Finite Group Databasebeta

← #45655804

Subgroups — #45655804

Order 32576.

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×#22550861C211yesmaximal, characteristic, complemented
2C2×D4×D509C211yesmaximal, characteristic, complemented
3#30654211C212yesmaximal, complemented
4#30654201C212yesmaximal, complemented
5#30654190C211yesmaximal, characteristic, complemented
6#30654189C211yesmaximal, characteristic, complemented
7#30654186C211yesmaximal, characteristic, complemented
8#30654185C212yesmaximal, complemented
9#30654183C212yesmaximal, complemented
10#30654173C211yesmaximal, characteristic, complemented
11#121×C509C211yesmaximal, characteristic, complemented
12C23×D509C2212yes—
13#22550861—24no—
14C2×Dic1018C2211yescharacteristic, complemented
15C2×D2036C2211yescharacteristic
16C2×D2036C2211yescharacteristic
17C2×D2036C2212yescomplemented
18C2×C22⋊D509C2212yes—
19C2×C22⋊D509C2211yescharacteristic
20C2×C22⋊D509C2211yescharacteristic, complemented
21C2×C22⋊D509C2212yescomplemented
22C2×C22⋊D509C2212yescomplemented
23D4×D509—24no—
24C2×C4×D509C2211yescharacteristic
25C2×C4×D509C2211yescharacteristic, complemented
26C4⋊Dic509C2211yescharacteristic, complemented
27C22⋊Dic509C2212yescomplemented
28#22550838C2212yes—
29#22550838C2212yescomplemented
30#22550838C2212yescomplemented
31C4×Dic509C2212yescomplemented
32C2×D4×C509C2211yescharacteristic
33C2×D4×C509C2212yescomplemented
34C22×C4×C509C2211yescharacteristic, complemented
35C4⋊C4×C509C2211yescharacteristic, complemented
36C22⋊C4×C509C2212yescomplemented
37C22×D509—24no—
38C22×D509—42no—
39C22×D509—42no—
40C22×D509C2312yes—
41C22×D509C2311yescharacteristic
42C22×D509C2311yescharacteristic
43Dic1018—22no—
44D2036—22no—
45D2036—22no—
46D2036—42no—
47C22⋊D509—22no—
48C22⋊D509—24no—
49C22⋊D509—22no—
50C22⋊D509—42no—
51C22⋊D509—42no—
52C2×Dic509C2312yes—
53C2×Dic509C2311yescharacteristic
54C2×Dic509C2311yescharacteristic
55C4×D509—22no—
56C4×D509D414yescomplemented
57C23×C509C2312yes—
58C23×C509C2311yescharacteristic
59D4×C509—22no—
60D4×C509—42no—
61C2×C4×C509—22no—
62C2×C4×C509C2311yescharacteristic
63C2×C4×C509C2312yes—
64C2×C4×C509C2311yescharacteristic
65C22×C509—21no—
66C22×C509—22no—
67C22×C509—21no—
68C22×C509—22no—
69C22×C509—21no—
70C22×C509—22no—
71C22×C509C2411yescharacteristic
72C2×D509—21no—
73C2×D509—21no—
74C2×D509—21no—
75C2×D509—22no—
76C2×D509—21no—
77C2×D509—22no—
78C2×D509—21no—
79C2×D509—22no—
80C2×D509—48no—
81C2×D509C2×D412yes—
82C4×C509—21no—
83C4×C509—22no—
84C4×C509C2×D412yes—
85Dic509—21no—
86Dic509—22no—
87Dic509C2×D412yes—
88C2×C509—42no—
89C2×C509—41no—
90C2×C509C22×D411yescharacteristic
91C2×C509H(5,𝔽2)11yescharacteristic
92C2×C509H(5,𝔽2)11yescharacteristic
93D509—22no—
94D509—41no—
95D509—42no—
96C509#53511yescharacteristic, complemented, Sylow
97#535—5091nomaximal, Sylow
98C2×#41—5091no—
99C22×D4—5091no—
100#127—5092no—
101#124—5092no—
102#121—5091no—
103#121—5091no—
104#121—5091no—
105#121D50911yescharacteristic, complemented
106#120—5092no—
107#120—5092no—
108#117—5091no—
109C24—5092no—
110#41—10184no—
111C2×Q8—5091no—
112C2×D4—10184no—
113C2×D4—5092no—
114C2×D4—5092no—
115C2×D4—5091no—
116C2×D4—5092no—
117C2×D4—5091no—
118C2×D4—5092no—
119C2×D4—5091no—
120C2×D4—5091no—
121C2×D4C2×D50911yescharacteristic
122C2×D4C2×D50912yescomplemented
123C22×C4—5091no—
124C22×C4—5091no—
125C22×C4C2×D50911yescharacteristic, complemented
126C4⋊C4—5091no—
127C4⋊C4C2×D50911yescharacteristic, complemented
128C22⋊C4—5092no—
129C22⋊C4—5092no—
130C22⋊C4—5092no—
131C22⋊C4—5092no—
132C22⋊C4C2×D50912yescomplemented
133C42—5092no—
134C23—20362no—
135C23—20362no—
136C23—10184no—
137C23—5092no—
138C23—5091no—
139C23—5091no—
140C23C22×D50912yes—
141C23C22×D50911yescharacteristic
142Q8—10182no—
143D4—22no—
144D4—42no—
145D4—20362no—
146D4—20362no—
147D4—20362no—
148D4—10182no—
149D4—10182no—
150D4—10184no—
151D4—10182no—
152D4—10182no—
153C2×C4—22no—
154C2×C4—10182no—
155C2×C4—5092no—
156C2×C4—5091no—
157C2×C4—5091no—
158C2×C4—5094no—
159C2×C4C22×D50912yes—
160C2×C4C22×D50911yescharacteristic
161C2×C4C22×D50911yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—20368no—
169C22—10181no—
170C22—10182no—
171C22—10181no—
172C22—10182no—
173C22—10181no—
174C22—10182no—
175C22—10181no—
176C22—10181no—
177C22—5092no—
178C22C23×D50911yescharacteristic
179C4—21no—
180C4—22no—
181C4—10181no—
182C4—10182no—
183C4—5092no—
184C4D4×D50912yes—
185C2—42no—
186C2—41no—
187C2—20362no—
188C2—20361no—
189C2—10182no—
190C2C2×D4×D50911yescharacteristic
191C2#3065426011yescharacteristic
192C2#3065426111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#121×C509C2
hypercenter#121D509
hypocenterC509#535
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#22550861C2semidirect
2. C2×D4×D509C2semidirect
3. #30654211C2semidirect
4. #30654201C2semidirect
5. #30654190C2semidirect
6. #30654189C2semidirect
7. #30654186C2semidirect
8. #30654185C2semidirect
9. #30654183C2semidirect
10. #30654173C2semidirect
11. #121×C509C2semidirect
14. C2×Dic1018C22semidirect
17. C2×D2036C22semidirect
20. C2×C22⋊D509C22semidirect
21. C2×C22⋊D509C22semidirect
22. C2×C22⋊D509C22semidirect
25. C2×C4×D509C22semidirect
26. C4⋊Dic509C22semidirect
27. C22⋊Dic509C22semidirect
29. #22550838C22semidirect
30. #22550838C22semidirect
31. C4×Dic509C22semidirect
33. C2×D4×C509C22semidirect
34. C22×C4×C509C22semidirect
35. C4⋊C4×C509C22semidirect
36. C22⋊C4×C509C22semidirect
56. C4×D509D4semidirect
96. C509#535semidirect
105. #121D509semidirect
122. C2×D4C2×D509semidirect
125. C22×C4C2×D509semidirect
127. C4⋊C4C2×D509semidirect
132. C22⋊C4C2×D509semidirect

Maximal subgroups

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

subgroupindex
C2×#225508612
C2×D4×D5092
#306542112
#306542012
#306541902
#306541892
#306541862
#306541852
#306541832
#306541732
#121×C5092
#535509