Finite Group Databasebeta

← #26012123

Subgroups — #26012123

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×C22⋊D173C212yesmaximal, complemented
2C2×D4×D173C212yesmaximal, complemented
3#19149675C211yesmaximal, characteristic, complemented
4#19149652C212yesmaximal, complemented
5#19149651C212yesmaximal, complemented
6#19149649C212yesmaximal, complemented
7#19149647C211yesmaximal, characteristic, complemented
8#19149639C212yesmaximal, complemented
9#120×C173C211yesmaximal, characteristic, complemented
10C23×D173C2211yescharacteristic, complemented
11C23×D173C2212yescomplemented
12C2×D692C2212yescomplemented
13C2×D692C2211yescharacteristic, complemented
14C2×C22⋊D173—28no—
15C2×C22⋊D173C2212yes—
16C2×C22⋊D173C2212yescomplemented
17C2×C22⋊D173C2212yescomplemented
18C2×C22⋊D173C2212yescomplemented
19C2×C22⋊D173C2212yescomplemented
20D4×D173—28no—
21C22×Dic173C2212yescomplemented
22C2×C4×D173C2212yescomplemented
23#14583805C2212yescomplemented
24C22⋊Dic173C2211yescharacteristic, complemented
25#14583803C2212yescomplemented
26#14583803C2212yescomplemented
27C4×Dic173C2211yescharacteristic, complemented
28C24×C173C2211yescharacteristic, complemented
29C2×D4×C173C2212yescomplemented
30C2×D4×C173C2211yescharacteristic, complemented
31C22⋊C4×C173C2212yescomplemented
32C22⋊C4×C173C2211yescharacteristic, complemented
33C22×D173—24no—
34C22×D173—24no—
35C22×D173—24no—
36C22×D173—41no—
37C22×D173—42no—
38C22×D173C2312yes—
39C22×D173C2312yes—
40D692—24no—
41D692—41no—
42C22⋊D173—24no—
43C22⋊D173—24no—
44C22⋊D173—24no—
45C22⋊D173—24no—
46C22⋊D173D418yescomplemented
47C2×Dic173—24no—
48C2×Dic173C2312yes—
49C2×Dic173C2312yes—
50C4×D173—24no—
51C23×C173—24no—
52C23×C173—41no—
53C23×C173C2311yescharacteristic
54C23×C173C2311yescharacteristic
55C23×C173C2312yes—
56D4×C173—24no—
57D4×C173—41no—
58C2×C4×C173C2312yes—
59C2×C4×C173C2311yescharacteristic
60C22×C173—22no—
61C22×C173—22no—
62C22×C173—22no—
63C22×C173—22no—
64C22×C173—21no—
65C22×C173—21no—
66C22×C173—44no—
67C22×C173C2×D414yescomplemented
68C22×C173C2411yescharacteristic
69C2×D173—22no—
70C2×D173—22no—
71C2×D173—22no—
72C2×D173—22no—
73C2×D173—22no—
74C2×D173—44no—
75C2×D173—48no—
76C2×D173C2×D414yescomplemented
77C4×C173—22no—
78C4×C173—21no—
79Dic173—22no—
80Dic173C2×D414yescomplemented
81C2×C173—24no—
82C2×C173—41no—
83C2×C173—41no—
84C2×C173C22×D412yes—
85C2×C173H(5,𝔽2)11yescharacteristic
86D173—24no—
87D173—42no—
88C173D4211yescharacteristic, complemented, Sylow
89D42—1731nomaximal, Sylow
90C22×D4—1732no—
91C22×D4—1732no—
92#127—1731no—
93#121—1732no—
94#121—1732no—
95#120—1731no—
96#120—1732no—
97#120D17311yescharacteristic, complemented
98C4×D4—1732no—
99C24—1731no—
100C24—1732no—
101C24C2×D17311yescharacteristic, complemented
102C2×D4—3468no—
103C2×D4—3468no—
104C2×D4—1732no—
105C2×D4—1732no—
106C2×D4—1732no—
107C2×D4—1732no—
108C2×D4—1732no—
109C2×D4—1732no—
110C2×D4—1731no—
111C2×D4C2×D17312yescomplemented
112C2×D4C2×D17311yescharacteristic, complemented
113C22×C4—1732no—
114C22×C4—1732no—
115C4⋊C4—1732no—
116C22⋊C4—1731no—
117C22⋊C4—1732no—
118C22⋊C4—1732no—
119C22⋊C4C2×D17312yescomplemented
120C22⋊C4C2×D17311yescharacteristic, complemented
121C42—1731no—
122C23—24no—
123C23—41no—
124C23—3464no—
125C23—3464no—
126C23—3464no—
127C23—1732no—
128C23—1732no—
129C23—6922no—
130C23—6921no—
131C23C22×D17311yescharacteristic
132C23C22×D17311yescharacteristic
133C23C22×D17312yes—
134D4—24no—
135D4—41no—
136D4—3464no—
137D4—3464no—
138D4—3464no—
139D4—3464no—
140D4—3464no—
141D4—1738no—
142D4—6921no—
143C2×C4—3464no—
144C2×C4—3464no—
145C2×C4—1732no—
146C2×C4—1732no—
147C2×C4C22×D17311yescharacteristic
148C2×C4C22×D17312yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—3462no—
157C22—3462no—
158C22—3462no—
159C22—3462no—
160C22—3462no—
161C22—1734no—
162C22—6928no—
163C22—6924no—
164C22D4×D17314yescomplemented
165C22C23×D17311yescharacteristic
166C4—22no—
167C4—21no—
168C4—3462no—
169C4—1734no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—3464no—
174C2—6922no—
175C2C2×D4×D17312yes—
176C2#1914972411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#120×C173C2
hypercenter#120D173
hypocenterC173D42
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. #19149675C2semidirect
4. #19149652C2semidirect
5. #19149651C2semidirect
6. #19149649C2semidirect
7. #19149647C2semidirect
8. #19149639C2semidirect
9. #120×C173C2semidirect
10. C23×D173C22semidirect
11. C23×D173C22semidirect
12. C2×D692C22semidirect
13. C2×D692C22semidirect
16. C2×C22⋊D173C22semidirect
17. C2×C22⋊D173C22semidirect
18. C2×C22⋊D173C22semidirect
19. C2×C22⋊D173C22semidirect
21. C22×Dic173C22semidirect
22. C2×C4×D173C22semidirect
23. #14583805C22semidirect
24. C22⋊Dic173C22semidirect
25. #14583803C22semidirect
26. #14583803C22semidirect
27. C4×Dic173C22semidirect
28. C24×C173C22semidirect
29. C2×D4×C173C22semidirect
30. C2×D4×C173C22semidirect
31. C22⋊C4×C173C22semidirect
32. C22⋊C4×C173C22semidirect
46. C22⋊D173D4semidirect
67. C22×C173C2×D4semidirect
76. C2×D173C2×D4semidirect
80. Dic173C2×D4semidirect
88. C173D42semidirect
97. #120D173semidirect
101. C24C2×D173semidirect
111. C2×D4C2×D173semidirect
112. C2×D4C2×D173semidirect
119. C22⋊C4C2×D173semidirect
120. C22⋊C4C2×D173semidirect
164. C22D4×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
#191496752
#191496522
#191496512
#191496492
#191496472
#191496392
#120×C1732
D42173