Finite Group Databasebeta

← #45655798

Subgroups — #45655798

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×C22⋊D509C211yesmaximal, characteristic, complemented
2C2×D4×D509C211yesmaximal, characteristic, complemented
3#30654199C211yesmaximal, characteristic, complemented
4#30654197C212yesmaximal, complemented
5#30654196C211yesmaximal, characteristic, complemented
6#30654188C212yesmaximal, complemented
7#30654186C212yesmaximal, complemented
8#30654184C211yesmaximal, characteristic, complemented
9#30654183C212yesmaximal, complemented
10C22⋊C4×D509C211yesmaximal, characteristic, complemented
11#120×C509C211yesmaximal, characteristic, complemented
12C23×D509C2211yescharacteristic
13C23×D509C2211yescharacteristic, complemented
14C2×D2036C2211yescharacteristic, complemented
15C2×C22⋊D509—24no—
16C2×C22⋊D509C2211yescharacteristic
17C2×C22⋊D509C2212yes—
18C2×C22⋊D509C2211yescharacteristic, complemented
19C2×C22⋊D509C2212yescomplemented
20C2×C22⋊D509C2212yescomplemented
21D4×D509—24no—
22C22×Dic509C2211yescharacteristic, complemented
23C2×C4×D509C2211yescharacteristic, complemented
24C2×C4×D509C2212yescomplemented
25C4⋊Dic509C2212yescomplemented
26#22550840C2212yescomplemented
27C22⋊Dic509C2211yescharacteristic, complemented
28C22⋊Dic509C2212yescomplemented
29C22⋊Dic509C2212yescomplemented
30#22550838C2212yescomplemented
31#22550838C2212yescomplemented
32C24×C509C2211yescharacteristic, complemented
33C2×D4×C509C2211yescharacteristic
34C2×D4×C509C2212yescomplemented
35C22⋊C4×C509C2211yescharacteristic, complemented
36C22⋊C4×C509C2212yescomplemented
37C22×D509—22no—
38C22×D509—42no—
39C22×D509—42no—
40C22×D509D414yescomplemented
41C22×D509C2311yescharacteristic
42C22×D509C2311yescharacteristic
43C22×D509C2311yescharacteristic
44D2036—22no—
45C22⋊D509—22no—
46C22⋊D509—22no—
47C22⋊D509—24no—
48C22⋊D509—24no—
49C22⋊D509—42no—
50C2×Dic509—22no—
51C2×Dic509C2312yes—
52C2×Dic509C2312yes—
53C2×Dic509C2311yescharacteristic
54C4×D509—22no—
55C4×D509—42no—
56C23×C509—22no—
57C23×C509—42no—
58C23×C509C2312yes—
59C23×C509C2311yescharacteristic
60C23×C509C2311yescharacteristic
61D4×C509—22no—
62D4×C509—42no—
63C2×C4×C509C2311yescharacteristic
64C2×C4×C509C2312yes—
65C22×C509—22no—
66C22×C509—21no—
67C22×C509—22no—
68C22×C509—22no—
69C22×C509—22no—
70C22×C509—22no—
71C22×C509—44no—
72C22×C509C2×D412yes—
73C22×C509C2411yescharacteristic
74C2×D509—22no—
75C2×D509—22no—
76C2×D509—21no—
77C2×D509—22no—
78C2×D509—44no—
79C2×D509—44no—
80C2×D509C2×D412yes—
81C2×D509C2×D412yes—
82C4×C509—21no—
83C4×C509—22no—
84Dic509—21no—
85Dic509—22no—
86Dic509—22no—
87C2×C509—22no—
88C2×C509—41no—
89C2×C509—42no—
90C2×C509C22×D411yescharacteristic
91C2×C509H(5,𝔽2)12yes—
92D509—22no—
93D509—22no—
94D509—41no—
95C509#53411yescharacteristic, complemented, Sylow
96#534—5091nomaximal, Sylow
97C22×D4—5091no—
98C22×D4—5091no—
99#123—5091no—
100#123—5092no—
101#123—5091no—
102#121—5092no—
103#121—5092no—
104#120—5091no—
105#120—5092no—
106#120D50911yescharacteristic, complemented
107C2×C22⋊C4—5091no—
108C24—5091no—
109C24—5091no—
110C24C2×D50911yescharacteristic, complemented
111C2×D4—10184no—
112C2×D4—10184no—
113C2×D4—5092no—
114C2×D4—5092no—
115C2×D4—5091no—
116C2×D4—5091no—
117C2×D4—5092no—
118C2×D4—5091no—
119C2×D4C2×D50911yescharacteristic
120C2×D4C2×D50912yescomplemented
121C22×C4—5092no—
122C22×C4—5091no—
123C22×C4—5091no—
124C4⋊C4—5092no—
125C4⋊C4—5092no—
126C22⋊C4—5091no—
127C22⋊C4—5092no—
128C22⋊C4—5092no—
129C22⋊C4—5092no—
130C22⋊C4—5092no—
131C22⋊C4C2×D50911yescharacteristic, complemented
132C22⋊C4C2×D50912yescomplemented
133C23—22no—
134C23—42no—
135C23—20362no—
136C23—20362no—
137C23—10182no—
138C23—5094no—
139C23—5091no—
140C23—5091no—
141C23—5091no—
142C23C22×D50911yescharacteristic
143C23C22×D50912yes—
144C23C22×D50911yescharacteristic
145D4—22no—
146D4—42no—
147D4—20362no—
148D4—10182no—
149D4—10182no—
150D4—10184no—
151D4—10184no—
152D4—10182no—
153C2×C4—20362no—
154C2×C4—10182no—
155C2×C4—10182no—
156C2×C4—5092no—
157C2×C4—5091no—
158C2×C4—5092no—
159C2×C4C22×D50912yes—
160C2×C4C22×D50911yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—20364no—
169C22—20364no—
170C22—10181no—
171C22—10182no—
172C22—10182no—
173C22—10182no—
174C22—5092no—
175C22—5092no—
176C22D4×D50912yes—
177C22C23×D50911yescharacteristic
178C4—21no—
179C4—22no—
180C4—10181no—
181C4—10182no—
182C4—10182no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—20361no—
187C2—10182no—
188C2—10182no—
189C2C2×D4×D50911yescharacteristic
190C2#3065426012yes—
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#120×C509C2
hypercenter#120D509
hypocenterC509#534
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D509C2semidirect
2. C2×D4×D509C2semidirect
3. #30654199C2semidirect
4. #30654197C2semidirect
5. #30654196C2semidirect
6. #30654188C2semidirect
7. #30654186C2semidirect
8. #30654184C2semidirect
9. #30654183C2semidirect
10. C22⋊C4×D509C2semidirect
11. #120×C509C2semidirect
13. C23×D509C22semidirect
14. C2×D2036C22semidirect
18. C2×C22⋊D509C22semidirect
19. C2×C22⋊D509C22semidirect
20. C2×C22⋊D509C22semidirect
22. C22×Dic509C22semidirect
23. C2×C4×D509C22semidirect
24. C2×C4×D509C22semidirect
25. C4⋊Dic509C22semidirect
26. #22550840C22semidirect
27. C22⋊Dic509C22semidirect
28. C22⋊Dic509C22semidirect
29. C22⋊Dic509C22semidirect
30. #22550838C22semidirect
31. #22550838C22semidirect
32. C24×C509C22semidirect
34. C2×D4×C509C22semidirect
35. C22⋊C4×C509C22semidirect
36. C22⋊C4×C509C22semidirect
40. C22×D509D4semidirect
95. C509#534semidirect
106. #120D509semidirect
110. C24C2×D509semidirect
120. C2×D4C2×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×C22⋊D5092
C2×D4×D5092
#306541992
#306541972
#306541962
#306541882
#306541862
#306541842
#306541832
C22⋊C4×D5092
#120×C5092
#534509