Finite Group Databasebeta

← #26012129

Subgroups — #26012129

Order 11072.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×D173C211yesmaximal, characteristic, complemented
2#19149668C211yesmaximal, characteristic, complemented
3#19149662C212yesmaximal, complemented
4#19149655C211yesmaximal, characteristic, complemented
5#19149653C211yesmaximal, characteristic, complemented
6#19149651C212yesmaximal, complemented
7#19149647C212yesmaximal, complemented
8D4×Dic173C211yesmaximal, characteristic, complemented
9C4×C22⋊D173C211yesmaximal, characteristic, complemented
10C2×#14583803C212yesmaximal, complemented
11C2×#41×C173C211yesmaximal, characteristic, complemented
12C23×D173C2212yes—
13C2×D692C2211yescharacteristic
14C2×C22⋊D173C2211yescharacteristic
15C2×C22⋊D173C2212yes—
16C2×C22⋊D173C2212yescomplemented
17D4×D173—24no—
18C22×Dic173C2212yescomplemented
19C2×C4×D173C2211yescharacteristic
20C4⋊Dic173C2211yescharacteristic, complemented
21#14583805C2211yescharacteristic, complemented
22#14583805C2212yescomplemented
23C22⋊Dic173C2211yescharacteristic, complemented
24C22⋊Dic173C2212yescomplemented
25#14583803C2211yescharacteristic
26#14583803C2212yes—
27#14583803C2212yescomplemented
28#14583803C2212yescomplemented
29#14583803C2212yescomplemented
30C4×Dic173C2211yescharacteristic, complemented
31#41×C173—24no—
32C2×Q8×C173C2211yescharacteristic, complemented
33C2×D4×C173C2211yescharacteristic
34C2×D4×C173C2212yescomplemented
35C22×C4×C173C2212yescomplemented
36C22×C4×C173C2211yescharacteristic, complemented
37C22×D173—24no—
38C22×D173—24no—
39C22×D173—42no—
40C22×D173C2312yes—
41C22×D173C2311yescharacteristic
42D692—22no—
43C22⋊D173—24no—
44C22⋊D173—22no—
45C22⋊D173—42no—
46C2×Dic173—42no—
47C2×Dic173C2311yescharacteristic
48C2×Dic173C2311yescharacteristic
49C2×Dic173C2312yes—
50C2×Dic173C2311yescharacteristic
51C4×D173—22no—
52C23×C173C2311yescharacteristic
53C23×C173C2312yes—
54Q8×C173—22no—
55D4×C173—24no—
56D4×C173D414yescomplemented
57C2×C4×C173—24no—
58C2×C4×C173—22no—
59C2×C4×C173C2311yescharacteristic
60C2×C4×C173C2312yes—
61C2×C4×C173C2311yescharacteristic
62C22×C173—22no—
63C22×C173—22no—
64C22×C173—21no—
65C22×C173—21no—
66C22×C173—21no—
67C22×C173C2×D414yes—
68C22×C173C2411yescharacteristic
69C2×D173—21no—
70C2×D173—21no—
71C2×D173—22no—
72C2×D173—22no—
73C2×D173—22no—
74C2×D173—48no—
75C2×D173#4112yescomplemented
76C4×C173—21no—
77C4×C173—22no—
78C4×C173C2×D412yes—
79Dic173—21no—
80Dic173—21no—
81Dic173—21no—
82Dic173—22no—
83C2×C173—24no—
84C2×C173—41no—
85C2×C173C22×D411yescharacteristic
86C2×C173C2×#4111yescharacteristic
87C2×C173H(5,𝔽2)11yescharacteristic
88D173—22no—
89D173—42no—
90C173#54611yescharacteristic, complemented, Sylow
91#546—1731nomaximal, Sylow
92C2×#41D17311yescharacteristic, complemented
93C22×D4—1731no—
94#124—1731no—
95#123—1732no—
96C22⋊Q8—1731no—
97#121—1732no—
98#121—1731no—
99#120—1732no—
100C4×D4—1731no—
101C4×D4—1731no—
102C2×C22⋊C4—1732no—
103C24—1732no—
104#41—24no—
105C2×Q8C2×D17311yescharacteristic, complemented
106C2×D4—3464no—
107C2×D4—1731no—
108C2×D4—1732no—
109C2×D4—1732no—
110C2×D4—1731no—
111C2×D4C2×D17311yescharacteristic
112C2×D4C2×D17312yescomplemented
113C22×C4—1731no—
114C22×C4—1732no—
115C22×C4C2×D17312yescomplemented
116C22×C4C2×D17311yescharacteristic, complemented
117C4⋊C4—1731no—
118C4⋊C4—1731no—
119C4⋊C4—1732no—
120C22⋊C4—1732no—
121C22⋊C4—1731no—
122C22⋊C4—1731no—
123C22⋊C4—1732no—
124C22⋊C4—1732no—
125C22⋊C4—1732no—
126C22⋊C4—1732no—
127C42—1731no—
128C23—3464no—
129C23—3464no—
130C23—1732no—
131C23—1731no—
132C23—6922no—
133C23C22×D17311yescharacteristic
134C23C22×D17312yes—
135Q8—22no—
136D4—24no—
137D4—3462no—
138D4—3464no—
139D4—3462no—
140D4—6922no—
141D4C22⋊D17314yescomplemented
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—3462no—
145C2×C4—1731no—
146C2×C4—1731no—
147C2×C4—1732no—
148C2×C4—1731no—
149C2×C4—6922no—
150C2×C4C22×D17311yescharacteristic
151C2×C4C22×D17312yes—
152C2×C4C22×D17311yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—3461no—
159C22—3461no—
160C22—3462no—
161C22—3462no—
162C22—3462no—
163C22—1732no—
164C22—6928no—
165C22C2×C22⋊D17314yes—
166C22C23×D17311yescharacteristic
167C4—21no—
168C4—22no—
169C4—3461no—
170C4—3461no—
171C4—3461no—
172C4—3462no—
173C4C2×C22⋊D17312yes—
174C2—24no—
175C2—41no—
176C2—3462no—
177C2—6922no—
178C2C22×C22⋊D17311yescharacteristic
179C2#41×D17311yescharacteristic
180C2#1914972511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
FittingC2×#41×C173C2
hypercenterC2×#41D173
hypocenterC173#546
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D4×D173C2semidirect
2. #19149668C2semidirect
3. #19149662C2semidirect
4. #19149655C2semidirect
5. #19149653C2semidirect
6. #19149651C2semidirect
7. #19149647C2semidirect
8. D4×Dic173C2semidirect
9. C4×C22⋊D173C2semidirect
10. C2×#14583803C2semidirect
11. C2×#41×C173C2semidirect
16. C2×C22⋊D173C22semidirect
18. C22×Dic173C22semidirect
20. C4⋊Dic173C22semidirect
21. #14583805C22semidirect
22. #14583805C22semidirect
23. C22⋊Dic173C22semidirect
24. C22⋊Dic173C22semidirect
27. #14583803C22semidirect
28. #14583803C22semidirect
29. #14583803C22semidirect
30. C4×Dic173C22semidirect
32. C2×Q8×C173C22semidirect
34. C2×D4×C173C22semidirect
35. C22×C4×C173C22semidirect
36. C22×C4×C173C22semidirect
56. D4×C173D4semidirect
75. C2×D173#41semidirect
90. C173#546semidirect
92. C2×#41D173semidirect
105. C2×Q8C2×D173semidirect
112. C2×D4C2×D173semidirect
115. C22×C4C2×D173semidirect
116. C22×C4C2×D173semidirect
141. D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D1732
#191496682
#191496622
#191496552
#191496532
#191496512
#191496472
D4×Dic1732
C4×C22⋊D1732
C2×#145838032
C2×#41×C1732
#546173