Finite Group Databasebeta

← #26012001

Subgroups — #26012001

Order 11072.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C22×C4×D173C211yesmaximal, characteristic, complemented
3#19149661C212yesmaximal, complemented
4#19149659C211yesmaximal, characteristic, complemented
5#19149658C211yesmaximal, characteristic, complemented
6#19149650C211yesmaximal, characteristic, complemented
7#19149647C212yesmaximal, complemented
8D4×Dic173C212yesmaximal, complemented
9#19149639C212yesmaximal, complemented
10#19149637C211yesmaximal, characteristic, complemented
11#121×C173C211yesmaximal, characteristic, complemented
12C23×D173C2211yescharacteristic
13#14583825—24no—
14C2×Dic346C2211yescharacteristic, complemented
15C2×C22⋊D173C2212yes—
16C2×C22⋊D173C2212yes—
17C22×Dic173C2211yescharacteristic
18C22×Dic173C2212yescomplemented
19C2×C4×D173—24no—
20C2×C4×D173C2211yescharacteristic
21C2×C4×D173C2211yescharacteristic
22C2×C4×D173C2211yescharacteristic
23C2×C4×D173C2211yescharacteristic
24C4⋊Dic173C2211yescharacteristic, complemented
25C4⋊Dic173C2212yescomplemented
26#14583805C2212yescomplemented
27C22⋊Dic173C2212yes—
28C22⋊Dic173C2212yescomplemented
29#14583803C2212yes—
30#14583803C2212yes—
31C4×Dic173C2212yescomplemented
32C2×D4×C173C2212yes—
33C2×D4×C173C2211yescharacteristic, complemented
34C22×C4×C173C2211yescharacteristic
35C4⋊C4×C173C2211yescharacteristic, complemented
36C22⋊C4×C173C2212yescomplemented
37C22×D173—22no—
38C22×D173—22no—
39C22×D173—22no—
40C22×D173C2311yescharacteristic
41C22×D173C2311yescharacteristic
42Dic346—22no—
43C22⋊D173—24no—
44C22⋊D173—24no—
45C2×Dic173—22no—
46C2×Dic173—24no—
47C2×Dic173C2311yescharacteristic
48C2×Dic173C2312yes—
49C2×Dic173C2312yes—
50C2×Dic173C2311yescharacteristic
51C4×D173—22no—
52C4×D173—22no—
53C4×D173—22no—
54C4×D173D414yescomplemented
55C23×C173C2312yes—
56C23×C173C2311yescharacteristic
57D4×C173—24no—
58D4×C173—22no—
59C2×C4×C173—22no—
60C2×C4×C173C2311yescharacteristic
61C2×C4×C173C2312yes—
62C2×C4×C173C2311yescharacteristic
63C22×C173—21no—
64C22×C173—21no—
65C22×C173—22no—
66C22×C173—22no—
67C22×C173—22no—
68C22×C173#4112yescomplemented
69C22×C173C2411yescharacteristic
70C2×D173—28no—
71C2×D173—21no—
72C2×D173—21no—
73C2×D173—21no—
74C2×D173—21no—
75C2×D173C2×D412yes—
76C2×D173#4112yescomplemented
77C4×C173—21no—
78C4×C173—22no—
79C4×C173C2×D412yes—
80Dic173—21no—
81Dic173—22no—
82Dic173—22no—
83Dic173C2×D412yes—
84C2×C173—22no—
85C2×C173—42no—
86C2×C173C22×D411yescharacteristic
87C2×C173C2×#4111yescharacteristic
88C2×C173C2×#4111yescharacteristic
89D173—22no—
90D173—22no—
91C173#52511yescharacteristic, complemented, Sylow
92#525—1731nomaximal, Sylow
93C2×#41—1731no—
94C23×C4—1731no—
95#123—1732no—
96C22⋊Q8—1731no—
97C22⋊Q8—1731no—
98#121—1731no—
99#121D17311yescharacteristic, complemented
100#120—1732no—
101C4×D4—1732no—
102C4×D4—1732no—
103#117—1731no—
104C24—1731no—
105#41—3464no—
106C2×Q8—1731no—
107C2×D4—1732no—
108C2×D4—1732no—
109C2×D4C2×D17312yes—
110C2×D4C2×D17311yescharacteristic, complemented
111C22×C4—3464no—
112C22×C4—1731no—
113C22×C4—1732no—
114C22×C4—1731no—
115C22×C4—1731no—
116C22×C4—1731no—
117C22×C4—1731no—
118C22×C4C2×D17311yescharacteristic
119C4⋊C4—1731no—
120C4⋊C4—1732no—
121C4⋊C4—1732no—
122C4⋊C4C2×D17311yescharacteristic, complemented
123C22⋊C4—1732no—
124C22⋊C4—1732no—
125C22⋊C4—1732no—
126C22⋊C4—1732no—
127C22⋊C4C2×D17312yescomplemented
128C42—1732no—
129C23—3462no—
130C23—3462no—
131C23—3462no—
132C23—1731no—
133C23—1731no—
134C23C22×D17312yes—
135C23C22×D17311yescharacteristic
136Q8—3462no—
137D4—24no—
138D4—22no—
139D4—3464no—
140D4—3464no—
141C2×C4—22no—
142C2×C4—3462no—
143C2×C4—3462no—
144C2×C4—3462no—
145C2×C4—3464no—
146C2×C4—3462no—
147C2×C4—1732no—
148C2×C4—1731no—
149C2×C4—1732no—
150C2×C4—1731no—
151C2×C4—1734no—
152C2×C4C22×D17312yes—
153C2×C4C22×D17311yescharacteristic
154C2×C4C22×D17311yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3468no—
161C22—3461no—
162C22—3461no—
163C22—3461no—
164C22—3461no—
165C22—1732no—
166C22—1732no—
167C22#1458382512yescomplemented
168C22C23×D17311yescharacteristic
169C4—21no—
170C4—22no—
171C4—3461no—
172C4—3462no—
173C4—3462no—
174C4—1732no—
175C4D4×D17312yes—
176C2—22no—
177C2—42no—
178C2—3462no—
179C2—3462no—
180C2C2×D4×D17311yescharacteristic
181C2#41×D17311yescharacteristic
182C2C2×#1458382511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#121×C173C2
hypercenter#121D173
hypocenterC173#525
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#14583825C2semidirect
2. C22×C4×D173C2semidirect
3. #19149661C2semidirect
4. #19149659C2semidirect
5. #19149658C2semidirect
6. #19149650C2semidirect
7. #19149647C2semidirect
8. D4×Dic173C2semidirect
9. #19149639C2semidirect
10. #19149637C2semidirect
11. #121×C173C2semidirect
14. C2×Dic346C22semidirect
18. C22×Dic173C22semidirect
24. C4⋊Dic173C22semidirect
25. C4⋊Dic173C22semidirect
26. #14583805C22semidirect
28. C22⋊Dic173C22semidirect
31. C4×Dic173C22semidirect
33. C2×D4×C173C22semidirect
35. C4⋊C4×C173C22semidirect
36. C22⋊C4×C173C22semidirect
54. C4×D173D4semidirect
68. C22×C173#41semidirect
76. C2×D173#41semidirect
91. C173#525semidirect
99. #121D173semidirect
110. C2×D4C2×D173semidirect
122. C4⋊C4C2×D173semidirect
127. C22⋊C4C2×D173semidirect
167. C22#14583825semidirect

Maximal subgroups

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

subgroupindex
C2×#145838252
C22×C4×D1732
#191496612
#191496592
#191496582
#191496502
#191496472
D4×Dic1732
#191496392
#191496372
#121×C1732
#525173