Finite Group Databasebeta

← #37527003

Subgroups — #37527003

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×C22⋊D359C211yesmaximal, characteristic, complemented
2C2×D4×D359C211yesmaximal, characteristic, complemented
3#26039999C211yesmaximal, characteristic, complemented
4#26039997C212yesmaximal, complemented
5#26039996C211yesmaximal, characteristic, complemented
6#26039988C212yesmaximal, complemented
7#26039986C212yesmaximal, complemented
8#26039984C211yesmaximal, characteristic, complemented
9#26039983C212yesmaximal, complemented
10C22⋊C4×D359C211yesmaximal, characteristic, complemented
11#120×C359C211yesmaximal, characteristic, complemented
12C23×D359C2211yescharacteristic
13C23×D359C2211yescharacteristic, complemented
14C2×D1436C2211yescharacteristic, complemented
15C2×C22⋊D359—24no—
16C2×C22⋊D359C2212yes—
17C2×C22⋊D359C2211yescharacteristic
18C2×C22⋊D359C2211yescharacteristic, complemented
19C2×C22⋊D359C2212yescomplemented
20C2×C22⋊D359C2212yescomplemented
21D4×D359—24no—
22C22×Dic359C2211yescharacteristic, complemented
23C2×C4×D359C2211yescharacteristic, complemented
24C2×C4×D359C2212yescomplemented
25C4⋊Dic359C2212yescomplemented
26Dic359⋊C4C2212yescomplemented
27C22⋊Dic359C2211yescharacteristic, complemented
28C22⋊Dic359C2212yescomplemented
29C22⋊Dic359C2212yescomplemented
30(C2×D359)⋊C4C2212yescomplemented
31(C2×D359)⋊C4C2212yescomplemented
32C24×C359C2211yescharacteristic, complemented
33C2×D4×C359C2211yescharacteristic
34C2×D4×C359C2212yescomplemented
35C22⋊C4×C359C2211yescharacteristic, complemented
36C22⋊C4×C359C2212yescomplemented
37C22×D359—22no—
38C22×D359—42no—
39C22×D359—42no—
40C22×D359D414yescomplemented
41C22×D359C2311yescharacteristic
42C22×D359C2311yescharacteristic
43C22×D359C2311yescharacteristic
44D1436—22no—
45C22⋊D359—22no—
46C22⋊D359—22no—
47C22⋊D359—24no—
48C22⋊D359—24no—
49C22⋊D359—42no—
50C2×Dic359—22no—
51C2×Dic359C2312yes—
52C2×Dic359C2312yes—
53C2×Dic359C2311yescharacteristic
54C4×D359—22no—
55C4×D359—42no—
56C23×C359—22no—
57C23×C359—42no—
58C23×C359C2312yes—
59C23×C359C2311yescharacteristic
60C23×C359C2311yescharacteristic
61D4×C359—22no—
62D4×C359—42no—
63C2×C4×C359C2311yescharacteristic
64C2×C4×C359C2312yes—
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—22no—
75C2×D359—22no—
76C2×D359—21no—
77C2×D359—22no—
78C2×D359—44no—
79C2×D359—44no—
80C2×D359C2×D412yes—
81C2×D359C2×D412yes—
82C4×C359—21no—
83C4×C359—22no—
84Dic359—21no—
85Dic359—22no—
86Dic359—22no—
87C2×C359—22no—
88C2×C359—41no—
89C2×C359—42no—
90C2×C359C22×D411yescharacteristic
91C2×C359H(5,𝔽2)12yes—
92D359—22no—
93D359—22no—
94D359—41no—
95C359#53411yescharacteristic, complemented, Sylow
96#534—3591nomaximal, Sylow
97C22×D4—3591no—
98C22×D4—3591no—
99#123—3591no—
100#123—3592no—
101#123—3591no—
102#121—3592no—
103#121—3592no—
104#120—3591no—
105#120—3592no—
106#120D35911yescharacteristic, complemented
107C2×C22⋊C4—3591no—
108C24—3591no—
109C24—3591no—
110C24C2×D35911yescharacteristic, complemented
111C2×D4—3592no—
112C2×D4—3592no—
113C2×D4—3591no—
114C2×D4—3591no—
115C2×D4—3592no—
116C2×D4—3591no—
117C2×D4—7184no—
118C2×D4—7184no—
119C2×D4C2×D35911yescharacteristic
120C2×D4C2×D35912yescomplemented
121C22×C4—3591no—
122C22×C4—3592no—
123C22×C4—3591no—
124C4⋊C4—3592no—
125C4⋊C4—3592no—
126C22⋊C4—3591no—
127C22⋊C4—3592no—
128C22⋊C4—3592no—
129C22⋊C4—3592no—
130C22⋊C4—3592no—
131C22⋊C4C2×D35911yescharacteristic, complemented
132C22⋊C4C2×D35912yescomplemented
133C23—22no—
134C23—42no—
135C23—3594no—
136C23—3591no—
137C23—3591no—
138C23—3591no—
139C23—14362no—
140C23—14362no—
141C23—7182no—
142C23C22×D35911yescharacteristic
143C23C22×D35912yes—
144C23C22×D35911yescharacteristic
145D4—22no—
146D4—42no—
147D4—14362no—
148D4—7182no—
149D4—7182no—
150D4—7184no—
151D4—7184no—
152D4—7182no—
153C2×C4—3592no—
154C2×C4—3591no—
155C2×C4—3592no—
156C2×C4—14362no—
157C2×C4—7182no—
158C2×C4—7182no—
159C2×C4C22×D35912yes—
160C2×C4C22×D35911yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—3592no—
169C22—3592no—
170C22—14364no—
171C22—14364no—
172C22—7181no—
173C22—7182no—
174C22—7182no—
175C22—7182no—
176C22D4×D35912yes—
177C22C23×D35911yescharacteristic
178C4—21no—
179C4—22no—
180C4—7181no—
181C4—7182no—
182C4—7182no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—14361no—
187C2—7182no—
188C2—7182no—
189C2C2×D4×D35911yescharacteristic
190C2#2604006012yes—
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#120×C359C2
hypercenter#120D359
hypocenterC359#534
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D359C2semidirect
2. C2×D4×D359C2semidirect
3. #26039999C2semidirect
4. #26039997C2semidirect
5. #26039996C2semidirect
6. #26039988C2semidirect
7. #26039986C2semidirect
8. #26039984C2semidirect
9. #26039983C2semidirect
10. C22⋊C4×D359C2semidirect
11. #120×C359C2semidirect
13. C23×D359C22semidirect
14. C2×D1436C22semidirect
18. C2×C22⋊D359C22semidirect
19. C2×C22⋊D359C22semidirect
20. C2×C22⋊D359C22semidirect
22. C22×Dic359C22semidirect
23. C2×C4×D359C22semidirect
24. C2×C4×D359C22semidirect
25. C4⋊Dic359C22semidirect
26. Dic359⋊C4C22semidirect
27. C22⋊Dic359C22semidirect
28. C22⋊Dic359C22semidirect
29. C22⋊Dic359C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. (C2×D359)⋊C4C22semidirect
32. C24×C359C22semidirect
34. C2×D4×C359C22semidirect
35. C22⋊C4×C359C22semidirect
36. C22⋊C4×C359C22semidirect
40. C22×D359D4semidirect
95. C359#534semidirect
106. #120D359semidirect
110. C24C2×D359semidirect
120. C2×D4C2×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×C22⋊D3592
C2×D4×D3592
#260399992
#260399972
#260399962
#260399882
#260399862
#260399842
#260399832
C22⋊C4×D3592
#120×C3592
#534359