C4—1731no—
127C4⋊C4C2×D17311yescharacteristic, complemented
128C22⋊C4—1732no—
129C22⋊C4—1732no—
130C22⋊C4—1732no—
131C22⋊C4—1732no—
132C22⋊C4C2×D17312yescomplemented
133C42—1732no—
134C23—3464no—
135C23—1732no—
136C23—1731no—
137C23—1731no—
138C23—6922no—
139C23—6922no—
140C23C22×D17312yes—
141C23C22×D17311yescharacteristic
142Q8—3462no—
143D4—22no—
144D4—42no—
145D4—3462no—
146D4—3462no—
147D4—3464no—
148D4—3462no—
149D4—3462no—
150D4—6922no—
151D4—6922no—
152D4—6922no—
153C2×C4—22no—
154C2×C4—3462no—
155C2×C4—1732no—
156C2×C4—1731no—
157C2×C4—1731no—
158C2×C4—1734no—
159C2×C4C22×D17312yes—
160C2×C4C22×D17311yescharacteristic
161C2×C4C22×D17311yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—3461no—
169C22—3461no—
170C22—3461no—
171C22—3462no—
172C22—3461no—
173C22—3462no—
174C22—3461no—
175C22—3462no—
176C22—1732no—
177C22—6928no—
178C22C23×D17311yescharacteristic
179C4—21no—
180C4—22no—
181C4—3461no—
182C4—3462no—
183C4—1732no—
184C4D4×D17312yes—
185C2—41no—
186C2—42no—
187C2—3462no—
188C2—6921no—
189C2—6922no—
190C2C2×D4×D17311yescharacteristic
191C2#1914972411yescharacteristic
192C2#1914972511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#121×C173C2
hypercenter#121D173
hypocenterC173#535
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#14583826C2semidirect
2. C2×D4×D173C2semidirect
3. #19149675C2semidirect
4. #19149665C2semidirect
5. #19149654C2semidirect
6. #19149653C2semidirect
7. #19149650C2semidirect
8. #19149649C2semidirect
9. #19149647C2semidirect
10. #19149637C2semidirect
11. #121×C173C2semidirect
14. C2×Dic346C22semidirect
17. C2×D692C22semidirect
20. C2×C22⋊D173C22semidirect
21. C2×C22⋊D173C22semidirect
22. C2×C22⋊D173C22semidirect
25. C2×C4×D173C22semidirect
26. C4⋊Dic173C22semidirect
27. C22⋊Dic173C22semidirect
29. #14583803C22semidirect
30. #14583803C22semidirect
31. C4×Dic173C22semidirect
33. C2×D4×C173C22semidirect
34. C22×C4×C173C22semidirect
35. C4⋊C4×C173C22semidirect
36. C22⋊C4×C173C22semidirect
56. C4×D173D4semidirect
96. C173#535semidirect
105. #121D173semidirect
122. C2×D4C2×D173semidirect
125. C22×C4C2×D173semidirect
127. C4⋊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
C2×#145838262
C2×D4×D1732
#191496752
#191496652
#191496542
#191496532
#191496502
#191496492
#191496472
#191496372
#121×C1732
#535173