Finite Group Databasebeta

← #37527004

Subgroups — #37527004

Order 22976.

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

normal subgroups only

This group has 346 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
1C22×D1436C211yesmaximal, characteristic, complemented
2C2×D4×D359C211yesmaximal, characteristic, complemented
3#26040000C212yesmaximal, complemented
4#26039998C211yesmaximal, characteristic, complemented
5#26039990C212yesmaximal, complemented
6#26039988C212yesmaximal, complemented
7#26039985C211yesmaximal, characteristic, complemented
8#26039985C212yesmaximal, complemented
9#26039983C211yesmaximal, characteristic, complemented
10C22⋊C4×D359C211yesmaximal, characteristic, complemented
11#123×C359C211yesmaximal, characteristic, complemented
12C23×D359C2211yescharacteristic
13C23×D359C2211yescharacteristic, complemented
14C23×D359C2211yescharacteristic, complemented
15C2×D1436—24no—
16C2×D1436C2212yes—
17C2×D1436C2212yescomplemented
18C2×D1436C2211yescharacteristic, complemented
19C2×D1436C2212yescomplemented
20C2×C22⋊D359C2211yescharacteristic
21C2×C22⋊D359C2211yescharacteristic, complemented
22C2×C22⋊D359C2212yescomplemented
23D4×D359—24no—
24C2×C4×D359C2211yescharacteristic, complemented
25C2×C4×D359C2212yescomplemented
26Dic359⋊C4C2212yescomplemented
27C22⋊Dic359C2211yescharacteristic, complemented
28(C2×D359)⋊C4C2212yescomplemented
29(C2×D359)⋊C4C2212yescomplemented
30(C2×D359)⋊C4C2212yescomplemented
31(C2×D359)⋊C4C2212yescomplemented
32C2×D4×C359C2211yescharacteristic
33C22×C4×C359C2211yescharacteristic, complemented
34C4⋊C4×C359C2212yescomplemented
35C22⋊C4×C359C2211yescharacteristic, complemented
36C22⋊C4×C359C2212yescomplemented
37C22×D359—22no—
38C22×D359—22no—
39C22×D359—42no—
40C22×D359—42no—
41C22×D359—42no—
42C22×D359D414yescomplemented
43C22×D359C2311yescharacteristic
44C22×D359C2312yes—
45C22×D359C2311yescharacteristic
46C22×D359C2311yescharacteristic
47D1436—22no—
48D1436—24no—
49D1436—24no—
50D1436—42no—
51C22⋊D359—22no—
52C22⋊D359—22no—
53C22⋊D359—42no—
54C2×Dic359C2312yes—
55C2×Dic359C2311yescharacteristic
56C4×D359—22no—
57C4×D359—42no—
58C23×C359C2311yescharacteristic
59C23×C359C2311yescharacteristic
60D4×C359—22no—
61C2×C4×C359—22no—
62C2×C4×C359C2312yes—
63C2×C4×C359C2312yes—
64C2×C4×C359C2311yescharacteristic
65C22×C359—22no—
66C22×C359—21no—
67C22×C359—22no—
68C22×C359C2×D412yes—
69C22×C359C2411yescharacteristic
70C2×D359—22no—
71C2×D359—22no—
72C2×D359—21no—
73C2×D359—22no—
74C2×D359—22no—
75C2×D359—22no—
76C2×D359—22no—
77C2×D359—44no—
78C2×D359—44no—
79C2×D359—44no—
80C2×D359C2×D412yes—
81C2×D359C2×D412yes—
82C4×C359—22no—
83C4×C359—21no—
84C4×C359—22no—
85Dic359—21no—
86Dic359—22no—
87C2×C359—22no—
88C2×C359—41no—
89C2×C359C22×D411yescharacteristic
90C2×C359H(5,𝔽2)12yes—
91D359—22no—
92D359—22no—
93D359—41no—
94D359—42no—
95C359#53411yescharacteristic, complemented, Sylow
96#534—3591nomaximal, Sylow
97C22×D4—3591no—
98C22×D4—3591no—
99#123—3591no—
100#123—3592no—
101#123D35911yescharacteristic, complemented
102#121—3592no—
103#121—3592no—
104#120—3591no—
105#120—3592no—
106#120—3591no—
107C2×C22⋊C4—3591no—
108C24—3591no—
109C24—3591no—
110C24—3591no—
111C2×D4—3592no—
112C2×D4—3591no—
113C2×D4—3591no—
114C2×D4—3592no—
115C2×D4—3592no—
116C2×D4—3591no—
117C2×D4—3592no—
118C2×D4—7184no—
119C2×D4—7184no—
120C2×D4C2×D35911yescharacteristic
121C22×C4—3592no—
122C22×C4—3591no—
123C22×C4C2×D35911yescharacteristic, complemented
124C4⋊C4—3592no—
125C4⋊C4C2×D35912yescomplemented
126C22⋊C4—3591no—
127C22⋊C4—3592no—
128C22⋊C4—3592no—
129C22⋊C4—3592no—
130C22⋊C4—3592no—
131C22⋊C4C2×D35911yescharacteristic, complemented
132C22⋊C4C2×D35912yescomplemented
133C23—3594no—
134C23—3591no—
135C23—3591no—
136C23—3592no—
137C23—3591no—
138C23—14362no—
139C23—14362no—
140C23—14362no—
141C23—7182no—
142C23—7182no—
143C23C22×D35911yescharacteristic
144C23C22×D35911yescharacteristic
145D4—22no—
146D4—14362no—
147D4—14362no—
148D4—7184no—
149D4—7184no—
150D4—7182no—
151D4—7182no—
152D4—7182no—
153C2×C4—22no—
154C2×C4—3592no—
155C2×C4—3591no—
156C2×C4—14362no—
157C2×C4—7182no—
158C2×C4C22×D35912yes—
159C2×C4C22×D35911yescharacteristic
160C2×C4C22×D35912yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—3592no—
165C22—3592no—
166C22—14364no—
167C22—14364no—
168C22—14364no—
169C22—7182no—
170C22—7181no—
171C22—7182no—
172C22—7182no—
173C22—7182no—
174C22—7182no—
175C22—7182no—
176C22D4×D35912yes—
177C22C23×D35911yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—7181no—
182C4—7182no—
183C2—22no—
184C2—41no—
185C2—14362no—
186C2—14361no—
187C2—7182no—
188C2—7182no—
189C2C2×D4×D35911yescharacteristic
190C2#2604006112yes—
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#123×C359C2
hypercenter#123D359
hypocenterC359#534
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D1436C2semidirect
2. C2×D4×D359C2semidirect
3. #26040000C2semidirect
4. #26039998C2semidirect
5. #26039990C2semidirect
6. #26039988C2semidirect
7. #26039985C2semidirect
8. #26039985C2semidirect
9. #26039983C2semidirect
10. C22⋊C4×D359C2semidirect
11. #123×C359C2semidirect
13. C23×D359C22semidirect
14. C23×D359C22semidirect
17. C2×D1436C22semidirect
18. C2×D1436C22semidirect
19. C2×D1436C22semidirect
21. C2×C22⋊D359C22semidirect
22. C2×C22⋊D359C22semidirect
24. C2×C4×D359C22semidirect
25. C2×C4×D359C22semidirect
26. Dic359⋊C4C22semidirect
27. C22⋊Dic359C22semidirect
28. (C2×D359)⋊C4C22semidirect
29. (C2×D359)⋊C4C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. (C2×D359)⋊C4C22semidirect
33. C22×C4×C359C22semidirect
34. C4⋊C4×C359C22semidirect
35. C22⋊C4×C359C22semidirect
36. C22⋊C4×C359C22semidirect
42. C22×D359D4semidirect
95. C359#534semidirect
101. #123D359semidirect
123. C22×C4C2×D359semidirect
125. C4⋊C4C2×D359semidirect
131. C22⋊C4C2×D359semidirect
132. C22⋊C4C2×D359semidirect

Maximal subgroups

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

subgroupindex
C22×D14362
C2×D4×D3592
#260400002
#260399982
#260399902
#260399882
#260399852
#260399832
C22⋊C4×D3592
#123×C3592
#534359