Finite Group Databasebeta

← #37527009

Subgroups — #37527009

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×#19153341C211yesmaximal, characteristic, complemented
2C2×D4×D359C211yesmaximal, characteristic, complemented
3#26040011C212yesmaximal, complemented
4#26040001C212yesmaximal, complemented
5#26039990C211yesmaximal, characteristic, complemented
6#26039989C211yesmaximal, characteristic, complemented
7#26039986C211yesmaximal, characteristic, complemented
8#26039985C212yesmaximal, complemented
9#26039983C212yesmaximal, complemented
10#26039973C211yesmaximal, characteristic, complemented
11#121×C359C211yesmaximal, characteristic, complemented
12C23×D359C2212yes—
13#19153341—24no—
14C2×Dic718C2211yescharacteristic, complemented
15C2×D1436C2211yescharacteristic
16C2×D1436C2211yescharacteristic
17C2×D1436C2212yescomplemented
18C2×C22⋊D359C2211yescharacteristic
19C2×C22⋊D359C2212yes—
20C2×C22⋊D359C2211yescharacteristic, complemented
21C2×C22⋊D359C2212yescomplemented
22C2×C22⋊D359C2212yescomplemented
23D4×D359—24no—
24C2×C4×D359C2211yescharacteristic
25C2×C4×D359C2211yescharacteristic, complemented
26C4⋊Dic359C2211yescharacteristic, complemented
27C22⋊Dic359C2212yescomplemented
28(C2×D359)⋊C4C2212yes—
29(C2×D359)⋊C4C2212yescomplemented
30(C2×D359)⋊C4C2212yescomplemented
31C4×Dic359C2212yescomplemented
32C2×D4×C359C2211yescharacteristic
33C2×D4×C359C2212yescomplemented
34C22×C4×C359C2211yescharacteristic, complemented
35C4⋊C4×C359C2211yescharacteristic, complemented
36C22⋊C4×C359C2212yescomplemented
37C22×D359—24no—
38C22×D359—42no—
39C22×D359—42no—
40C22×D359C2312yes—
41C22×D359C2311yescharacteristic
42C22×D359C2311yescharacteristic
43Dic718—22no—
44D1436—22no—
45D1436—22no—
46D1436—42no—
47C22⋊D359—22no—
48C22⋊D359—24no—
49C22⋊D359—22no—
50C22⋊D359—42no—
51C22⋊D359—42no—
52C2×Dic359C2312yes—
53C2×Dic359C2311yescharacteristic
54C2×Dic359C2311yescharacteristic
55C4×D359—22no—
56C4×D359D414yescomplemented
57C23×C359C2312yes—
58C23×C359C2311yescharacteristic
59D4×C359—22no—
60D4×C359—42no—
61C2×C4×C359—22no—
62C2×C4×C359C2311yescharacteristic
63C2×C4×C359C2312yes—
64C2×C4×C359C2311yescharacteristic
65C22×C359—21no—
66C22×C359—22no—
67C22×C359—21no—
68C22×C359—22no—
69C22×C359—21no—
70C22×C359—22no—
71C22×C359C2411yescharacteristic
72C2×D359—21no—
73C2×D359—21no—
74C2×D359—21no—
75C2×D359—22no—
76C2×D359—21no—
77C2×D359—22no—
78C2×D359—21no—
79C2×D359—22no—
80C2×D359—48no—
81C2×D359C2×D412yes—
82C4×C359—21no—
83C4×C359—22no—
84C4×C359C2×D412yes—
85Dic359—21no—
86Dic359—22no—
87Dic359C2×D412yes—
88C2×C359—42no—
89C2×C359—41no—
90C2×C359C22×D411yescharacteristic
91C2×C359H(5,𝔽2)11yescharacteristic
92C2×C359H(5,𝔽2)11yescharacteristic
93D359—22no—
94D359—41no—
95D359—42no—
96C359#53511yescharacteristic, complemented, Sylow
97#535—3591nomaximal, Sylow
98C2×#41—3591no—
99C22×D4—3591no—
100#127—3592no—
101#124—3592no—
102#121—3591no—
103#121—3591no—
104#121—3591no—
105#121D35911yescharacteristic, complemented
106#120—3592no—
107#120—3592no—
108#117—3591no—
109C24—3592no—
110#41—7184no—
111C2×Q8—3591no—
112C2×D4—3592no—
113C2×D4—3592no—
114C2×D4—3591no—
115C2×D4—3592no—
116C2×D4—3591no—
117C2×D4—3592no—
118C2×D4—3591no—
119C2×D4—3591no—
120C2×D4—7184no—
121C2×D4C2×D35911yescharacteristic
122C2×D4C2×D35912yescomplemented
123C22×C4—3591no—
124C22×C4—3591no—
125C22×C4C2×D35911yescharacteristic, complemented
126C4⋊C4—3591no—
127C4⋊C4C2×D35911yescharacteristic, complemented
128C22⋊C4—3592no—
129C22⋊C4—3592no—
130C22⋊C4—3592no—
131C22⋊C4—3592no—
132C22⋊C4C2×D35912yescomplemented
133C42—3592no—
134C23—3591no—
135C23—3592no—
136C23—3591no—
137C23—14362no—
138C23—14362no—
139C23—7184no—
140C23C22×D35912yes—
141C23C22×D35911yescharacteristic
142Q8—7182no—
143D4—22no—
144D4—42no—
145D4—14362no—
146D4—14362no—
147D4—14362no—
148D4—7182no—
149D4—7184no—
150D4—7182no—
151D4—7182no—
152D4—7182no—
153C2×C4—22no—
154C2×C4—3591no—
155C2×C4—3594no—
156C2×C4—3592no—
157C2×C4—3591no—
158C2×C4—7182no—
159C2×C4C22×D35912yes—
160C2×C4C22×D35911yescharacteristic
161C2×C4C22×D35911yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—3592no—
169C22—14368no—
170C22—7181no—
171C22—7182no—
172C22—7181no—
173C22—7182no—
174C22—7181no—
175C22—7182no—
176C22—7181no—
177C22—7181no—
178C22C23×D35911yescharacteristic
179C4—21no—
180C4—22no—
181C4—3592no—
182C4—7181no—
183C4—7182no—
184C4D4×D35912yes—
185C2—42no—
186C2—41no—
187C2—14362no—
188C2—14361no—
189C2—7182no—
190C2C2×D4×D35911yescharacteristic
191C2#2604006011yescharacteristic
192C2#2604006111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#121×C359C2
hypercenter#121D359
hypocenterC359#535
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19153341C2semidirect
2. C2×D4×D359C2semidirect
3. #26040011C2semidirect
4. #26040001C2semidirect
5. #26039990C2semidirect
6. #26039989C2semidirect
7. #26039986C2semidirect
8. #26039985C2semidirect
9. #26039983C2semidirect
10. #26039973C2semidirect
11. #121×C359C2semidirect
14. C2×Dic718C22semidirect
17. C2×D1436C22semidirect
20. C2×C22⋊D359C22semidirect
21. C2×C22⋊D359C22semidirect
22. C2×C22⋊D359C22semidirect
25. C2×C4×D359C22semidirect
26. C4⋊Dic359C22semidirect
27. C22⋊Dic359C22semidirect
29. (C2×D359)⋊C4C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. C4×Dic359C22semidirect
33. C2×D4×C359C22semidirect
34. C22×C4×C359C22semidirect
35. C4⋊C4×C359C22semidirect
36. C22⋊C4×C359C22semidirect
56. C4×D359D4semidirect
96. C359#535semidirect
105. #121D359semidirect
122. C2×D4C2×D359semidirect
125. C22×C4C2×D359semidirect
127. C4⋊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
C2×#191533412
C2×D4×D3592
#260400112
#260400012
#260399902
#260399892
#260399862
#260399852
#260399832
#260399732
#121×C3592
#535359