Finite Group Databasebeta

← #26012132

Subgroups — #26012132

Order 11072.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14583827C211yesmaximal, characteristic, complemented
2C22×D692C211yesmaximal, characteristic, complemented
3#19149668C211yesmaximal, characteristic, complemented
4#19149664C212yesmaximal, complemented
5#19149654C211yesmaximal, characteristic, complemented
6#19149654C212yesmaximal, complemented
7#19149649C212yesmaximal, complemented
8#19149641C211yesmaximal, characteristic, complemented
9C4×C22⋊D173C211yesmaximal, characteristic, complemented
10C22⋊C4×D173C212yesmaximal, complemented
11C22⋊Q8×C173C211yesmaximal, characteristic, complemented
12C23×D173C2212yes—
13#14583827—24no—
14C2×D692—24no—
15C2×D692C2212yes—
16C2×D692C2211yescharacteristic
17C2×D692C2211yescharacteristic
18C2×D692C2212yescomplemented
19C2×D692C2212yescomplemented
20C2×C22⋊D173C2211yescharacteristic
21C2×C4×D173C2212yescomplemented
22C2×C4×D173C2211yescharacteristic, complemented
23C2×C4×D173C2212yescomplemented
24#14583805C2211yescharacteristic, complemented
25C22⋊Dic173C2211yescharacteristic
26#14583803C2212yescomplemented
27#14583803C2212yescomplemented
28#14583803C2211yescharacteristic, complemented
29#14583803C2212yescomplemented
30#14583803C2212yescomplemented
31C4×Dic173C2211yescharacteristic, complemented
32C2×Q8×C173C2211yescharacteristic, complemented
33C22×C4×C173C2211yescharacteristic
34C4⋊C4×C173C2211yescharacteristic, complemented
35C4⋊C4×C173C2212yescomplemented
36C22⋊C4×C173C2212yes—
37C22×D173—24no—
38C22×D173—24no—
39C22×D173—42no—
40C22×D173C2311yescharacteristic
41C22×D173C2312yes—
42C22×D173C2312yes—
43D692—24no—
44D692—24no—
45D692—22no—
46D692—42no—
47D692D414yescomplemented
48C22⋊D173—22no—
49C2×Dic173C2311yescharacteristic
50C2×Dic173C2311yescharacteristic
51C2×Dic173C2311yescharacteristic
52C4×D173—24no—
53C4×D173—22no—
54C4×D173—42no—
55C23×C173C2311yescharacteristic
56Q8×C173—22no—
57C2×C4×C173—22no—
58C2×C4×C173C2312yes—
59C2×C4×C173C2312yes—
60C2×C4×C173C2311yescharacteristic
61C2×C4×C173C2311yescharacteristic
62C22×C173—21no—
63C22×C173—21no—
64C22×C173#4112yescomplemented
65C22×C173C2411yescharacteristic
66C2×D173—22no—
67C2×D173—22no—
68C2×D173—21no—
69C2×D173—22no—
70C2×D173—21no—
71C2×D173—22no—
72C2×D173—21no—
73C2×D173—22no—
74C2×D173—48no—
75C2×D173C2×D414yes—
76C4×C173—21no—
77C4×C173—22no—
78C4×C173—22no—
79C4×C173C2×D412yes—
80Dic173—21no—
81Dic173—21no—
82Dic173—21no—
83C2×C173—22no—
84C2×C173C22×D411yescharacteristic
85C2×C173C2×#4111yescharacteristic
86C2×C173H(5,𝔽2)11yescharacteristic
87D173—24no—
88D173—41no—
89D173—42no—
90C173#54611yescharacteristic, complemented, Sylow
91#546—1731nomaximal, Sylow
92C2×#41—1731no—
93C22×D4—1731no—
94#124—1731no—
95#123—1732no—
96C22⋊Q8D17311yescharacteristic, complemented
97#121—1731no—
98#121—1732no—
99#120—1732no—
100C4×D4—1731no—
101C4×D4—1731no—
102C2×C22⋊C4—1732no—
103C24—1732no—
104#41—3464no—
105C2×Q8C2×D17311yescharacteristic, complemented
106C2×D4—3464no—
107C2×D4—1731no—
108C2×D4—1732no—
109C2×D4—1731no—
110C2×D4—1732no—
111C2×D4—1731no—
112C2×D4—1732no—
113C22×C4—1732no—
114C22×C4—1731no—
115C22×C4—1732no—
116C22×C4C2×D17311yescharacteristic
117C4⋊C4—1731no—
118C4⋊C4C2×D17311yescharacteristic, complemented
119C4⋊C4C2×D17312yescomplemented
120C22⋊C4—1731no—
121C22⋊C4—1732no—
122C22⋊C4—1732no—
123C22⋊C4—1732no—
124C22⋊C4—1731no—
125C22⋊C4—1732no—
126C22⋊C4C2×D17312yes—
127C42—1731no—
128C23—3464no—
129C23—3464no—
130C23—1732no—
131C23—1731no—
132C23—1732no—
133C23—6922no—
134C23C22×D17311yescharacteristic
135Q8—22no—
136D4—3464no—
137D4—3464no—
138D4—3462no—
139D4—3462no—
140D4—1734no—
141D4—6922no—
142C2×C4—22no—
143C2×C4—3464no—
144C2×C4—3462no—
145C2×C4—1731no—
146C2×C4—1731no—
147C2×C4—1731no—
148C2×C4—6922no—
149C2×C4C22×D17312yes—
150C2×C4C22×D17312yes—
151C2×C4C22×D17311yescharacteristic
152C2×C4C22×D17311yescharacteristic
153C22—21no—
154C22—21no—
155C22—3462no—
156C22—3462no—
157C22—3461no—
158C22—3462no—
159C22—3461no—
160C22—3462no—
161C22—3461no—
162C22—3462no—
163C22—1734no—
164C22—6928no—
165C22#1458382712yescomplemented
166C22C23×D17311yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—3461no—
171C4—3461no—
172C4—3461no—
173C4D4×D17312yes—
174C2—22no—
175C2—3464no—
176C2—6921no—
177C2—6922no—
178C2C2×D4×D17311yescharacteristic
179C2C2×#1458382711yescharacteristic
180C2#1914972511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
FittingC22⋊Q8×C173C2
hypercenterC22⋊Q8D173
hypocenterC173#546
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14583827C2semidirect
2. C22×D692C2semidirect
3. #19149668C2semidirect
4. #19149664C2semidirect
5. #19149654C2semidirect
6. #19149654C2semidirect
7. #19149649C2semidirect
8. #19149641C2semidirect
9. C4×C22⋊D173C2semidirect
10. C22⋊C4×D173C2semidirect
11. C22⋊Q8×C173C2semidirect
18. C2×D692C22semidirect
19. C2×D692C22semidirect
21. C2×C4×D173C22semidirect
22. C2×C4×D173C22semidirect
23. C2×C4×D173C22semidirect
24. #14583805C22semidirect
26. #14583803C22semidirect
27. #14583803C22semidirect
28. #14583803C22semidirect
29. #14583803C22semidirect
30. #14583803C22semidirect
31. C4×Dic173C22semidirect
32. C2×Q8×C173C22semidirect
34. C4⋊C4×C173C22semidirect
35. C4⋊C4×C173C22semidirect
47. D692D4semidirect
64. C22×C173#41semidirect
90. C173#546semidirect
96. C22⋊Q8D173semidirect
105. C2×Q8C2×D173semidirect
118. C4⋊C4C2×D173semidirect
119. C4⋊C4C2×D173semidirect
165. C22#14583827semidirect

Maximal subgroups

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

subgroupindex
C2×#145838272
C22×D6922
#191496682
#191496642
#191496542
#191496492
#191496412
C4×C22⋊D1732
C22⋊C4×D1732
C22⋊Q8×C1732
#546173