Finite Group Databasebeta

← #26012056

Subgroups — #26012056

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×#14583825C211yesmaximal, characteristic, complemented
2C22×C22⋊D173C211yesmaximal, characteristic, complemented
3#19149675C212yesmaximal, complemented
4#19149665C212yesmaximal, complemented
5#19149652C212yesmaximal, complemented
6#19149651C212yesmaximal, complemented
7#19149649C211yesmaximal, characteristic, complemented
8#19149648C211yesmaximal, characteristic, complemented
9#19149647C211yesmaximal, characteristic, complemented
10#19149635C211yesmaximal, characteristic, complemented
11#120×C173C211yesmaximal, characteristic, complemented
12C23×D173C2211yescharacteristic
13#14583825—24no—
14C2×Dic346C2211yescharacteristic, complemented
15C2×D692C2212yescomplemented
16C2×C22⋊D173—24no—
17C2×C22⋊D173C2211yescharacteristic
18C2×C22⋊D173C2212yes—
19C2×C22⋊D173C2212yes—
20C2×C22⋊D173C2211yescharacteristic, complemented
21C2×C22⋊D173C2212yescomplemented
22C2×C22⋊D173C2212yescomplemented
23C22×Dic173C2211yescharacteristic
24C22×Dic173C2211yescharacteristic, complemented
25C2×C4×D173C2211yescharacteristic, complemented
26#14583805C2212yescomplemented
27C22⋊Dic173C2211yescharacteristic
28C22⋊Dic173C2212yescomplemented
29#14583803C2212yescomplemented
30#14583803C2212yescomplemented
31C4×Dic173C2212yescomplemented
32C24×C173C2211yescharacteristic
33C2×D4×C173C2211yescharacteristic
34C2×D4×C173C2212yescomplemented
35C22⋊C4×C173C2211yescharacteristic
36C22⋊C4×C173C2212yescomplemented
37C22×D173—22no—
38C22×D173—42no—
39C22×D173C2311yescharacteristic
40C22×D173C2312yes—
41Dic346—22no—
42D692—42no—
43C22⋊D173—22no—
44C22⋊D173—24no—
45C22⋊D173—24no—
46C22⋊D173—22no—
47C22⋊D173—42no—
48C22⋊D173—42no—
49C2×Dic173—22no—
50C2×Dic173D414yescomplemented
51C2×Dic173C2311yescharacteristic
52C2×Dic173C2312yes—
53C2×Dic173C2311yescharacteristic
54C2×Dic173C2311yescharacteristic
55C4×D173—22no—
56C23×C173—22no—
57C23×C173—42no—
58C23×C173C2311yescharacteristic
59C23×C173C2312yes—
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—21no—
75C2×D173—22no—
76C2×D173—22no—
77C2×D173—22no—
78C2×D173—22no—
79C2×D173—44no—
80C4×C173—21no—
81C4×C173—22no—
82Dic173—21no—
83Dic173—22no—
84Dic173C2×D412yes—
85Dic173C2×D412yes—
86C2×C173—22no—
87C2×C173—41no—
88C2×C173—42no—
89C2×C173C22×D411yescharacteristic
90C2×C173H(5,𝔽2)12yes—
91D173—41no—
92D173—42no—
93C173#53511yescharacteristic, complemented, Sylow
94#535—1731nomaximal, Sylow
95C2×#41—1731no—
96C22×D4—1731no—
97#127—1732no—
98#124—1732no—
99#121—1732no—
100#121—1732no—
101#120—1731no—
102#120—1731no—
103#120—1731no—
104#120D17311yescharacteristic, complemented
105#117—1731no—
106C24—1731no—
107C24C2×D17311yescharacteristic
108#41—3464no—
109C2×Q8—1731no—
110C2×D4—3464no—
111C2×D4—1731no—
112C2×D4—1732no—
113C2×D4—1732no—
114C2×D4—1731no—
115C2×D4—1732no—
116C2×D4—1732no—
117C2×D4—1732no—
118C2×D4C2×D17311yescharacteristic
119C2×D4C2×D17312yescomplemented
120C22×C4—1731no—
121C22×C4—1731no—
122C22×C4—1731no—
123C4⋊C4—1732no—
124C22⋊C4—1731no—
125C22⋊C4—1732no—
126C22⋊C4—1732no—
127C22⋊C4—1732no—
128C22⋊C4C2×D17311yescharacteristic
129C22⋊C4C2×D17312yescomplemented
130C42—1732no—
131C23—22no—
132C23—42no—
133C23—3462no—
134C23—1732no—
135C23—1731no—
136C23—6922no—
137C23C22×D17312yes—
138C23C22×D17311yescharacteristic
139C23C22×D17311yescharacteristic
140Q8—3462no—
141D4—22no—
142D4—42no—
143D4—3462no—
144D4—3464no—
145D4—3464no—
146D4—3462no—
147D4—6922no—
148D4—6922no—
149D4—6922no—
150C2×C4—3462no—
151C2×C4—3462no—
152C2×C4—1732no—
153C2×C4—1731no—
154C2×C4—1734no—
155C2×C4—1731no—
156C2×C4—1731no—
157C2×C4C22×D17312yes—
158C2×C4C22×D17311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—3461no—
167C22—3462no—
168C22—3462no—
169C22—3462no—
170C22—3462no—
171C22—6924no—
172C22D4×D17312yes—
173C22C23×D17311yescharacteristic
174C4—21no—
175C4—22no—
176C4—3461no—
177C4—3462no—
178C4—1732no—
179C4—1732no—
180C2—22no—
181C2—41no—
182C2—42no—
183C2—6921no—
184C2—6922no—
185C2C2×D4×D17311yescharacteristic
186C2#1914972412yes—
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#120×C173C2
hypercenter#120D173
hypocenterC173#535
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#14583825C2semidirect
2. C22×C22⋊D173C2semidirect
3. #19149675C2semidirect
4. #19149665C2semidirect
5. #19149652C2semidirect
6. #19149651C2semidirect
7. #19149649C2semidirect
8. #19149648C2semidirect
9. #19149647C2semidirect
10. #19149635C2semidirect
11. #120×C173C2semidirect
14. C2×Dic346C22semidirect
15. C2×D692C22semidirect
20. C2×C22⋊D173C22semidirect
21. C2×C22⋊D173C22semidirect
22. C2×C22⋊D173C22semidirect
24. C22×Dic173C22semidirect
25. C2×C4×D173C22semidirect
26. #14583805C22semidirect
28. C22⋊Dic173C22semidirect
29. #14583803C22semidirect
30. #14583803C22semidirect
31. C4×Dic173C22semidirect
34. C2×D4×C173C22semidirect
36. C22⋊C4×C173C22semidirect
50. C2×Dic173D4semidirect
93. C173#535semidirect
104. #120D173semidirect
119. C2×D4C2×D173semidirect
129. C22⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#145838252
C22×C22⋊D1732
#191496752
#191496652
#191496522
#191496512
#191496492
#191496482
#191496472
#191496352
#120×C1732
#535173