Finite Group Databasebeta

← #39909985

Subgroups — #39909985

Order 26176.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#27189713C211yesmaximal, characteristic, complemented
2#27189712C211yesmaximal, characteristic, complemented
3#27189710C211yesmaximal, characteristic, complemented
4#27189702C211yesmaximal, characteristic, complemented
5#27189701C211yesmaximal, characteristic, complemented
6#27189700C211yesmaximal, characteristic, complemented
7#27189691C211yesmaximal, characteristic, complemented
8#27189690C211yesmaximal, characteristic, complemented
9#27189689C211yesmaximal, characteristic, complemented
10#27189689C211yesmaximal, characteristic, complemented
11#27189688C211yesmaximal, characteristic, complemented
12#27189687C211yesmaximal, characteristic, complemented
13#27189685C211yesmaximal, characteristic, complemented
14#27189685C211yesmaximal, characteristic, complemented
15#121×C409C211yesmaximal, characteristic, complemented
16C23×D409C2211yescharacteristic
17C2×D1636C2211yescharacteristic
18C2×D1636C2211yescharacteristic
19C2×C22⋊D409C2211yescharacteristic
20C2×C22⋊D409C2211yescharacteristic
21C2×C22⋊D409C2211yescharacteristic
22C2×C22⋊D409C2211yescharacteristic
23C2×C22⋊D409C2211yescharacteristic
24C2×C22⋊D409C2211yescharacteristic, complemented
25C2×C22⋊D409C2211yescharacteristic, complemented
26C22×Dic409C2211yescharacteristic, complemented
27C2×C4×D409C2211yescharacteristic, complemented
28C4⋊Dic409C2211yescharacteristic, complemented
29C4⋊Dic409C2211yescharacteristic, complemented
30#20266967C2211yescharacteristic, complemented
31#20266967C2211yescharacteristic, complemented
32#20266967C2211yescharacteristic, complemented
33C22⋊Dic409C2211yescharacteristic
34C22⋊Dic409C2211yescharacteristic
35C22⋊Dic409C2211yescharacteristic, complemented
36#20266965C2211yescharacteristic
37#20266965C2211yescharacteristic
38#20266965C2211yescharacteristic
39#20266965C2211yescharacteristic
40#20266965C2211yescharacteristic
41#20266965C2211yescharacteristic
42#20266965C2211yescharacteristic, complemented
43C4×Dic409C2211yescharacteristic, complemented
44C2×D4×C409C2211yescharacteristic
45C2×D4×C409C2211yescharacteristic
46C2×D4×C409C2211yescharacteristic, complemented
47C22×C4×C409C2211yescharacteristic, complemented
48C4⋊C4×C409C2211yescharacteristic, complemented
49C22⋊C4×C409C2211yescharacteristic
50C22⋊C4×C409C2211yescharacteristic, complemented
51C22×D409—41no—
52C22×D409—41no—
53C22×D409—41no—
54C22×D409C2311yescharacteristic
55C22×D409C2311yescharacteristic
56C22×D409C2311yescharacteristic
57D1636—41no—
58D1636—41no—
59C22⋊D409—41no—
60C22⋊D409—41no—
61C22⋊D409—41no—
62C22⋊D409—41no—
63C22⋊D409—41no—
64C22⋊D409—41no—
65C22⋊D409—41no—
66C2×Dic409—41no—
67C2×Dic409C2311yescharacteristic
68C2×Dic409C2311yescharacteristic
69C2×Dic409C2311yescharacteristic
70C2×Dic409C2311yescharacteristic
71C2×Dic409C2311yescharacteristic
72C4×D409—41no—
73C23×C409C2311yescharacteristic
74C23×C409C2311yescharacteristic
75C23×C409C2311yescharacteristic
76D4×C409—41no—
77D4×C409—41no—
78D4×C409—41no—
79C2×C4×C409—41no—
80C2×C4×C409C2311yescharacteristic
81C2×C4×C409C2311yescharacteristic
82C2×C4×C409C2311yescharacteristic
83C2×C4×C409C2311yescharacteristic
84C22×C409—21no—
85C22×C409—21no—
86C22×C409—21no—
87C22×C409—21no—
88C22×C409—21no—
89C22×C409—21no—
90C22×C409—21no—
91C22×C409—21no—
92C22×C409—21no—
93C22×C409C2411yescharacteristic
94C2×D409—21no—
95C2×D409—21no—
96C2×D409—21no—
97C2×D409—21no—
98C2×D409—21no—
99C2×D409—21no—
100C2×D409—21no—
101C2×D409—21no—
102C2×D409—21no—
103C2×D409—44no—
104C4×C409—21no—
105C4×C409—21no—
106C4×C409—21no—
107C4×C409—21no—
108Dic409—21no—
109Dic409—21no—
110Dic409—21no—
111Dic409—21no—
112Dic409—21no—
113C2×C409—41no—
114C2×C409—41no—
115C2×C409—41no—
116C2×C409H(5,𝔽2)11yescharacteristic
117C2×C409H(5,𝔽2)11yescharacteristic
118C2×C409H(5,𝔽2)11yescharacteristic
119D409—41no—
120D409—41no—
121D409—41no—
122C409#56011yescharacteristic, complemented, Sylow
123#560—4091nomaximal, Sylow
124#127—4091no—
125#126—4091no—
126#126—4091no—
127#123—4091no—
128#123—4091no—
129#123—4091no—
130#121—4091no—
131#121—4091no—
132#121—4091no—
133#121—4091no—
134#121—4091no—
135#121D40911yescharacteristic, complemented
136#120—4091no—
137#120—4091no—
138#120—4091no—
139C24—4091no—
140C2×D4—4091no—
141C2×D4—4091no—
142C2×D4—4091no—
143C2×D4—4091no—
144C2×D4—4091no—
145C2×D4—4091no—
146C2×D4—4091no—
147C2×D4—4091no—
148C2×D4—4091no—
149C2×D4C2×D40911yescharacteristic
150C2×D4C2×D40911yescharacteristic
151C2×D4C2×D40911yescharacteristic, complemented
152C22×C4—4091no—
153C22×C4—4091no—
154C22×C4C2×D40911yescharacteristic, complemented
155C4⋊C4—4091no—
156C4⋊C4—4091no—
157C4⋊C4—4091no—
158C4⋊C4—4091no—
159C4⋊C4—4091no—
160C4⋊C4C2×D40911yescharacteristic, complemented
161C22⋊C4—4091no—
162C22⋊C4—4091no—
163C22⋊C4—4091no—
164C22⋊C4—4091no—
165C22⋊C4—4091no—
166C22⋊C4—4091no—
167C22⋊C4—4091no—
168C22⋊C4—4091no—
169C22⋊C4—4091no—
170C22⋊C4—4091no—
171C22⋊C4C2×D40911yescharacteristic
172C22⋊C4C2×D40911yescharacteristic, complemented
173C42—4091no—
174C23—16361no—
175C23—16361no—
176C23—16361no—
177C23—4091no—
178C23—4091no—
179C23—4091no—
180C23C22×D40911yescharacteristic
181C23C22×D40911yescharacteristic
182C23C22×D40911yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—16361no—
187D4—16361no—
188D4—16361no—
189D4—16361no—
190D4—16361no—
191D4—16361no—
192D4—16361no—
193D4—16361no—
194D4—16361no—
195C2×C4—41no—
196C2×C4—16361no—
197C2×C4—16361no—
198C2×C4—4091no—
199C2×C4—4091no—
200C2×C4—4091no—
201C2×C4—4091no—
202C2×C4—4091no—
203C2×C4C22×D40911yescharacteristic
204C2×C4C22×D40911yescharacteristic
205C2×C4C22×D40911yescharacteristic
206C2×C4C22×D40911yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—8181no—
217C22—8181no—
218C22—8181no—
219C22—8181no—
220C22—8181no—
221C22—8181no—
222C22—8181no—
223C22—8181no—
224C22—8181no—
225C22—16364no—
226C22C23×D40911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—8181no—
232C4—8181no—
233C4—8181no—
234C4—8181no—
235C4—8181no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—16361no—
240C2—16361no—
241C2—16361no—
242C2#2718976211yescharacteristic
243C2#2718976211yescharacteristic
244C2#2718976311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D409
commutator, socle, cosocleC22×C409C24
Fitting#121×C409C2
hypercenter#121D409
hypocenterC409#560
LCSG ▷ C22×C409 ▷ C409
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #27189713C2semidirect
2. #27189712C2semidirect
3. #27189710C2semidirect
4. #27189702C2semidirect
5. #27189701C2semidirect
6. #27189700C2semidirect
7. #27189691C2semidirect
8. #27189690C2semidirect
9. #27189689C2semidirect
10. #27189689C2semidirect
11. #27189688C2semidirect
12. #27189687C2semidirect
13. #27189685C2semidirect
14. #27189685C2semidirect
15. #121×C409C2semidirect
24. C2×C22⋊D409C22semidirect
25. C2×C22⋊D409C22semidirect
26. C22×Dic409C22semidirect
27. C2×C4×D409C22semidirect
28. C4⋊Dic409C22semidirect
29. C4⋊Dic409C22semidirect
30. #20266967C22semidirect
31. #20266967C22semidirect
32. #20266967C22semidirect
35. C22⋊Dic409C22semidirect
42. #20266965C22semidirect
43. C4×Dic409C22semidirect
46. C2×D4×C409C22semidirect
47. C22×C4×C409C22semidirect
48. C4⋊C4×C409C22semidirect
50. C22⋊C4×C409C22semidirect
122. C409#560semidirect
135. #121D409semidirect
151. C2×D4C2×D409semidirect
154. C22×C4C2×D409semidirect
160. C4⋊C4C2×D409semidirect
172. C22⋊C4C2×D409semidirect

Maximal subgroups

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

subgroupindex
#271897132
#271897122
#271897102
#271897022
#271897012
#271897002
#271896912
#271896902
#271896892
#271896882
#271896872
#271896852
#121×C4092
#560409