Finite Group Databasebeta

← #37527090

Subgroups — #37527090

Order 22976.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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#26040002C211yesmaximal, characteristic, complemented
4#26039997C212yesmaximal, complemented
5#26039995C211yesmaximal, characteristic, complemented
6#26039986C212yesmaximal, complemented
7#26039983C212yesmaximal, complemented
8#26039977C211yesmaximal, characteristic, complemented
9C4×C22⋊D359C211yesmaximal, characteristic, complemented
10C22⋊C4×D359C212yesmaximal, complemented
11#121×C359C211yesmaximal, characteristic, complemented
12C23×D359C2212yes—
13#19153341—24no—
14C2×Dic718C2211yescharacteristic, complemented
15C2×D1436C2211yescharacteristic
16C2×C22⋊D359C2211yescharacteristic
17C2×C22⋊D359C2212yes—
18C2×C22⋊D359C2212yescomplemented
19D4×D359—24no—
20C2×C4×D359C2211yescharacteristic
21C2×C4×D359C2212yescomplemented
22C2×C4×D359C2212yescomplemented
23C4⋊Dic359C2212yescomplemented
24Dic359⋊C4C2211yescharacteristic, complemented
25C22⋊Dic359C2212yescomplemented
26C22⋊Dic359C2211yescharacteristic, complemented
27C22⋊Dic359C2212yescomplemented
28(C2×D359)⋊C4C2211yescharacteristic
29(C2×D359)⋊C4C2212yes—
30(C2×D359)⋊C4C2212yescomplemented
31C4×Dic359C2211yescharacteristic, complemented
32C2×D4×C359C2211yescharacteristic
33C2×D4×C359C2212yescomplemented
34C22×C4×C359C2211yescharacteristic, complemented
35C4⋊C4×C359C2211yescharacteristic, complemented
36C22⋊C4×C359C2212yescomplemented
37C22×D359—24no—
38C22×D359—24no—
39C22×D359—42no—
40C22×D359C2312yes—
41C22×D359C2311yescharacteristic
42Dic718—22no—
43D1436D414yescomplemented
44C22⋊D359—24no—
45C22⋊D359—24no—
46C22⋊D359—22no—
47C2×Dic359C2311yescharacteristic
48C2×Dic359C2311yescharacteristic
49C2×Dic359C2312yes—
50C2×Dic359C2311yescharacteristic
51C4×D359—24no—
52C4×D359—22no—
53C4×D359—42no—
54C23×C359C2312yes—
55C23×C359C2311yescharacteristic
56D4×C359—22no—
57D4×C359—42no—
58C2×C4×C359—22no—
59C2×C4×C359C2311yescharacteristic
60C2×C4×C359C2312yes—
61C2×C4×C359C2311yescharacteristic
62C22×C359—21no—
63C22×C359—22no—
64C22×C359—21no—
65C22×C359—22no—
66C22×C359—21no—
67C22×C359—22no—
68C22×C359C2411yescharacteristic
69C2×D359—21no—
70C2×D359—22no—
71C2×D359—22no—
72C2×D359—21no—
73C2×D359—48no—
74C2×D359C2×D414yes—
75C2×D359#4112yescomplemented
76C4×C359—21no—
77C4×C359—22no—
78C4×C359C2×D412yes—
79Dic359—21no—
80Dic359—22no—
81Dic359—21no—
82Dic359—21no—
83C2×C359—42no—
84C2×C359—41no—
85C2×C359C22×D411yescharacteristic
86C2×C359C2×#4111yescharacteristic
87C2×C359H(5,𝔽2)11yescharacteristic
88D359—22no—
89D359—24no—
90C359#54611yescharacteristic, complemented, Sylow
91#546—3591nomaximal, Sylow
92C2×#41—3591no—
93C22×D4—3591no—
94#124—3591no—
95#123—3592no—
96C22⋊Q8—3591no—
97#121—3592no—
98#121D35911yescharacteristic, complemented
99#120—3592no—
100C4×D4—3591no—
101C4×D4—3591no—
102C2×C22⋊C4—3592no—
103C24—3592no—
104#41—7184no—
105C2×Q8—3591no—
106C2×D4—3591no—
107C2×D4—3592no—
108C2×D4—3592no—
109C2×D4—3591no—
110C2×D4—7184no—
111C2×D4C2×D35911yescharacteristic
112C2×D4C2×D35912yescomplemented
113C22×C4—3592no—
114C22×C4—3592no—
115C22×C4—3591no—
116C22×C4C2×D35911yescharacteristic, complemented
117C4⋊C4—3591no—
118C4⋊C4—3592no—
119C4⋊C4C2×D35911yescharacteristic, complemented
120C22⋊C4—3592no—
121C22⋊C4—3591no—
122C22⋊C4—3592no—
123C22⋊C4—3591no—
124C22⋊C4—3592no—
125C22⋊C4—3592no—
126C22⋊C4C2×D35912yescomplemented
127C42—3591no—
128C23—3591no—
129C23—3592no—
130C23—14362no—
131C23—7184no—
132C23—7184no—
133C23C22×D35912yes—
134C23C22×D35911yescharacteristic
135Q8—7182no—
136D4—22no—
137D4—42no—
138D4—3594no—
139D4—7184no—
140D4—7184no—
141D4—7182no—
142C2×C4—22no—
143C2×C4—3591no—
144C2×C4—3591no—
145C2×C4—3592no—
146C2×C4—3591no—
147C2×C4—14362no—
148C2×C4—7184no—
149C2×C4—7182no—
150C2×C4C22×D35912yes—
151C2×C4C22×D35911yescharacteristic
152C2×C4C22×D35911yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—3594no—
160C22—3592no—
161C22—14368no—
162C22—7181no—
163C22—7182no—
164C22—7182no—
165C22—7181no—
166C22C23×D35911yescharacteristic
167C4—21no—
168C4—22no—
169C4—7181no—
170C4—7182no—
171C4—7181no—
172C4—7181no—
173C4D4×D35912yes—
174C2—42no—
175C2—41no—
176C2—7182no—
177C2—7184no—
178C2C2×D4×D35911yescharacteristic
179C2#41×D35911yescharacteristic
180C2#2604006011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#121×C359C2
hypercenter#121D359
hypocenterC359#546
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. #26040002C2semidirect
4. #26039997C2semidirect
5. #26039995C2semidirect
6. #26039986C2semidirect
7. #26039983C2semidirect
8. #26039977C2semidirect
9. C4×C22⋊D359C2semidirect
10. C22⋊C4×D359C2semidirect
11. #121×C359C2semidirect
14. C2×Dic718C22semidirect
18. C2×C22⋊D359C22semidirect
21. C2×C4×D359C22semidirect
22. C2×C4×D359C22semidirect
23. C4⋊Dic359C22semidirect
24. Dic359⋊C4C22semidirect
25. C22⋊Dic359C22semidirect
26. C22⋊Dic359C22semidirect
27. C22⋊Dic359C22semidirect
30. (C2×D359)⋊C4C22semidirect
31. C4×Dic359C22semidirect
33. C2×D4×C359C22semidirect
34. C22×C4×C359C22semidirect
35. C4⋊C4×C359C22semidirect
36. C22⋊C4×C359C22semidirect
43. D1436D4semidirect
75. C2×D359#41semidirect
90. C359#546semidirect
98. #121D359semidirect
112. C2×D4C2×D359semidirect
116. C22×C4C2×D359semidirect
119. C4⋊C4C2×D359semidirect
126. 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
#260400022
#260399972
#260399952
#260399862
#260399832
#260399772
C4×C22⋊D3592
C22⋊C4×D3592
#121×C3592
#546359