Finite Group Databasebeta

← #36378352

Subgroups — #36378352

Order 22592.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 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×#19150736C211yesmaximal, characteristic, complemented
2C22×C22⋊D353C211yesmaximal, characteristic, complemented
3#26014988C212yesmaximal, complemented
4#26014978C212yesmaximal, complemented
5#26014965C212yesmaximal, complemented
6#26014964C212yesmaximal, complemented
7#26014962C211yesmaximal, characteristic, complemented
8#26014961C211yesmaximal, characteristic, complemented
9#26014960C211yesmaximal, characteristic, complemented
10#26014948C211yesmaximal, characteristic, complemented
11#120×C353C211yesmaximal, characteristic, complemented
12C23×D353C2211yescharacteristic
13#19150736—24no—
14C2×Dic706C2211yescharacteristic, complemented
15C2×D1412C2212yescomplemented
16C2×C22⋊D353—24no—
17C2×C22⋊D353C2212yes—
18C2×C22⋊D353C2212yes—
19C2×C22⋊D353C2211yescharacteristic
20C2×C22⋊D353C2211yescharacteristic, complemented
21C2×C22⋊D353C2212yescomplemented
22C2×C22⋊D353C2212yescomplemented
23C22×Dic353C2211yescharacteristic
24C22×Dic353C2211yescharacteristic, complemented
25C2×C4×D353C2211yescharacteristic, complemented
26#19150716C2212yescomplemented
27C22⋊Dic353C2211yescharacteristic
28C22⋊Dic353C2212yescomplemented
29#19150714C2212yescomplemented
30#19150714C2212yescomplemented
31C4×Dic353C2212yescomplemented
32C24×C353C2211yescharacteristic
33C2×D4×C353C2211yescharacteristic
34C2×D4×C353C2212yescomplemented
35C22⋊C4×C353C2211yescharacteristic
36C22⋊C4×C353C2212yescomplemented
37C22×D353—22no—
38C22×D353—42no—
39C22×D353C2312yes—
40C22×D353C2311yescharacteristic
41Dic706—22no—
42D1412—42no—
43C22⋊D353—22no—
44C22⋊D353—24no—
45C22⋊D353—24no—
46C22⋊D353—22no—
47C22⋊D353—42no—
48C22⋊D353—42no—
49C2×Dic353—22no—
50C2×Dic353D414yescomplemented
51C2×Dic353C2312yes—
52C2×Dic353C2311yescharacteristic
53C2×Dic353C2311yescharacteristic
54C2×Dic353C2311yescharacteristic
55C4×D353—22no—
56C23×C353—22no—
57C23×C353—42no—
58C23×C353C2312yes—
59C23×C353C2311yescharacteristic
60C23×C353C2311yescharacteristic
61D4×C353—22no—
62D4×C353—42no—
63C2×C4×C353C2312yes—
64C2×C4×C353C2311yescharacteristic
65C22×C353—22no—
66C22×C353—21no—
67C22×C353—22no—
68C22×C353—22no—
69C22×C353—22no—
70C22×C353—22no—
71C22×C353—44no—
72C22×C353C2×D412yes—
73C22×C353C2411yescharacteristic
74C2×D353—21no—
75C2×D353—22no—
76C2×D353—22no—
77C2×D353—22no—
78C2×D353—22no—
79C2×D353—44no—
80C4×C353—21no—
81C4×C353—22no—
82Dic353—21no—
83Dic353—22no—
84Dic353C2×D412yes—
85Dic353C2×D412yes—
86C2×C353—22no—
87C2×C353—41no—
88C2×C353—42no—
89C2×C353C22×D411yescharacteristic
90C2×C353H(5,𝔽2)12yes—
91D353—41no—
92D353—42no—
93C353#53511yescharacteristic, complemented, Sylow
94#535—3531nomaximal, Sylow
95C2×#41—3531no—
96C22×D4—3531no—
97#127—3532no—
98#124—3532no—
99#121—3532no—
100#121—3532no—
101#120—3531no—
102#120—3531no—
103#120—3531no—
104#120D35311yescharacteristic, complemented
105#117—3531no—
106C24—3531no—
107C24C2×D35311yescharacteristic
108#41—7064no—
109C2×Q8—3531no—
110C2×D4—3532no—
111C2×D4—3532no—
112C2×D4—3531no—
113C2×D4—3532no—
114C2×D4—3532no—
115C2×D4—3531no—
116C2×D4—3532no—
117C2×D4—7064no—
118C2×D4C2×D35311yescharacteristic
119C2×D4C2×D35312yescomplemented
120C22×C4—3531no—
121C22×C4—3531no—
122C22×C4—3531no—
123C4⋊C4—3532no—
124C22⋊C4—3531no—
125C22⋊C4—3532no—
126C22⋊C4—3532no—
127C22⋊C4—3532no—
128C22⋊C4C2×D35311yescharacteristic
129C22⋊C4C2×D35312yescomplemented
130C42—3532no—
131C23—22no—
132C23—42no—
133C23—3532no—
134C23—3531no—
135C23—14122no—
136C23—7062no—
137C23C22×D35312yes—
138C23C22×D35311yescharacteristic
139C23C22×D35311yescharacteristic
140Q8—7062no—
141D4—22no—
142D4—42no—
143D4—14122no—
144D4—14122no—
145D4—14122no—
146D4—7064no—
147D4—7064no—
148D4—7062no—
149D4—7062no—
150C2×C4—3534no—
151C2×C4—3531no—
152C2×C4—3531no—
153C2×C4—3532no—
154C2×C4—3531no—
155C2×C4—7062no—
156C2×C4—7062no—
157C2×C4C22×D35312yes—
158C2×C4C22×D35311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—14124no—
167C22—7061no—
168C22—7062no—
169C22—7062no—
170C22—7062no—
171C22—7062no—
172C22D4×D35312yes—
173C22C23×D35311yescharacteristic
174C4—21no—
175C4—22no—
176C4—3532no—
177C4—3532no—
178C4—7061no—
179C4—7062no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—14122no—
184C2—14121no—
185C2C2×D4×D35311yescharacteristic
186C2#2601503712yes—
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
Fitting#120×C353C2
hypercenter#120D353
hypocenterC353#535
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#19150736C2semidirect
2. C22×C22⋊D353C2semidirect
3. #26014988C2semidirect
4. #26014978C2semidirect
5. #26014965C2semidirect
6. #26014964C2semidirect
7. #26014962C2semidirect
8. #26014961C2semidirect
9. #26014960C2semidirect
10. #26014948C2semidirect
11. #120×C353C2semidirect
14. C2×Dic706C22semidirect
15. C2×D1412C22semidirect
20. C2×C22⋊D353C22semidirect
21. C2×C22⋊D353C22semidirect
22. C2×C22⋊D353C22semidirect
24. C22×Dic353C22semidirect
25. C2×C4×D353C22semidirect
26. #19150716C22semidirect
28. C22⋊Dic353C22semidirect
29. #19150714C22semidirect
30. #19150714C22semidirect
31. C4×Dic353C22semidirect
34. C2×D4×C353C22semidirect
36. C22⋊C4×C353C22semidirect
50. C2×Dic353D4semidirect
93. C353#535semidirect
104. #120D353semidirect
119. C2×D4C2×D353semidirect
129. C22⋊C4C2×D353semidirect

Maximal subgroups

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

subgroupindex
C2×#191507362
C22×C22⋊D3532
#260149882
#260149782
#260149652
#260149642
#260149622
#260149612
#260149602
#260149482
#120×C3532
#535353