Finite Group Databasebeta

← #37527081

Subgroups — #37527081

Order 22976.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×D4×D359C211yesmaximal, characteristic, complemented
2#26040004C211yesmaximal, characteristic, complemented
3#26039998C212yesmaximal, complemented
4#26039991C211yesmaximal, characteristic, complemented
5#26039989C211yesmaximal, characteristic, complemented
6#26039987C212yesmaximal, complemented
7#26039983C212yesmaximal, complemented
8D4×Dic359C211yesmaximal, characteristic, complemented
9C4×C22⋊D359C211yesmaximal, characteristic, complemented
10C2×(C2×D359)⋊C4C212yesmaximal, complemented
11C2×#41×C359C211yesmaximal, characteristic, complemented
12C23×D359C2212yes—
13C2×D1436C2211yescharacteristic
14C2×C22⋊D359C2211yescharacteristic
15C2×C22⋊D359C2212yes—
16C2×C22⋊D359C2212yescomplemented
17D4×D359—24no—
18C22×Dic359C2212yescomplemented
19C2×C4×D359C2211yescharacteristic
20C4⋊Dic359C2211yescharacteristic, complemented
21Dic359⋊C4C2211yescharacteristic, complemented
22Dic359⋊C4C2212yescomplemented
23C22⋊Dic359C2211yescharacteristic, complemented
24C22⋊Dic359C2212yescomplemented
25(C2×D359)⋊C4C2211yescharacteristic
26(C2×D359)⋊C4C2212yes—
27(C2×D359)⋊C4C2212yescomplemented
28(C2×D359)⋊C4C2212yescomplemented
29(C2×D359)⋊C4C2212yescomplemented
30C4×Dic359C2211yescharacteristic, complemented
31#41×C359—24no—
32C2×Q8×C359C2211yescharacteristic, complemented
33C2×D4×C359C2211yescharacteristic
34C2×D4×C359C2212yescomplemented
35C22×C4×C359C2212yescomplemented
36C22×C4×C359C2211yescharacteristic, complemented
37C22×D359—24no—
38C22×D359—24no—
39C22×D359—42no—
40C22×D359C2312yes—
41C22×D359C2311yescharacteristic
42D1436—22no—
43C22⋊D359—24no—
44C22⋊D359—22no—
45C22⋊D359—42no—
46C2×Dic359—42no—
47C2×Dic359C2311yescharacteristic
48C2×Dic359C2311yescharacteristic
49C2×Dic359C2312yes—
50C2×Dic359C2311yescharacteristic
51C4×D359—22no—
52C23×C359C2311yescharacteristic
53C23×C359C2312yes—
54Q8×C359—22no—
55D4×C359—24no—
56D4×C359D414yescomplemented
57C2×C4×C359—24no—
58C2×C4×C359—22no—
59C2×C4×C359C2311yescharacteristic
60C2×C4×C359C2312yes—
61C2×C4×C359C2311yescharacteristic
62C22×C359—22no—
63C22×C359—22no—
64C22×C359—21no—
65C22×C359—21no—
66C22×C359—21no—
67C22×C359C2×D414yes—
68C22×C359C2411yescharacteristic
69C2×D359—21no—
70C2×D359—21no—
71C2×D359—22no—
72C2×D359—22no—
73C2×D359—22no—
74C2×D359—48no—
75C2×D359#4112yescomplemented
76C4×C359—21no—
77C4×C359—22no—
78C4×C359C2×D412yes—
79Dic359—21no—
80Dic359—21no—
81Dic359—21no—
82Dic359—22no—
83C2×C359—24no—
84C2×C359—41no—
85C2×C359C22×D411yescharacteristic
86C2×C359C2×#4111yescharacteristic
87C2×C359H(5,𝔽2)11yescharacteristic
88D359—22no—
89D359—42no—
90C359#54611yescharacteristic, complemented, Sylow
91#546—3591nomaximal, Sylow
92C2×#41D35911yescharacteristic, complemented
93C22×D4—3591no—
94#124—3591no—
95#123—3592no—
96C22⋊Q8—3591no—
97#121—3592no—
98#121—3591no—
99#120—3592no—
100C4×D4—3591no—
101C4×D4—3591no—
102C2×C22⋊C4—3592no—
103C24—3592no—
104#41—24no—
105C2×Q8C2×D35911yescharacteristic, complemented
106C2×D4—3592no—
107C2×D4—3591no—
108C2×D4—3592no—
109C2×D4—3591no—
110C2×D4—7184no—
111C2×D4C2×D35911yescharacteristic
112C2×D4C2×D35912yescomplemented
113C22×C4—3592no—
114C22×C4—3591no—
115C22×C4C2×D35912yescomplemented
116C22×C4C2×D35911yescharacteristic, complemented
117C4⋊C4—3591no—
118C4⋊C4—3591no—
119C4⋊C4—3592no—
120C22⋊C4—3592no—
121C22⋊C4—3591no—
122C22⋊C4—3591no—
123C22⋊C4—3592no—
124C22⋊C4—3592no—
125C22⋊C4—3592no—
126C22⋊C4—3592no—
127C42—3591no—
128C23—3591no—
129C23—3592no—
130C23—14362no—
131C23—7184no—
132C23—7184no—
133C23C22×D35911yescharacteristic
134C23C22×D35912yes—
135Q8—22no—
136D4—24no—
137D4—14362no—
138D4—7182no—
139D4—7184no—
140D4—7182no—
141D4C22⋊D35914yescomplemented
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—3591no—
145C2×C4—3591no—
146C2×C4—3592no—
147C2×C4—3591no—
148C2×C4—14362no—
149C2×C4—7182no—
150C2×C4C22×D35911yescharacteristic
151C2×C4C22×D35912yes—
152C2×C4C22×D35911yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—3592no—
159C22—14368no—
160C22—7182no—
161C22—7182no—
162C22—7182no—
163C22—7181no—
164C22—7181no—
165C22C2×C22⋊D35914yes—
166C22C23×D35911yescharacteristic
167C4—21no—
168C4—22no—
169C4—7181no—
170C4—7181no—
171C4—7181no—
172C4—7182no—
173C4C2×C22⋊D35912yes—
174C2—24no—
175C2—41no—
176C2—14362no—
177C2—7182no—
178C2C22×C22⋊D35911yescharacteristic
179C2#41×D35911yescharacteristic
180C2#2604006111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
FittingC2×#41×C359C2
hypercenterC2×#41D359
hypocenterC359#546
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D4×D359C2semidirect
2. #26040004C2semidirect
3. #26039998C2semidirect
4. #26039991C2semidirect
5. #26039989C2semidirect
6. #26039987C2semidirect
7. #26039983C2semidirect
8. D4×Dic359C2semidirect
9. C4×C22⋊D359C2semidirect
10. C2×(C2×D359)⋊C4C2semidirect
11. C2×#41×C359C2semidirect
16. C2×C22⋊D359C22semidirect
18. C22×Dic359C22semidirect
20. C4⋊Dic359C22semidirect
21. Dic359⋊C4C22semidirect
22. Dic359⋊C4C22semidirect
23. C22⋊Dic359C22semidirect
24. C22⋊Dic359C22semidirect
27. (C2×D359)⋊C4C22semidirect
28. (C2×D359)⋊C4C22semidirect
29. (C2×D359)⋊C4C22semidirect
30. C4×Dic359C22semidirect
32. C2×Q8×C359C22semidirect
34. C2×D4×C359C22semidirect
35. C22×C4×C359C22semidirect
36. C22×C4×C359C22semidirect
56. D4×C359D4semidirect
75. C2×D359#41semidirect
90. C359#546semidirect
92. C2×#41D359semidirect
105. C2×Q8C2×D359semidirect
112. C2×D4C2×D359semidirect
115. C22×C4C2×D359semidirect
116. C22×C4C2×D359semidirect
141. D4C22⋊D359semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3592
#260400042
#260399982
#260399912
#260399892
#260399872
#260399832
D4×Dic3592
C4×C22⋊D3592
C2×(C2×D359)⋊C42
C2×#41×C3592
#546359