Finite Group Databasebeta

← #45655799

Subgroups — #45655799

Order 32576.

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×D2036C211yesmaximal, characteristic, complemented
2C2×D4×D509C211yesmaximal, characteristic, complemented
3#30654200C212yesmaximal, complemented
4#30654198C211yesmaximal, characteristic, complemented
5#30654190C212yesmaximal, complemented
6#30654188C212yesmaximal, complemented
7#30654185C211yesmaximal, characteristic, complemented
8#30654185C212yesmaximal, complemented
9#30654183C211yesmaximal, characteristic, complemented
10C22⋊C4×D509C211yesmaximal, characteristic, complemented
11#123×C509C211yesmaximal, characteristic, complemented
12C23×D509C2211yescharacteristic
13C23×D509C2211yescharacteristic, complemented
14C23×D509C2211yescharacteristic, complemented
15C2×D2036—24no—
16C2×D2036C2212yes—
17C2×D2036C2211yescharacteristic, complemented
18C2×D2036C2212yescomplemented
19C2×D2036C2212yescomplemented
20C2×C22⋊D509C2211yescharacteristic
21C2×C22⋊D509C2211yescharacteristic, complemented
22C2×C22⋊D509C2212yescomplemented
23D4×D509—24no—
24C2×C4×D509C2211yescharacteristic, complemented
25C2×C4×D509C2212yescomplemented
26#22550840C2212yescomplemented
27C22⋊Dic509C2211yescharacteristic, complemented
28#22550838C2212yescomplemented
29#22550838C2212yescomplemented
30#22550838C2212yescomplemented
31#22550838C2212yescomplemented
32C2×D4×C509C2211yescharacteristic
33C22×C4×C509C2211yescharacteristic, complemented
34C4⋊C4×C509C2212yescomplemented
35C22⋊C4×C509C2211yescharacteristic, complemented
36C22⋊C4×C509C2212yescomplemented
37C22×D509—22no—
38C22×D509—22no—
39C22×D509—42no—
40C22×D509—42no—
41C22×D509—42no—
42C22×D509D414yescomplemented
43C22×D509C2311yescharacteristic
44C22×D509C2312yes—
45C22×D509C2311yescharacteristic
46C22×D509C2311yescharacteristic
47D2036—22no—
48D2036—24no—
49D2036—24no—
50D2036—42no—
51C22⋊D509—22no—
52C22⋊D509—22no—
53C22⋊D509—42no—
54C2×Dic509C2312yes—
55C2×Dic509C2311yescharacteristic
56C4×D509—22no—
57C4×D509—42no—
58C23×C509C2311yescharacteristic
59C23×C509C2311yescharacteristic
60D4×C509—22no—
61C2×C4×C509—22no—
62C2×C4×C509C2312yes—
63C2×C4×C509C2312yes—
64C2×C4×C509C2311yescharacteristic
65C22×C509—22no—
66C22×C509—21no—
67C22×C509—22no—
68C22×C509C2×D412yes—
69C22×C509C2411yescharacteristic
70C2×D509—22no—
71C2×D509—22no—
72C2×D509—21no—
73C2×D509—22no—
74C2×D509—22no—
75C2×D509—22no—
76C2×D509—22no—
77C2×D509—44no—
78C2×D509—44no—
79C2×D509—44no—
80C2×D509C2×D412yes—
81C2×D509C2×D412yes—
82C4×C509—22no—
83C4×C509—21no—
84C4×C509—22no—
85Dic509—21no—
86Dic509—22no—
87C2×C509—22no—
88C2×C509—41no—
89C2×C509C22×D411yescharacteristic
90C2×C509H(5,𝔽2)12yes—
91D509—22no—
92D509—22no—
93D509—41no—
94D509—42no—
95C509#53411yescharacteristic, complemented, Sylow
96#534—5091nomaximal, Sylow
97C22×D4—5091no—
98C22×D4—5091no—
99#123—5091no—
100#123—5092no—
101#123D50911yescharacteristic, complemented
102#121—5092no—
103#121—5092no—
104#120—5091no—
105#120—5092no—
106#120—5091no—
107C2×C22⋊C4—5091no—
108C24—5091no—
109C24—5091no—
110C24—5091no—
111C2×D4—10184no—
112C2×D4—10184no—
113C2×D4—5092no—
114C2×D4—5091no—
115C2×D4—5091no—
116C2×D4—5092no—
117C2×D4—5092no—
118C2×D4—5091no—
119C2×D4—5092no—
120C2×D4C2×D50911yescharacteristic
121C22×C4—5092no—
122C22×C4—5091no—
123C22×C4C2×D50911yescharacteristic, complemented
124C4⋊C4—5092no—
125C4⋊C4C2×D50912yescomplemented
126C22⋊C4—5091no—
127C22⋊C4—5092no—
128C22⋊C4—5092no—
129C22⋊C4—5092no—
130C22⋊C4—5092no—
131C22⋊C4C2×D50911yescharacteristic, complemented
132C22⋊C4C2×D50912yescomplemented
133C23—20362no—
134C23—20362no—
135C23—20362no—
136C23—10182no—
137C23—10182no—
138C23—5094no—
139C23—5092no—
140C23—5091no—
141C23—5091no—
142C23—5091no—
143C23C22×D50911yescharacteristic
144C23C22×D50911yescharacteristic
145D4—22no—
146D4—20362no—
147D4—20362no—
148D4—10184no—
149D4—10184no—
150D4—10182no—
151D4—10182no—
152D4—10182no—
153C2×C4—22no—
154C2×C4—20362no—
155C2×C4—10182no—
156C2×C4—5092no—
157C2×C4—5091no—
158C2×C4C22×D50912yes—
159C2×C4C22×D50911yescharacteristic
160C2×C4C22×D50912yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—20364no—
165C22—20364no—
166C22—20364no—
167C22—10182no—
168C22—10181no—
169C22—10182no—
170C22—10182no—
171C22—10182no—
172C22—10182no—
173C22—10182no—
174C22—5092no—
175C22—5092no—
176C22D4×D50912yes—
177C22C23×D50911yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—10181no—
182C4—10182no—
183C2—22no—
184C2—41no—
185C2—20362no—
186C2—20361no—
187C2—10182no—
188C2—10182no—
189C2C2×D4×D50911yescharacteristic
190C2#3065426112yes—
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#123×C509C2
hypercenter#123D509
hypocenterC509#534
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D2036C2semidirect
2. C2×D4×D509C2semidirect
3. #30654200C2semidirect
4. #30654198C2semidirect
5. #30654190C2semidirect
6. #30654188C2semidirect
7. #30654185C2semidirect
8. #30654185C2semidirect
9. #30654183C2semidirect
10. C22⋊C4×D509C2semidirect
11. #123×C509C2semidirect
13. C23×D509C22semidirect
14. C23×D509C22semidirect
17. C2×D2036C22semidirect
18. C2×D2036C22semidirect
19. C2×D2036C22semidirect
21. C2×C22⋊D509C22semidirect
22. C2×C22⋊D509C22semidirect
24. C2×C4×D509C22semidirect
25. C2×C4×D509C22semidirect
26. #22550840C22semidirect
27. C22⋊Dic509C22semidirect
28. #22550838C22semidirect
29. #22550838C22semidirect
30. #22550838C22semidirect
31. #22550838C22semidirect
33. C22×C4×C509C22semidirect
34. C4⋊C4×C509C22semidirect
35. C22⋊C4×C509C22semidirect
36. C22⋊C4×C509C22semidirect
42. C22×D509D4semidirect
95. C509#534semidirect
101. #123D509semidirect
123. C22×C4C2×D509semidirect
125. C4⋊C4C2×D509semidirect
131. C22⋊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
C22×D20362
C2×D4×D5092
#306542002
#306541982
#306541902
#306541882
#306541852
#306541832
C22⋊C4×D5092
#123×C5092
#534509