Finite Group Databasebeta

← #37527008

Subgroups — #37527008

Order 22976.

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×#19153340C211yesmaximal, characteristic, complemented
2C22×C22⋊D359C211yesmaximal, characteristic, complemented
3#26040011C212yesmaximal, complemented
4#26040001C212yesmaximal, complemented
5#26039988C212yesmaximal, complemented
6#26039987C212yesmaximal, complemented
7#26039985C211yesmaximal, characteristic, complemented
8#26039984C211yesmaximal, characteristic, complemented
9#26039983C211yesmaximal, characteristic, complemented
10#26039971C211yesmaximal, characteristic, complemented
11#120×C359C211yesmaximal, characteristic, complemented
12C23×D359C2211yescharacteristic
13#19153340—24no—
14C2×Dic718C2211yescharacteristic, complemented
15C2×D1436C2212yescomplemented
16C2×C22⋊D359—24no—
17C2×C22⋊D359C2211yescharacteristic
18C2×C22⋊D359C2212yes—
19C2×C22⋊D359C2212yes—
20C2×C22⋊D359C2211yescharacteristic, complemented
21C2×C22⋊D359C2212yescomplemented
22C2×C22⋊D359C2212yescomplemented
23C22×Dic359C2211yescharacteristic
24C22×Dic359C2211yescharacteristic, complemented
25C2×C4×D359C2211yescharacteristic, complemented
26Dic359⋊C4C2212yescomplemented
27C22⋊Dic359C2211yescharacteristic
28C22⋊Dic359C2212yescomplemented
29(C2×D359)⋊C4C2212yescomplemented
30(C2×D359)⋊C4C2212yescomplemented
31C4×Dic359C2212yescomplemented
32C24×C359C2211yescharacteristic
33C2×D4×C359C2211yescharacteristic
34C2×D4×C359C2212yescomplemented
35C22⋊C4×C359C2211yescharacteristic
36C22⋊C4×C359C2212yescomplemented
37C22×D359—22no—
38C22×D359—42no—
39C22×D359C2312yes—
40C22×D359C2311yescharacteristic
41Dic718—22no—
42D1436—42no—
43C22⋊D359—22no—
44C22⋊D359—24no—
45C22⋊D359—24no—
46C22⋊D359—22no—
47C22⋊D359—42no—
48C22⋊D359—42no—
49C2×Dic359—22no—
50C2×Dic359D414yescomplemented
51C2×Dic359C2312yes—
52C2×Dic359C2311yescharacteristic
53C2×Dic359C2311yescharacteristic
54C2×Dic359C2311yescharacteristic
55C4×D359—22no—
56C23×C359—22no—
57C23×C359—42no—
58C23×C359C2312yes—
59C23×C359C2311yescharacteristic
60C23×C359C2311yescharacteristic
61D4×C359—22no—
62D4×C359—42no—
63C2×C4×C359C2312yes—
64C2×C4×C359C2311yescharacteristic
65C22×C359—22no—
66C22×C359—21no—
67C22×C359—22no—
68C22×C359—22no—
69C22×C359—22no—
70C22×C359—22no—
71C22×C359—44no—
72C22×C359C2×D412yes—
73C22×C359C2411yescharacteristic
74C2×D359—21no—
75C2×D359—22no—
76C2×D359—22no—
77C2×D359—22no—
78C2×D359—22no—
79C2×D359—44no—
80C4×C359—21no—
81C4×C359—22no—
82Dic359—21no—
83Dic359—22no—
84Dic359C2×D412yes—
85Dic359C2×D412yes—
86C2×C359—22no—
87C2×C359—41no—
88C2×C359—42no—
89C2×C359C22×D411yescharacteristic
90C2×C359H(5,𝔽2)12yes—
91D359—41no—
92D359—42no—
93C359#53511yescharacteristic, complemented, Sylow
94#535—3591nomaximal, Sylow
95C2×#41—3591no—
96C22×D4—3591no—
97#127—3592no—
98#124—3592no—
99#121—3592no—
100#121—3592no—
101#120—3591no—
102#120—3591no—
103#120—3591no—
104#120D35911yescharacteristic, complemented
105#117—3591no—
106C24—3591no—
107C24C2×D35911yescharacteristic
108#41—7184no—
109C2×Q8—3591no—
110C2×D4—3592no—
111C2×D4—3592no—
112C2×D4—3591no—
113C2×D4—3592no—
114C2×D4—3592no—
115C2×D4—3591no—
116C2×D4—3592no—
117C2×D4—7184no—
118C2×D4C2×D35911yescharacteristic
119C2×D4C2×D35912yescomplemented
120C22×C4—3591no—
121C22×C4—3591no—
122C22×C4—3591no—
123C4⋊C4—3592no—
124C22⋊C4—3591no—
125C22⋊C4—3592no—
126C22⋊C4—3592no—
127C22⋊C4—3592no—
128C22⋊C4C2×D35911yescharacteristic
129C22⋊C4C2×D35912yescomplemented
130C42—3592no—
131C23—22no—
132C23—42no—
133C23—3592no—
134C23—3591no—
135C23—14362no—
136C23—7182no—
137C23C22×D35912yes—
138C23C22×D35911yescharacteristic
139C23C22×D35911yescharacteristic
140Q8—7182no—
141D4—22no—
142D4—42no—
143D4—14362no—
144D4—14362no—
145D4—14362no—
146D4—7184no—
147D4—7184no—
148D4—7182no—
149D4—7182no—
150C2×C4—3594no—
151C2×C4—3591no—
152C2×C4—3591no—
153C2×C4—3592no—
154C2×C4—3591no—
155C2×C4—7182no—
156C2×C4—7182no—
157C2×C4C22×D35912yes—
158C2×C4C22×D35911yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—14364no—
167C22—7181no—
168C22—7182no—
169C22—7182no—
170C22—7182no—
171C22—7182no—
172C22D4×D35912yes—
173C22C23×D35911yescharacteristic
174C4—21no—
175C4—22no—
176C4—3592no—
177C4—3592no—
178C4—7181no—
179C4—7182no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—14362no—
184C2—14361no—
185C2C2×D4×D35911yescharacteristic
186C2#2604006012yes—
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#120×C359C2
hypercenter#120D359
hypocenterC359#535
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#19153340C2semidirect
2. C22×C22⋊D359C2semidirect
3. #26040011C2semidirect
4. #26040001C2semidirect
5. #26039988C2semidirect
6. #26039987C2semidirect
7. #26039985C2semidirect
8. #26039984C2semidirect
9. #26039983C2semidirect
10. #26039971C2semidirect
11. #120×C359C2semidirect
14. C2×Dic718C22semidirect
15. C2×D1436C22semidirect
20. C2×C22⋊D359C22semidirect
21. C2×C22⋊D359C22semidirect
22. C2×C22⋊D359C22semidirect
24. C22×Dic359C22semidirect
25. C2×C4×D359C22semidirect
26. Dic359⋊C4C22semidirect
28. C22⋊Dic359C22semidirect
29. (C2×D359)⋊C4C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. C4×Dic359C22semidirect
34. C2×D4×C359C22semidirect
36. C22⋊C4×C359C22semidirect
50. C2×Dic359D4semidirect
93. C359#535semidirect
104. #120D359semidirect
119. C2×D4C2×D359semidirect
129. C22⋊C4C2×D359semidirect

Maximal subgroups

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

subgroupindex
C2×#191533402
C22×C22⋊D3592
#260400112
#260400012
#260399882
#260399872
#260399852
#260399842
#260399832
#260399712
#120×C3592
#535359