Finite Group Databasebeta

← #45655822

Subgroups — #45655822

Order 32576.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#30654209C211yesmaximal, characteristic, complemented
2#30654208C211yesmaximal, characteristic, complemented
3#30654204C211yesmaximal, characteristic, complemented
4#30654199C211yesmaximal, characteristic, complemented
5#30654197C211yesmaximal, characteristic, complemented
6#30654191C211yesmaximal, characteristic, complemented
7#30654188C211yesmaximal, characteristic, complemented
8#30654188C211yesmaximal, characteristic, complemented
9#30654187C211yesmaximal, characteristic, complemented
10#30654185C211yesmaximal, characteristic, complemented
11#30654183C211yesmaximal, characteristic, complemented
12C4×D2036C211yesmaximal, characteristic, complemented
13#30654175C211yesmaximal, characteristic, complemented
14C22⋊C4×D509C211yesmaximal, characteristic, complemented
15#124×C509C211yesmaximal, characteristic, complemented
16C23×D509C2211yescharacteristic
17C2×D2036C2211yescharacteristic
18C2×D2036C2211yescharacteristic
19C2×C22⋊D509C2211yescharacteristic
20C2×C22⋊D509C2211yescharacteristic
21C2×C22⋊D509C2211yescharacteristic
22C2×C22⋊D509C2211yescharacteristic, complemented
23C22×Dic509C2211yescharacteristic, complemented
24C2×C4×D509C2211yescharacteristic
25C2×C4×D509C2211yescharacteristic
26C2×C4×D509C2211yescharacteristic, complemented
27C4⋊Dic509C2211yescharacteristic, complemented
28C4⋊Dic509C2211yescharacteristic, complemented
29#22550840C2211yescharacteristic, complemented
30#22550840C2211yescharacteristic, complemented
31#22550840C2211yescharacteristic, complemented
32#22550840C2211yescharacteristic, complemented
33C22⋊Dic509C2211yescharacteristic
34C22⋊Dic509C2211yescharacteristic
35#22550838C2211yescharacteristic
36#22550838C2211yescharacteristic
37#22550838C2211yescharacteristic
38#22550838C2211yescharacteristic
39#22550838C2211yescharacteristic
40#22550838C2211yescharacteristic
41#22550838C2211yescharacteristic, complemented
42#22550838C2211yescharacteristic, complemented
43C4×Dic509C2211yescharacteristic, complemented
44C2×Q8×C509C2211yescharacteristic, complemented
45C2×D4×C509C2211yescharacteristic
46C22⋊C4×C509C2211yescharacteristic
47C22⋊C4×C509C2211yescharacteristic
48C22⋊C4×C509C2211yescharacteristic, complemented
49C22⋊C4×C509C2211yescharacteristic, complemented
50C42×C509C2211yescharacteristic, complemented
51C22×D509—22no—
52C22×D509—41no—
53C22×D509—41no—
54C22×D509C2311yescharacteristic
55C22×D509C2311yescharacteristic
56C22×D509C2311yescharacteristic
57D2036—22no—
58D2036—41no—
59C22⋊D509—22no—
60C22⋊D509—41no—
61C22⋊D509—41no—
62C22⋊D509—41no—
63C2×Dic509—41no—
64C2×Dic509C2311yescharacteristic
65C2×Dic509C2311yescharacteristic
66C2×Dic509C2311yescharacteristic
67C2×Dic509C2311yescharacteristic
68C2×Dic509C2311yescharacteristic
69C4×D509—41no—
70C4×D509—41no—
71C4×D509—41no—
72C23×C509C2311yescharacteristic
73C23×C509C2311yescharacteristic
74Q8×C509—41no—
75D4×C509—41no—
76C2×C4×C509C2311yescharacteristic
77C2×C4×C509C2311yescharacteristic
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×C509C2411yescharacteristic
88C2×D509—21no—
89C2×D509—21no—
90C2×D509—21no—
91C2×D509—21no—
92C2×D509—21no—
93C2×D509—21no—
94C2×D509—21no—
95C2×D509—21no—
96C2×D509—44no—
97C2×D509#4112yes—
98C4×C509—21no—
99C4×C509—21no—
100C4×C509—21no—
101C4×C509—21no—
102C4×C509—21no—
103Dic509—21no—
104Dic509—21no—
105Dic509—21no—
106Dic509—21no—
107Dic509—21no—
108C2×C509—41no—
109C2×C509—41no—
110C2×C509C2×#4111yescharacteristic
111C2×C509H(5,𝔽2)11yescharacteristic
112C2×C509H(5,𝔽2)11yescharacteristic
113D509—22no—
114D509—41no—
115D509—41no—
116C509#53811yescharacteristic, complemented, Sylow
117#538—5091nomaximal, Sylow
118#126—5091no—
119#126—5091no—
120#124—5091no—
121#124D50911yescharacteristic, complemented
122#123—5091no—
123#123—5091no—
124C22⋊Q8—5091no—
125#121—5091no—
126#121—5091no—
127#121—5091no—
128#120—5091no—
129#120—5091no—
130C4×D4—5091no—
131C4×D4—5091no—
132C2×C22⋊C4—5091no—
133C24—5091no—
134C2×Q8C2×D50911yescharacteristic, complemented
135C2×D4—5091no—
136C2×D4—5091no—
137C2×D4—5091no—
138C2×D4—5091no—
139C2×D4—5091no—
140C2×D4—5091no—
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⋊C4C2×D50911yescharacteristic
163C22⋊C4C2×D50911yescharacteristic
164C22⋊C4C2×D50911yescharacteristic, complemented
165C22⋊C4C2×D50911yescharacteristic, complemented
166C42—5091no—
167C42C2×D50911yescharacteristic, complemented
168C23—20361no—
169C23—20361no—
170C23—10182no—
171C23—5091no—
172C23—5091no—
173C23—5091no—
174C23C22×D50911yescharacteristic
175C23C22×D50911yescharacteristic
176Q8—41no—
177D4—41no—
178D4—20361no—
179D4—20361no—
180D4—20361no—
181D4—20361no—
182D4—10182no—
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×C4C22×D50911yescharacteristic
194C2×C4C22×D50911yescharacteristic
195C2×C4C22×D50911yescharacteristic
196C2×C4C22×D50911yescharacteristic
197C2×C4C22×D50911yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—20364no—
205C22—10181no—
206C22—10181no—
207C22—10181no—
208C22—10181no—
209C22—10181no—
210C22—10181no—
211C22—10181no—
212C22—10181no—
213C22—5092no—
214C22C23×D50911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—10181no—
221C4—10181no—
222C4—10181no—
223C4—10181no—
224C4—10181no—
225C2—41no—
226C2—41no—
227C2—20361no—
228C2—20361no—
229C2—10182no—
230C2#41×D50911yescharacteristic
231C2#3065426011yescharacteristic
232C2#3065426111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#124×C509C2
hypercenter#124D509
hypocenterC509#538
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #30654209C2semidirect
2. #30654208C2semidirect
3. #30654204C2semidirect
4. #30654199C2semidirect
5. #30654197C2semidirect
6. #30654191C2semidirect
7. #30654188C2semidirect
8. #30654188C2semidirect
9. #30654187C2semidirect
10. #30654185C2semidirect
11. #30654183C2semidirect
12. C4×D2036C2semidirect
13. #30654175C2semidirect
14. C22⋊C4×D509C2semidirect
15. #124×C509C2semidirect
22. C2×C22⋊D509C22semidirect
23. C22×Dic509C22semidirect
26. C2×C4×D509C22semidirect
27. C4⋊Dic509C22semidirect
28. C4⋊Dic509C22semidirect
29. #22550840C22semidirect
30. #22550840C22semidirect
31. #22550840C22semidirect
32. #22550840C22semidirect
41. #22550838C22semidirect
42. #22550838C22semidirect
43. C4×Dic509C22semidirect
44. C2×Q8×C509C22semidirect
48. C22⋊C4×C509C22semidirect
49. C22⋊C4×C509C22semidirect
50. C42×C509C22semidirect
116. C509#538semidirect
121. #124D509semidirect
134. C2×Q8C2×D509semidirect
164. C22⋊C4C2×D509semidirect
165. C22⋊C4C2×D509semidirect
167. C42C2×D509semidirect

Maximal subgroups

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

subgroupindex
#306542092
#306542082
#306542042
#306541992
#306541972
#306541912
#306541882
#306541872
#306541852
#306541832
C4×D20362
#306541752
C22⋊C4×D5092
#124×C5092
#538509