Finite Group Databasebeta

← #45655803

Subgroups — #45655803

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×#22550860C211yesmaximal, characteristic, complemented
2C22×C22⋊D509C211yesmaximal, characteristic, complemented
3#30654211C212yesmaximal, complemented
4#30654201C212yesmaximal, complemented
5#30654188C212yesmaximal, complemented
6#30654187C212yesmaximal, complemented
7#30654185C211yesmaximal, characteristic, complemented
8#30654184C211yesmaximal, characteristic, complemented
9#30654183C211yesmaximal, characteristic, complemented
10#30654171C211yesmaximal, characteristic, complemented
11#120×C509C211yesmaximal, characteristic, complemented
12C23×D509C2211yescharacteristic
13#22550860—24no—
14C2×Dic1018C2211yescharacteristic, complemented
15C2×D2036C2212yescomplemented
16C2×C22⋊D509—24no—
17C2×C22⋊D509C2212yes—
18C2×C22⋊D509C2212yes—
19C2×C22⋊D509C2211yescharacteristic
20C2×C22⋊D509C2211yescharacteristic, complemented
21C2×C22⋊D509C2212yescomplemented
22C2×C22⋊D509C2212yescomplemented
23C22×Dic509C2211yescharacteristic
24C22×Dic509C2211yescharacteristic, complemented
25C2×C4×D509C2211yescharacteristic, complemented
26#22550840C2212yescomplemented
27C22⋊Dic509C2211yescharacteristic
28C22⋊Dic509C2212yescomplemented
29#22550838C2212yescomplemented
30#22550838C2212yescomplemented
31C4×Dic509C2212yescomplemented
32C24×C509C2211yescharacteristic
33C2×D4×C509C2211yescharacteristic
34C2×D4×C509C2212yescomplemented
35C22⋊C4×C509C2211yescharacteristic
36C22⋊C4×C509C2212yescomplemented
37C22×D509—22no—
38C22×D509—42no—
39C22×D509C2312yes—
40C22×D509C2311yescharacteristic
41Dic1018—22no—
42D2036—42no—
43C22⋊D509—22no—
44C22⋊D509—24no—
45C22⋊D509—24no—
46C22⋊D509—22no—
47C22⋊D509—42no—
48C22⋊D509—42no—
49C2×Dic509—22no—
50C2×Dic509D414yescomplemented
51C2×Dic509C2312yes—
52C2×Dic509C2311yescharacteristic
53C2×Dic509C2311yescharacteristic
54C2×Dic509C2311yescharacteristic
55C4×D509—22no—
56C23×C509—22no—
57C23×C509—42no—
58C23×C509C2312yes—
59C23×C509C2311yescharacteristic
60C23×C509C2311yescharacteristic
61D4×C509—22no—
62D4×C509—42no—
63C2×C4×C509C2312yes—
64C2×C4×C509C2311yescharacteristic
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—21no—
75C2×D509—22no—
76C2×D509—22no—
77C2×D509—22no—
78C2×D509—22no—
79C2×D509—44no—
80C4×C509—21no—
81C4×C509—22no—
82Dic509—21no—
83Dic509—22no—
84Dic509C2×D412yes—
85Dic509C2×D412yes—
86C2×C509—22no—
87C2×C509—41no—
88C2×C509—42no—
89C2×C509C22×D411yescharacteristic
90C2×C509H(5,𝔽2)12yes—
91D509—41no—
92D509—42no—
93C509#53511yescharacteristic, complemented, Sylow
94#535—5091nomaximal, Sylow
95C2×#41—5091no—
96C22×D4—5091no—
97#127—5092no—
98#124—5092no—
99#121—5092no—
100#121—5092no—
101#120—5091no—
102#120—5091no—
103#120—5091no—
104#120D50911yescharacteristic, complemented
105#117—5091no—
106C24—5091no—
107C24C2×D50911yescharacteristic
108#41—10184no—
109C2×Q8—5091no—
110C2×D4—10184no—
111C2×D4—5092no—
112C2×D4—5092no—
113C2×D4—5091no—
114C2×D4—5092no—
115C2×D4—5092no—
116C2×D4—5091no—
117C2×D4—5092no—
118C2×D4C2×D50911yescharacteristic
119C2×D4C2×D50912yescomplemented
120C22×C4—5091no—
121C22×C4—5091no—
122C22×C4—5091no—
123C4⋊C4—5092no—
124C22⋊C4—5091no—
125C22⋊C4—5092no—
126C22⋊C4—5092no—
127C22⋊C4—5092no—
128C22⋊C4C2×D50911yescharacteristic
129C22⋊C4C2×D50912yescomplemented
130C42—5092no—
131C23—22no—
132C23—42no—
133C23—20362no—
134C23—10182no—
135C23—5092no—
136C23—5091no—
137C23C22×D50912yes—
138C23C22×D50911yescharacteristic
139C23C22×D50911yescharacteristic
140Q8—10182no—
141D4—22no—
142D4—42no—
143D4—20362no—
144D4—20362no—
145D4—20362no—
146D4—10182no—
147D4—10184no—
148D4—10184no—
149D4—10182no—
150C2×C4—10182no—
151C2×C4—10182no—
152C2×C4—5092no—
153C2×C4—5091no—
154C2×C4—5094no—
155C2×C4—5091no—
156C2×C4—5091no—
157C2×C4C22×D50912yes—
158C2×C4C22×D50911yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—20364no—
167C22—10181no—
168C22—10182no—
169C22—10182no—
170C22—10182no—
171C22—10182no—
172C22D4×D50912yes—
173C22C23×D50911yescharacteristic
174C4—21no—
175C4—22no—
176C4—10181no—
177C4—10182no—
178C4—5092no—
179C4—5092no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—20362no—
184C2—20361no—
185C2C2×D4×D50911yescharacteristic
186C2#3065426012yes—
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#120×C509C2
hypercenter#120D509
hypocenterC509#535
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#22550860C2semidirect
2. C22×C22⋊D509C2semidirect
3. #30654211C2semidirect
4. #30654201C2semidirect
5. #30654188C2semidirect
6. #30654187C2semidirect
7. #30654185C2semidirect
8. #30654184C2semidirect
9. #30654183C2semidirect
10. #30654171C2semidirect
11. #120×C509C2semidirect
14. C2×Dic1018C22semidirect
15. C2×D2036C22semidirect
20. C2×C22⋊D509C22semidirect
21. C2×C22⋊D509C22semidirect
22. C2×C22⋊D509C22semidirect
24. C22×Dic509C22semidirect
25. C2×C4×D509C22semidirect
26. #22550840C22semidirect
28. C22⋊Dic509C22semidirect
29. #22550838C22semidirect
30. #22550838C22semidirect
31. C4×Dic509C22semidirect
34. C2×D4×C509C22semidirect
36. C22⋊C4×C509C22semidirect
50. C2×Dic509D4semidirect
93. C509#535semidirect
104. #120D509semidirect
119. C2×D4C2×D509semidirect
129. C22⋊C4C2×D509semidirect

Maximal subgroups

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

subgroupindex
C2×#225508602
C22×C22⋊D5092
#306542112
#306542012
#306541882
#306541872
#306541852
#306541842
#306541832
#306541712
#120×C5092
#535509