Finite Group Databasebeta

← #24924007

Subgroups — #24924007

Order 10688.

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#19128828C211yesmaximal, characteristic, complemented
2#19128827C211yesmaximal, characteristic, complemented
3#19128825C211yesmaximal, characteristic, complemented
4#19128817C211yesmaximal, characteristic, complemented
5#19128816C211yesmaximal, characteristic, complemented
6#19128815C211yesmaximal, characteristic, complemented
7#19128806C211yesmaximal, characteristic, complemented
8#19128805C211yesmaximal, characteristic, complemented
9#19128804C211yesmaximal, characteristic, complemented
10#19128804C211yesmaximal, characteristic, complemented
11#19128803C211yesmaximal, characteristic, complemented
12#19128802C211yesmaximal, characteristic, complemented
13#19128800C211yesmaximal, characteristic, complemented
14#19128800C211yesmaximal, characteristic, complemented
15#121×C167C211yesmaximal, characteristic, complemented
16C23×D167C2211yescharacteristic
17C2×D668C2211yescharacteristic
18C2×D668C2211yescharacteristic
19C2×C22⋊D167C2211yescharacteristic
20C2×C22⋊D167C2211yescharacteristic
21C2×C22⋊D167C2211yescharacteristic
22C2×C22⋊D167C2211yescharacteristic
23C2×C22⋊D167C2211yescharacteristic
24C2×C22⋊D167C2211yescharacteristic, complemented
25C2×C22⋊D167C2211yescharacteristic, complemented
26C22×Dic167C2211yescharacteristic, complemented
27C2×C4×D167C2211yescharacteristic, complemented
28C4⋊Dic167C2211yescharacteristic, complemented
29C4⋊Dic167C2211yescharacteristic, complemented
30Dic167⋊C4C2211yescharacteristic, complemented
31Dic167⋊C4C2211yescharacteristic, complemented
32Dic167⋊C4C2211yescharacteristic, complemented
33C22⋊Dic167C2211yescharacteristic
34C22⋊Dic167C2211yescharacteristic
35C22⋊Dic167C2211yescharacteristic, complemented
36(C2×D167)⋊C4C2211yescharacteristic
37(C2×D167)⋊C4C2211yescharacteristic
38(C2×D167)⋊C4C2211yescharacteristic
39(C2×D167)⋊C4C2211yescharacteristic
40(C2×D167)⋊C4C2211yescharacteristic
41(C2×D167)⋊C4C2211yescharacteristic
42(C2×D167)⋊C4C2211yescharacteristic, complemented
43C4×Dic167C2211yescharacteristic, complemented
44C2×D4×C167C2211yescharacteristic
45C2×D4×C167C2211yescharacteristic
46C2×D4×C167C2211yescharacteristic, complemented
47C22×C4×C167C2211yescharacteristic, complemented
48C4⋊C4×C167C2211yescharacteristic, complemented
49C22⋊C4×C167C2211yescharacteristic
50C22⋊C4×C167C2211yescharacteristic, complemented
51C22×D167—41no—
52C22×D167—41no—
53C22×D167—41no—
54C22×D167C2311yescharacteristic
55C22×D167C2311yescharacteristic
56C22×D167C2311yescharacteristic
57D668—41no—
58D668—41no—
59C22⋊D167—41no—
60C22⋊D167—41no—
61C22⋊D167—41no—
62C22⋊D167—41no—
63C22⋊D167—41no—
64C22⋊D167—41no—
65C22⋊D167—41no—
66C2×Dic167—41no—
67C2×Dic167C2311yescharacteristic
68C2×Dic167C2311yescharacteristic
69C2×Dic167C2311yescharacteristic
70C2×Dic167C2311yescharacteristic
71C2×Dic167C2311yescharacteristic
72C4×D167—41no—
73C23×C167C2311yescharacteristic
74C23×C167C2311yescharacteristic
75C23×C167C2311yescharacteristic
76D4×C167—41no—
77D4×C167—41no—
78D4×C167—41no—
79C2×C4×C167—41no—
80C2×C4×C167C2311yescharacteristic
81C2×C4×C167C2311yescharacteristic
82C2×C4×C167C2311yescharacteristic
83C2×C4×C167C2311yescharacteristic
84C22×C167—21no—
85C22×C167—21no—
86C22×C167—21no—
87C22×C167—21no—
88C22×C167—21no—
89C22×C167—21no—
90C22×C167—21no—
91C22×C167—21no—
92C22×C167—21no—
93C22×C167C2411yescharacteristic
94C2×D167—21no—
95C2×D167—21no—
96C2×D167—21no—
97C2×D167—21no—
98C2×D167—21no—
99C2×D167—21no—
100C2×D167—21no—
101C2×D167—21no—
102C2×D167—21no—
103C2×D167—44no—
104C4×C167—21no—
105C4×C167—21no—
106C4×C167—21no—
107C4×C167—21no—
108Dic167—21no—
109Dic167—21no—
110Dic167—21no—
111Dic167—21no—
112Dic167—21no—
113C2×C167—41no—
114C2×C167—41no—
115C2×C167—41no—
116C2×C167H(5,𝔽2)11yescharacteristic
117C2×C167H(5,𝔽2)11yescharacteristic
118C2×C167H(5,𝔽2)11yescharacteristic
119D167—41no—
120D167—41no—
121D167—41no—
122C167#56011yescharacteristic, complemented, Sylow
123#560—1671nomaximal, Sylow
124#127—1671no—
125#126—1671no—
126#126—1671no—
127#123—1671no—
128#123—1671no—
129#123—1671no—
130#121—1671no—
131#121—1671no—
132#121—1671no—
133#121—1671no—
134#121—1671no—
135#121D16711yescharacteristic, complemented
136#120—1671no—
137#120—1671no—
138#120—1671no—
139C24—1671no—
140C2×D4—1671no—
141C2×D4—1671no—
142C2×D4—1671no—
143C2×D4—1671no—
144C2×D4—1671no—
145C2×D4—1671no—
146C2×D4—1671no—
147C2×D4—1671no—
148C2×D4—1671no—
149C2×D4C2×D16711yescharacteristic
150C2×D4C2×D16711yescharacteristic
151C2×D4C2×D16711yescharacteristic, complemented
152C22×C4—1671no—
153C22×C4—1671no—
154C22×C4C2×D16711yescharacteristic, complemented
155C4⋊C4—1671no—
156C4⋊C4—1671no—
157C4⋊C4—1671no—
158C4⋊C4—1671no—
159C4⋊C4—1671no—
160C4⋊C4C2×D16711yescharacteristic, complemented
161C22⋊C4—1671no—
162C22⋊C4—1671no—
163C22⋊C4—1671no—
164C22⋊C4—1671no—
165C22⋊C4—1671no—
166C22⋊C4—1671no—
167C22⋊C4—1671no—
168C22⋊C4—1671no—
169C22⋊C4—1671no—
170C22⋊C4—1671no—
171C22⋊C4C2×D16711yescharacteristic
172C22⋊C4C2×D16711yescharacteristic, complemented
173C42—1671no—
174C23—6681no—
175C23—6681no—
176C23—6681no—
177C23—1671no—
178C23—1671no—
179C23—1671no—
180C23C22×D16711yescharacteristic
181C23C22×D16711yescharacteristic
182C23C22×D16711yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—6681no—
187D4—6681no—
188D4—6681no—
189D4—6681no—
190D4—6681no—
191D4—6681no—
192D4—6681no—
193D4—6681no—
194D4—6681no—
195C2×C4—41no—
196C2×C4—6681no—
197C2×C4—6681no—
198C2×C4—1671no—
199C2×C4—1671no—
200C2×C4—1671no—
201C2×C4—1671no—
202C2×C4—1671no—
203C2×C4C22×D16711yescharacteristic
204C2×C4C22×D16711yescharacteristic
205C2×C4C22×D16711yescharacteristic
206C2×C4C22×D16711yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—3341no—
217C22—3341no—
218C22—3341no—
219C22—3341no—
220C22—3341no—
221C22—3341no—
222C22—3341no—
223C22—3341no—
224C22—3341no—
225C22—6684no—
226C22C23×D16711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—3341no—
232C4—3341no—
233C4—3341no—
234C4—3341no—
235C4—3341no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—6681no—
240C2—6681no—
241C2—6681no—
242C2#1912887711yescharacteristic
243C2#1912887711yescharacteristic
244C2#1912887811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D167
commutator, socle, cosocleC22×C167C24
Fitting#121×C167C2
hypercenter#121D167
hypocenterC167#560
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #19128828C2semidirect
2. #19128827C2semidirect
3. #19128825C2semidirect
4. #19128817C2semidirect
5. #19128816C2semidirect
6. #19128815C2semidirect
7. #19128806C2semidirect
8. #19128805C2semidirect
9. #19128804C2semidirect
10. #19128804C2semidirect
11. #19128803C2semidirect
12. #19128802C2semidirect
13. #19128800C2semidirect
14. #19128800C2semidirect
15. #121×C167C2semidirect
24. C2×C22⋊D167C22semidirect
25. C2×C22⋊D167C22semidirect
26. C22×Dic167C22semidirect
27. C2×C4×D167C22semidirect
28. C4⋊Dic167C22semidirect
29. C4⋊Dic167C22semidirect
30. Dic167⋊C4C22semidirect
31. Dic167⋊C4C22semidirect
32. Dic167⋊C4C22semidirect
35. C22⋊Dic167C22semidirect
42. (C2×D167)⋊C4C22semidirect
43. C4×Dic167C22semidirect
46. C2×D4×C167C22semidirect
47. C22×C4×C167C22semidirect
48. C4⋊C4×C167C22semidirect
50. C22⋊C4×C167C22semidirect
122. C167#560semidirect
135. #121D167semidirect
151. C2×D4C2×D167semidirect
154. C22×C4C2×D167semidirect
160. C4⋊C4C2×D167semidirect
172. C22⋊C4C2×D167semidirect