Finite Group Databasebeta

← #45655823

Subgroups — #45655823

Order 32576.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#30654208C211yesmaximal, characteristic, complemented
2#30654208C211yesmaximal, characteristic, complemented
3#30654202C211yesmaximal, characteristic, complemented
4#30654201C211yesmaximal, characteristic, complemented
5#30654197C211yesmaximal, characteristic, complemented
6#30654196C211yesmaximal, characteristic, complemented
7#30654192C211yesmaximal, characteristic, complemented
8#30654187C211yesmaximal, characteristic, complemented
9#30654187C211yesmaximal, characteristic, complemented
10#30654186C211yesmaximal, characteristic, complemented
11#30654184C211yesmaximal, characteristic, complemented
12D4×Dic509C211yesmaximal, characteristic, complemented
13#30654175C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic509C211yesmaximal, characteristic, complemented
15#120×C509C211yesmaximal, characteristic, complemented
16C2×Dic1018C2211yescharacteristic, complemented
17C2×C22⋊D509C2211yescharacteristic
18C2×C22⋊D509C2211yescharacteristic
19C2×C22⋊D509C2211yescharacteristic
20C2×C22⋊D509C2211yescharacteristic, complemented
21C22×Dic509C2211yescharacteristic
22C22×Dic509C2211yescharacteristic
23C22×Dic509C2211yescharacteristic, complemented
24C2×C4×D509C2211yescharacteristic, complemented
25C4⋊Dic509C2211yescharacteristic, complemented
26C4⋊Dic509C2211yescharacteristic, complemented
27#22550840C2211yescharacteristic, complemented
28#22550840C2211yescharacteristic, complemented
29#22550840C2211yescharacteristic, complemented
30#22550840C2211yescharacteristic, complemented
31C22⋊Dic509C2211yescharacteristic
32C22⋊Dic509C2211yescharacteristic
33C22⋊Dic509C2211yescharacteristic
34C22⋊Dic509C2211yescharacteristic
35C22⋊Dic509C2211yescharacteristic
36C22⋊Dic509C2211yescharacteristic
37C22⋊Dic509C2211yescharacteristic
38C22⋊Dic509C2211yescharacteristic, complemented
39C22⋊Dic509C2211yescharacteristic, complemented
40#22550838C2211yescharacteristic, complemented
41#22550838C2211yescharacteristic, complemented
42C4×Dic509C2211yescharacteristic, complemented
43C4×Dic509C2211yescharacteristic, complemented
44C24×C509C2211yescharacteristic
45C2×D4×C509C2211yescharacteristic
46C2×D4×C509C2211yescharacteristic
47C2×D4×C509C2211yescharacteristic
48C22⋊C4×C509C2211yescharacteristic
49C22⋊C4×C509C2211yescharacteristic
50C22⋊C4×C509C2211yescharacteristic
51C22×D509C2311yescharacteristic
52Dic1018—41no—
53C22⋊D509—22no—
54C22⋊D509—41no—
55C22⋊D509—41no—
56C22⋊D509—41no—
57C2×Dic509—41no—
58C2×Dic509—41no—
59C2×Dic509—41no—
60C2×Dic509C2311yescharacteristic
61C2×Dic509C2311yescharacteristic
62C2×Dic509C2311yescharacteristic
63C2×Dic509C2311yescharacteristic
64C2×Dic509C2311yescharacteristic
65C2×Dic509C2311yescharacteristic
66C2×Dic509C2311yescharacteristic
67C4×D509—41no—
68C23×C509—22no—
69C23×C509—41no—
70C23×C509—41no—
71C23×C509C2311yescharacteristic
72C23×C509C2311yescharacteristic
73C23×C509C2311yescharacteristic
74C23×C509C2311yescharacteristic
75D4×C509—22no—
76D4×C509—41no—
77D4×C509—41no—
78C2×C4×C509C2311yescharacteristic
79C2×C4×C509C2311yescharacteristic
80C2×C4×C509C2311yescharacteristic
81C22×C509—21no—
82C22×C509—21no—
83C22×C509—21no—
84C22×C509—21no—
85C22×C509—21no—
86C22×C509—21no—
87C22×C509—21no—
88C22×C509—21no—
89C22×C509—21no—
90C22×C509—21no—
91C22×C509—21no—
92C22×C509—44no—
93C22×C509#4112yes—
94C22×C509C2411yescharacteristic
95C2×D509—21no—
96C2×D509—21no—
97C2×D509—21no—
98C4×C509—21no—
99C4×C509—21no—
100C4×C509—21no—
101Dic509—21no—
102Dic509—21no—
103Dic509—21no—
104Dic509—21no—
105Dic509—21no—
106Dic509—21no—
107Dic509—21no—
108C2×C509—22no—
109C2×C509—41no—
110C2×C509—41no—
111C2×C509—41no—
112C2×C509C2×#4111yescharacteristic
113C2×C509H(5,𝔽2)11yescharacteristic
114C2×C509H(5,𝔽2)11yescharacteristic
115D509—41no—
116C509#53811yescharacteristic, complemented, Sylow
117#538—5091nomaximal, Sylow
118#126—5091no—
119#126—5091no—
120#124—5091no—
121#124—5091no—
122#123—5091no—
123#123—5091no—
124C22⋊Q8—5091no—
125#121—5091no—
126#121—5091no—
127#121—5091no—
128#120—5091no—
129#120D50911yescharacteristic, complemented
130C4×D4—5091no—
131C4×D4—5091no—
132C2×C22⋊C4—5091no—
133C24C2×D50911yescharacteristic
134C2×Q8—5091no—
135C2×D4—5091no—
136C2×D4—5091no—
137C2×D4—5091no—
138C2×D4—5091no—
139C2×D4C2×D50911yescharacteristic
140C2×D4C2×D50911yescharacteristic
141C2×D4C2×D50911yescharacteristic
142C22×C4—5091no—
143C22×C4—5091no—
144C22×C4—5091no—
145C22×C4—5091no—
146C4⋊C4—5091no—
147C4⋊C4—5091no—
148C4⋊C4—5091no—
149C4⋊C4—5091no—
150C4⋊C4—5091no—
151C4⋊C4—5091no—
152C22⋊C4—5091no—
153C22⋊C4—5091no—
154C22⋊C4—5091no—
155C22⋊C4—5091no—
156C22⋊C4—5091no—
157C22⋊C4—5091no—
158C22⋊C4—5091no—
159C22⋊C4—5091no—
160C22⋊C4—5091no—
161C22⋊C4—5091no—
162C22⋊C4—5091no—
163C22⋊C4C2×D50911yescharacteristic
164C22⋊C4C2×D50911yescharacteristic
165C22⋊C4C2×D50911yescharacteristic
166C42—5091no—
167C42—5091no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—5091no—
172C23C22×D50911yescharacteristic
173C23C22×D50911yescharacteristic
174C23C22×D50911yescharacteristic
175C23C22×D50911yescharacteristic
176Q8—20361no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—20361no—
181D4—20361no—
182D4—20361no—
183D4—10182no—
184C2×C4—20361no—
185C2×C4—20361no—
186C2×C4—20361no—
187C2×C4—20361no—
188C2×C4—5091no—
189C2×C4—5091no—
190C2×C4—5091no—
191C2×C4—5091no—
192C2×C4—5091no—
193C2×C4—5091no—
194C2×C4—5091no—
195C2×C4C22×D50911yescharacteristic
196C2×C4C22×D50911yescharacteristic
197C2×C4C22×D50911yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—10181no—
211C22—10181no—
212C22—10181no—
213C22#2255086012yes—
214C22C23×D50911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—10181no—
219C4—10181no—
220C4—10181no—
221C4—10181no—
222C4—10181no—
223C4—10181no—
224C4—10181no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—20361no—
230C2C2×#2255086011yescharacteristic
231C2#3065426011yescharacteristic
232C2#3065426011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#120×C509C2
hypercenter#120D509
hypocenterC509#538
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #30654208C2semidirect
2. #30654208C2semidirect
3. #30654202C2semidirect
4. #30654201C2semidirect
5. #30654197C2semidirect
6. #30654196C2semidirect
7. #30654192C2semidirect
8. #30654187C2semidirect
9. #30654187C2semidirect
10. #30654186C2semidirect
11. #30654184C2semidirect
12. D4×Dic509C2semidirect
13. #30654175C2semidirect
14. C2×C22⋊Dic509C2semidirect
15. #120×C509C2semidirect
16. C2×Dic1018C22semidirect
20. C2×C22⋊D509C22semidirect
23. C22×Dic509C22semidirect
24. C2×C4×D509C22semidirect
25. C4⋊Dic509C22semidirect
26. C4⋊Dic509C22semidirect
27. #22550840C22semidirect
28. #22550840C22semidirect
29. #22550840C22semidirect
30. #22550840C22semidirect
38. C22⋊Dic509C22semidirect
39. C22⋊Dic509C22semidirect
40. #22550838C22semidirect
41. #22550838C22semidirect
42. C4×Dic509C22semidirect
43. C4×Dic509C22semidirect
116. C509#538semidirect
129. #120D509semidirect

Maximal subgroups

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

subgroupindex
#306542082
#306542022
#306542012
#306541972
#306541962
#306541922
#306541872
#306541862
#306541842
D4×Dic5092
#306541752
C2×C22⋊Dic5092
#120×C5092
#538509