Finite Group Databasebeta

← #26012049

Subgroups — #26012049

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
1C2×D4×D173C212yesmaximal, complemented
2#19149664C212yesmaximal, complemented
3#19149661C212yesmaximal, complemented
4#19149653C211yesmaximal, characteristic, complemented
5#19149651C212yesmaximal, complemented
6#19149649C212yesmaximal, complemented
7#19149647C212yesmaximal, complemented
8C2×#14583803C211yesmaximal, characteristic, complemented
9#121×C173C211yesmaximal, characteristic, complemented
10C23×D173C2211yescharacteristic
11C23×D173C2212yescomplemented
12C2×D692C2212yes—
13C2×D692C2211yescharacteristic, complemented
14C2×C22⋊D173C2212yes—
15C2×C22⋊D173C2212yescomplemented
16C2×C22⋊D173C2212yescomplemented
17D4×D173—28no—
18C22×Dic173C2211yescharacteristic, complemented
19C2×C4×D173C2212yescomplemented
20C4⋊Dic173C2211yescharacteristic, complemented
21#14583805C2212yescomplemented
22C22⋊Dic173C2212yescomplemented
23#14583803C2212yescomplemented
24#14583803C2212yescomplemented
25#14583803C2212yescomplemented
26#14583803C2212yescomplemented
27C2×D4×C173C2212yescomplemented
28C2×D4×C173C2211yescharacteristic, complemented
29C22×C4×C173C2211yescharacteristic, complemented
30C4⋊C4×C173C2211yescharacteristic, complemented
31C22⋊C4×C173C2212yescomplemented
32C22×D173—24no—
33C22×D173—41no—
34C22×D173—41no—
35C22×D173—42no—
36C22×D173—42no—
37C22×D173D414yescomplemented
38C22×D173C2312yes—
39C22×D173C2312yes—
40D692—24no—
41D692—41no—
42C22⋊D173—24no—
43C22⋊D173—24no—
44C22⋊D173—42no—
45C2×Dic173—41no—
46C2×Dic173C2312yes—
47C2×Dic173C2312yes—
48C4×D173—24no—
49C23×C173C2312yes—
50C23×C173C2311yescharacteristic
51D4×C173—24no—
52D4×C173—41no—
53C2×C4×C173—41no—
54C2×C4×C173C2312yes—
55C2×C4×C173C2311yescharacteristic
56C2×C4×C173C2311yescharacteristic
57C22×C173—21no—
58C22×C173—21no—
59C22×C173—22no—
60C22×C173—22no—
61C22×C173—22no—
62C22×C173C2×D412yes—
63C22×C173C2411yescharacteristic
64C2×D173—22no—
65C2×D173—22no—
66C2×D173—22no—
67C2×D173—22no—
68C2×D173—22no—
69C2×D173—44no—
70C2×D173—48no—
71C2×D173C2×D414yes—
72C4×C173—22no—
73C4×C173—21no—
74C4×C173—21no—
75Dic173—22no—
76Dic173—22no—
77C2×C173—22no—
78C2×C173—42no—
79C2×C173C22×D411yescharacteristic
80C2×C173H(5,𝔽2)11yescharacteristic
81C2×C173H(5,𝔽2)11yescharacteristic
82D173—24no—
83D173—42no—
84C173#53411yescharacteristic, complemented, Sylow
85#534—1731nomaximal, Sylow
86C22×D4—1732no—
87#123—1732no—
88#123—1732no—
89#121—1732no—
90#121—1731no—
91#121D17311yescharacteristic, complemented
92#120—1732no—
93#120—1732no—
94C2×C22⋊C4—1731no—
95C24—1732no—
96C24—1731no—
97C2×D4—3468no—
98C2×D4—1732no—
99C2×D4—1732no—
100C2×D4—1732no—
101C2×D4—1732no—
102C2×D4—1731no—
103C2×D4C2×D17312yescomplemented
104C2×D4C2×D17311yescharacteristic, complemented
105C22×C4—1732no—
106C22×C4—1731no—
107C22×C4C2×D17311yescharacteristic, complemented
108C4⋊C4—1732no—
109C4⋊C4—1731no—
110C4⋊C4C2×D17311yescharacteristic, complemented
111C22⋊C4—1732no—
112C22⋊C4—1732no—
113C22⋊C4—1732no—
114C22⋊C4—1732no—
115C22⋊C4—1732no—
116C22⋊C4C2×D17312yescomplemented
117C23—3464no—
118C23—1732no—
119C23—1732no—
120C23—1734no—
121C23—6921no—
122C23—6921no—
123C23—6922no—
124C23—6922no—
125C23C22×D17312yes—
126C23C22×D17311yescharacteristic
127D4—24no—
128D4—41no—
129D4—3464no—
130D4—3464no—
131D4—3464no—
132D4—6921no—
133D4—6922no—
134C2×C4—41no—
135C2×C4—3464no—
136C2×C4—1732no—
137C2×C4—1732no—
138C2×C4—6921no—
139C2×C4C22×D17311yescharacteristic
140C2×C4C22×D17311yescharacteristic
141C2×C4C22×D17312yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—3462no—
148C22—3462no—
149C22—3462no—
150C22—3462no—
151C22—3462no—
152C22—1734no—
153C22—6924no—
154C22—6928no—
155C22D4×D17312yes—
156C22C23×D17311yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—3462no—
161C4—3462no—
162C2—22no—
163C2—42no—
164C2—3464no—
165C2—6922no—
166C2C2×D4×D17311yescharacteristic
167C2#1914972411yescharacteristic
168C2#1914972511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#121×C173C2
hypercenter#121D173
hypocenterC173#534
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D173C2semidirect
2. #19149664C2semidirect
3. #19149661C2semidirect
4. #19149653C2semidirect
5. #19149651C2semidirect
6. #19149649C2semidirect
7. #19149647C2semidirect
8. C2×#14583803C2semidirect
9. #121×C173C2semidirect
11. C23×D173C22semidirect
13. C2×D692C22semidirect
15. C2×C22⋊D173C22semidirect
16. C2×C22⋊D173C22semidirect
18. C22×Dic173C22semidirect
19. C2×C4×D173C22semidirect
20. C4⋊Dic173C22semidirect
21. #14583805C22semidirect
22. C22⋊Dic173C22semidirect
23. #14583803C22semidirect
24. #14583803C22semidirect
25. #14583803C22semidirect
26. #14583803C22semidirect
27. C2×D4×C173C22semidirect
28. C2×D4×C173C22semidirect
29. C22×C4×C173C22semidirect
30. C4⋊C4×C173C22semidirect
31. C22⋊C4×C173C22semidirect
37. C22×D173D4semidirect
84. C173#534semidirect
91. #121D173semidirect
103. C2×D4C2×D173semidirect
104. C2×D4C2×D173semidirect
107. C22×C4C2×D173semidirect
110. C4⋊C4C2×D173semidirect
116. C22⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D1732
#191496642
#191496612
#191496532
#191496512
#191496492
#191496472
C2×#145838032
#121×C1732
#534173