Finite Group Databasebeta

← #45655870

Subgroups — #45655870

Order 32576.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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⋊D509C212yesmaximal, complemented
2C2×D4×D509C212yesmaximal, complemented
3#30654211C211yesmaximal, characteristic, complemented
4#30654188C212yesmaximal, complemented
5#30654187C212yesmaximal, complemented
6#30654185C212yesmaximal, complemented
7#30654183C211yesmaximal, characteristic, complemented
8#30654175C212yesmaximal, complemented
9#120×C509C211yesmaximal, characteristic, complemented
10C23×D509C2211yescharacteristic, complemented
11C23×D509C2212yescomplemented
12C2×D2036C2212yescomplemented
13C2×D2036C2211yescharacteristic, complemented
14C2×C22⋊D509—28no—
15C2×C22⋊D509C2212yes—
16C2×C22⋊D509C2212yescomplemented
17C2×C22⋊D509C2212yescomplemented
18C2×C22⋊D509C2212yescomplemented
19C2×C22⋊D509C2212yescomplemented
20D4×D509—28no—
21C22×Dic509C2212yescomplemented
22C2×C4×D509C2212yescomplemented
23#22550840C2212yescomplemented
24C22⋊Dic509C2211yescharacteristic, complemented
25#22550838C2212yescomplemented
26#22550838C2212yescomplemented
27C4×Dic509C2211yescharacteristic, complemented
28C24×C509C2211yescharacteristic, complemented
29C2×D4×C509C2212yescomplemented
30C2×D4×C509C2211yescharacteristic, complemented
31C22⋊C4×C509C2212yescomplemented
32C22⋊C4×C509C2211yescharacteristic, complemented
33C22×D509—24no—
34C22×D509—24no—
35C22×D509—24no—
36C22×D509—41no—
37C22×D509—42no—
38C22×D509C2312yes—
39C22×D509C2312yes—
40D2036—24no—
41D2036—41no—
42C22⋊D509—24no—
43C22⋊D509—24no—
44C22⋊D509—24no—
45C22⋊D509—24no—
46C22⋊D509D418yescomplemented
47C2×Dic509—24no—
48C2×Dic509C2312yes—
49C2×Dic509C2312yes—
50C4×D509—24no—
51C23×C509—24no—
52C23×C509—41no—
53C23×C509C2311yescharacteristic
54C23×C509C2311yescharacteristic
55C23×C509C2312yes—
56D4×C509—24no—
57D4×C509—41no—
58C2×C4×C509C2311yescharacteristic
59C2×C4×C509C2312yes—
60C22×C509—22no—
61C22×C509—22no—
62C22×C509—22no—
63C22×C509—22no—
64C22×C509—21no—
65C22×C509—21no—
66C22×C509—44no—
67C22×C509C2×D414yescomplemented
68C22×C509C2411yescharacteristic
69C2×D509—22no—
70C2×D509—22no—
71C2×D509—22no—
72C2×D509—22no—
73C2×D509—22no—
74C2×D509—44no—
75C2×D509—48no—
76C2×D509C2×D414yescomplemented
77C4×C509—22no—
78C4×C509—21no—
79Dic509—22no—
80Dic509C2×D414yescomplemented
81C2×C509—24no—
82C2×C509—41no—
83C2×C509—41no—
84C2×C509C22×D412yes—
85C2×C509H(5,𝔽2)11yescharacteristic
86D509—24no—
87D509—42no—
88C509D4211yescharacteristic, complemented, Sylow
89D42—5091nomaximal, Sylow
90C22×D4—5092no—
91C22×D4—5092no—
92#127—5091no—
93#121—5092no—
94#121—5092no—
95#120—5091no—
96#120—5092no—
97#120D50911yescharacteristic, complemented
98C4×D4—5092no—
99C24—5091no—
100C24—5092no—
101C24C2×D50911yescharacteristic, complemented
102C2×D4—10188no—
103C2×D4—10188no—
104C2×D4—5092no—
105C2×D4—5092no—
106C2×D4—5092no—
107C2×D4—5092no—
108C2×D4—5092no—
109C2×D4—5091no—
110C2×D4—5092no—
111C2×D4C2×D50912yescomplemented
112C2×D4C2×D50911yescharacteristic, complemented
113C22×C4—5092no—
114C22×C4—5092no—
115C4⋊C4—5092no—
116C22⋊C4—5091no—
117C22⋊C4—5092no—
118C22⋊C4—5092no—
119C22⋊C4C2×D50912yescomplemented
120C22⋊C4C2×D50911yescharacteristic, complemented
121C42—5091no—
122C23—24no—
123C23—41no—
124C23—20362no—
125C23—20361no—
126C23—10184no—
127C23—10184no—
128C23—10184no—
129C23—5092no—
130C23—5092no—
131C23C22×D50911yescharacteristic
132C23C22×D50911yescharacteristic
133C23C22×D50912yes—
134D4—24no—
135D4—41no—
136D4—20361no—
137D4—10184no—
138D4—10184no—
139D4—10184no—
140D4—10184no—
141D4—10184no—
142D4—5098no—
143C2×C4—10184no—
144C2×C4—10184no—
145C2×C4—5092no—
146C2×C4—5092no—
147C2×C4C22×D50911yescharacteristic
148C2×C4C22×D50912yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—20368no—
157C22—20364no—
158C22—10182no—
159C22—10182no—
160C22—10182no—
161C22—10182no—
162C22—10182no—
163C22—5094no—
164C22D4×D50914yescomplemented
165C22C23×D50911yescharacteristic
166C4—22no—
167C4—21no—
168C4—10182no—
169C4—5094no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—20362no—
174C2—10184no—
175C2C2×D4×D50912yes—
176C2#3065426011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#120×C509C2
hypercenter#120D509
hypocenterC509D42
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. #30654211C2semidirect
4. #30654188C2semidirect
5. #30654187C2semidirect
6. #30654185C2semidirect
7. #30654183C2semidirect
8. #30654175C2semidirect
9. #120×C509C2semidirect
10. C23×D509C22semidirect
11. C23×D509C22semidirect
12. C2×D2036C22semidirect
13. C2×D2036C22semidirect
16. C2×C22⋊D509C22semidirect
17. C2×C22⋊D509C22semidirect
18. C2×C22⋊D509C22semidirect
19. C2×C22⋊D509C22semidirect
21. C22×Dic509C22semidirect
22. C2×C4×D509C22semidirect
23. #22550840C22semidirect
24. C22⋊Dic509C22semidirect
25. #22550838C22semidirect
26. #22550838C22semidirect
27. C4×Dic509C22semidirect
28. C24×C509C22semidirect
29. C2×D4×C509C22semidirect
30. C2×D4×C509C22semidirect
31. C22⋊C4×C509C22semidirect
32. C22⋊C4×C509C22semidirect
46. C22⋊D509D4semidirect
67. C22×C509C2×D4semidirect
76. C2×D509C2×D4semidirect
80. Dic509C2×D4semidirect
88. C509D42semidirect
97. #120D509semidirect
101. C24C2×D509semidirect
111. C2×D4C2×D509semidirect
112. C2×D4C2×D509semidirect
119. C22⋊C4C2×D509semidirect
120. C22⋊C4C2×D509semidirect
164. C22D4×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
#306542112
#306541882
#306541872
#306541852
#306541832
#306541752
#120×C5092
D42509