Finite Group Databasebeta

← #26012136

Subgroups — #26012136

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×#14583825C211yesmaximal, characteristic, complemented
2C2×D4×D173C211yesmaximal, characteristic, complemented
3#19149667C211yesmaximal, characteristic, complemented
4#19149663C212yesmaximal, complemented
5#19149659C211yesmaximal, characteristic, complemented
6#19149654C211yesmaximal, characteristic, complemented
7#19149653C212yesmaximal, complemented
8#19149649C212yesmaximal, complemented
9C4×D692C211yesmaximal, characteristic, complemented
10C2×#14583803C212yesmaximal, complemented
11C4×D4×C173C211yesmaximal, characteristic, complemented
12C23×D173C2212yes—
13#14583825—24no—
14C2×Dic346C2211yescharacteristic, complemented
15C2×D692C2211yescharacteristic
16C2×D692C2211yescharacteristic
17C2×D692C2212yescomplemented
18C2×C22⋊D173C2212yes—
19C2×C22⋊D173C2212yescomplemented
20D4×D173—24no—
21C22×Dic173C2212yescomplemented
22C2×C4×D173C2211yescharacteristic
23C2×C4×D173C2211yescharacteristic, complemented
24C4⋊Dic173C2211yescharacteristic, complemented
25C4⋊Dic173C2212yescomplemented
26#14583803C2211yescharacteristic
27#14583803C2212yes—
28#14583803C2212yescomplemented
29#14583803C2212yescomplemented
30#14583803C2211yescharacteristic, complemented
31#14583803C2212yescomplemented
32C2×D4×C173C2211yescharacteristic
33C22×C4×C173C2212yescomplemented
34C4⋊C4×C173C2211yescharacteristic, complemented
35C22⋊C4×C173C2212yescomplemented
36C42×C173C2211yescharacteristic, complemented
37C22×D173—24no—
38C22×D173—24no—
39C22×D173—42no—
40C22×D173C2311yescharacteristic
41C22×D173C2312yes—
42C22×D173C2311yescharacteristic
43Dic346—22no—
44D692—22no—
45D692—22no—
46D692—42no—
47C22⋊D173—24no—
48C22⋊D173—24no—
49C2×Dic173—24no—
50C2×Dic173C2311yescharacteristic
51C2×Dic173C2311yescharacteristic
52C2×Dic173C2312yes—
53C4×D173—22no—
54C4×D173—22no—
55C23×C173C2312yes—
56D4×C173D414yescomplemented
57C2×C4×C173—42no—
58C2×C4×C173C2311yescharacteristic
59C2×C4×C173C2312yes—
60C2×C4×C173C2311yescharacteristic
61C2×C4×C173C2311yescharacteristic
62C22×C173—22no—
63C22×C173—22no—
64C22×C173C2×D414yes—
65C22×C173C2411yescharacteristic
66C2×D173—21no—
67C2×D173—21no—
68C2×D173—21no—
69C2×D173—22no—
70C2×D173—21no—
71C2×D173—22no—
72C2×D173—21no—
73C2×D173—22no—
74C2×D173—48no—
75C2×D173#4112yescomplemented
76C4×C173—21no—
77C4×C173—21no—
78C4×C173—22no—
79C4×C173C2×D412yes—
80Dic173—21no—
81Dic173—21no—
82Dic173—22no—
83C2×C173—24no—
84C2×C173C22×D411yescharacteristic
85C2×C173C2×#4111yescharacteristic
86C2×C173H(5,𝔽2)11yescharacteristic
87D173—22no—
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⋊Q8—1731no—
97#121—1732no—
98#121—1731no—
99#120—1732no—
100C4×D4—1731no—
101C4×D4D17311yescharacteristic, complemented
102C2×C22⋊C4—1732no—
103C24—1732no—
104#41—3464no—
105C2×Q8—1731no—
106C2×D4—3464no—
107C2×D4—1732no—
108C2×D4—1732no—
109C2×D4—1731no—
110C2×D4—1731no—
111C2×D4—1732no—
112C2×D4C2×D17311yescharacteristic
113C22×C4—1732no—
114C22×C4—1731no—
115C22×C4—1731no—
116C22×C4C2×D17312yescomplemented
117C4⋊C4—1731no—
118C4⋊C4—1732no—
119C4⋊C4C2×D17311yescharacteristic, complemented
120C22⋊C4—1731no—
121C22⋊C4—1732no—
122C22⋊C4—1732no—
123C22⋊C4—1732no—
124C22⋊C4—1731no—
125C22⋊C4—1732no—
126C22⋊C4C2×D17312yescomplemented
127C42C2×D17311yescharacteristic, complemented
128C23—3464no—
129C23—3464no—
130C23—1732no—
131C23—1731no—
132C23—1731no—
133C23—6922no—
134C23C22×D17312yes—
135Q8—3462no—
136D4—3462no—
137D4—3464no—
138D4—3464no—
139D4—3462no—
140D4—6922no—
141D4D69214yescomplemented
142C2×C4—42no—
143C2×C4—3464no—
144C2×C4—3462no—
145C2×C4—3462no—
146C2×C4—1731no—
147C2×C4—1732no—
148C2×C4—1731no—
149C2×C4C22×D17311yescharacteristic
150C2×C4C22×D17311yescharacteristic
151C2×C4C22×D17312yes—
152C2×C4C22×D17311yescharacteristic
153C22—22no—
154C22—22no—
155C22—3461no—
156C22—3461no—
157C22—3461no—
158C22—3462no—
159C22—3461no—
160C22—3462no—
161C22—3461no—
162C22—3462no—
163C22—1732no—
164C22—6928no—
165C22C2×D69214yes—
166C22C23×D17311yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—3461no—
171C4—3461no—
172C4—3462no—
173C4C2×D69212yes—
174C2—24no—
175C2—3462no—
176C2—6921no—
177C2—6922no—
178C2C22×D69211yescharacteristic
179C2#41×D17311yescharacteristic
180C2#1914972411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
FittingC4×D4×C173C2
hypercenterC4×D4D173
hypocenterC173#546
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14583825C2semidirect
2. C2×D4×D173C2semidirect
3. #19149667C2semidirect
4. #19149663C2semidirect
5. #19149659C2semidirect
6. #19149654C2semidirect
7. #19149653C2semidirect
8. #19149649C2semidirect
9. C4×D692C2semidirect
10. C2×#14583803C2semidirect
11. C4×D4×C173C2semidirect
14. C2×Dic346C22semidirect
17. C2×D692C22semidirect
19. C2×C22⋊D173C22semidirect
21. C22×Dic173C22semidirect
23. C2×C4×D173C22semidirect
24. C4⋊Dic173C22semidirect
25. C4⋊Dic173C22semidirect
28. #14583803C22semidirect
29. #14583803C22semidirect
30. #14583803C22semidirect
31. #14583803C22semidirect
33. C22×C4×C173C22semidirect
34. C4⋊C4×C173C22semidirect
35. C22⋊C4×C173C22semidirect
36. C42×C173C22semidirect
56. D4×C173D4semidirect
75. C2×D173#41semidirect
90. C173#546semidirect
101. C4×D4D173semidirect
116. C22×C4C2×D173semidirect
119. C4⋊C4C2×D173semidirect
126. C22⋊C4C2×D173semidirect
127. C42C2×D173semidirect
141. D4D692semidirect

Maximal subgroups

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

subgroupindex
C2×#145838252
C2×D4×D1732
#191496672
#191496632
#191496592
#191496542
#191496532
#191496492
C4×D6922
C2×#145838032
C4×D4×C1732
#546173