Finite Group Databasebeta

← #24923943

Subgroups — #24923943

Order 10688.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#19128827C211yesmaximal, characteristic, complemented
2#19128826C211yesmaximal, characteristic, complemented
3#19128820C211yesmaximal, characteristic, complemented
4#19128817C211yesmaximal, characteristic, complemented
5#19128817C211yesmaximal, characteristic, complemented
6#19128815C211yesmaximal, characteristic, complemented
7#19128810C211yesmaximal, characteristic, complemented
8#19128809C211yesmaximal, characteristic, complemented
9#19128802C211yesmaximal, characteristic
10C4×D668C211yesmaximal, characteristic, complemented
11(C22⋊D167)⋊C4C211yesmaximal, characteristic, complemented
12#19128789C211yesmaximal, characteristic, complemented
13C2×(C2×D167)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D167C211yesmaximal, characteristic, complemented
15#117×C167C211yesmaximal, characteristic, complemented
16C23×D167C2211yescharacteristic
17C2×Dic334C2211yescharacteristic, complemented
18C2×D668C2211yescharacteristic
19C2×D668C2211yescharacteristic
20C2×C22⋊D167C2211yescharacteristic
21C22×Dic167C2211yescharacteristic
22C2×C4×D167C2211yescharacteristic
23C2×C4×D167C2211yescharacteristic
24C2×C4×D167C2211yescharacteristic, complemented
25C4⋊Dic167C2211yescharacteristic, complemented
26Dic167⋊C4C2211yescharacteristic, complemented
27Dic167⋊C4C2211yescharacteristic, complemented
28Dic167⋊C4C2211yescharacteristic, complemented
29Dic167⋊C4C2211yescharacteristic, complemented
30Dic167⋊C4C2211yescharacteristic, complemented
31C22⋊Dic167C2211yescharacteristic
32(C2×D167)⋊C4C2211yescharacteristic
33(C2×D167)⋊C4C2211yescharacteristic
34(C2×D167)⋊C4C2211yescharacteristic
35(C2×D167)⋊C4C2211yescharacteristic
36(C2×D167)⋊C4C2211yescharacteristic
37(C2×D167)⋊C4C2211yescharacteristic
38(C2×D167)⋊C4C2211yescharacteristic
39(C2×D167)⋊C4C2211yescharacteristic
40(C2×D167)⋊C4C2211yescharacteristic, complemented
41(C2×D167)⋊C4C2211yescharacteristic, complemented
42(C2×D167)⋊C4C2211yescharacteristic, complemented
43C4×Dic167C2211yescharacteristic, complemented
44C22×C4×C167C2211yescharacteristic
45C4⋊C4×C167C2211yescharacteristic, complemented
46C4⋊C4×C167C2211yescharacteristic, complemented
47C22⋊C4×C167C2211yescharacteristic
48C22⋊C4×C167C2211yescharacteristic
49C42×C167C2211yescharacteristic, complemented
50C42×C167C2211yescharacteristic, complemented
51C22×D167—22no—
52C22×D167—22no—
53C22×D167—41no—
54C22×D167C2311yescharacteristic
55C22×D167C2311yescharacteristic
56C22×D167C2311yescharacteristic
57Dic334—41no—
58D668—22no—
59D668—41no—
60C22⋊D167—22no—
61C2×Dic167—41no—
62C2×Dic167C2311yescharacteristic
63C2×Dic167C2311yescharacteristic
64C2×Dic167C2311yescharacteristic
65C2×Dic167C2311yescharacteristic
66C2×Dic167C2311yescharacteristic
67C4×D167—22no—
68C4×D167—41no—
69C4×D167—41no—
70C23×C167C2311yescharacteristic
71C2×C4×C167—22no—
72C2×C4×C167C2311yescharacteristic
73C2×C4×C167C2311yescharacteristic
74C2×C4×C167C2311yescharacteristic
75C2×C4×C167C2311yescharacteristic
76C2×C4×C167C2311yescharacteristic
77C2×C4×C167C2311yescharacteristic
78C22×C167—21no—
79C22×C167—21no—
80C22×C167#4112yes—
81C22×C167C2411yescharacteristic
82C2×D167—21no—
83C2×D167—21no—
84C2×D167—21no—
85C2×D167—21no—
86C2×D167—21no—
87C2×D167—21no—
88C2×D167—21no—
89C2×D167—21no—
90C2×D167—44no—
91C2×D167#4112yes—
92C4×C167—21no—
93C4×C167—21no—
94C4×C167—21no—
95C4×C167—21no—
96C4×C167—21no—
97C4×C167—21no—
98Dic167—21no—
99Dic167—21no—
100Dic167—21no—
101Dic167—21no—
102Dic167—21no—
103C2×C167—22no—
104C2×C167C2×#4111yescharacteristic
105C2×C167C2×#4111yescharacteristic
106C2×C167H(5,𝔽2)11yescharacteristic
107D167—22no—
108D167—41no—
109D167—41no—
110C167#55111yescharacteristic, complemented, Sylow
111#551—1671nomaximal, Sylow
112#126—1671no—
113#126—1671no—
114#124—1671no—
115#123—1671no—
116#123—1671no—
117#123—1671no—
118C22⋊Q8—1671no—
119C22⋊Q8—1671no—
120#120—1671no—
121C4×D4—1671no—
122C4×D4—1671no—
123#117—1671no—
124#117D16711yescharacteristic, complemented
125C2×C22⋊C4—1671no—
126C2×C22⋊C4—1671no—
127C24—1671no—
128C2×Q8—1671no—
129C2×D4—1671no—
130C2×D4—1671no—
131C2×D4—1671no—
132C22×C4—1671no—
133C22×C4—1671no—
134C22×C4—1671no—
135C22×C4—1671no—
136C22×C4C2×D16711yescharacteristic
137C4⋊C4—1671no—
138C4⋊C4—1671no—
139C4⋊C4—1671no—
140C4⋊C4—1671no—
141C4⋊C4—1671no—
142C4⋊C4—1671no—
143C4⋊C4C2×D16711yescharacteristic, complemented
144C4⋊C4C2×D16711yescharacteristic, complemented
145C22⋊C4—1671no—
146C22⋊C4—1671no—
147C22⋊C4—1671no—
148C22⋊C4—1671no—
149C22⋊C4—1671no—
150C22⋊C4—1671no—
151C22⋊C4—1671no—
152C22⋊C4—1671no—
153C22⋊C4—1671no—
154C22⋊C4—1671no—
155C22⋊C4—1671no—
156C22⋊C4—1671no—
157C22⋊C4C2×D16711yescharacteristic
158C22⋊C4C2×D16711yescharacteristic
159C42—1671no—
160C42C2×D16711yescharacteristic, complemented
161C42C2×D16711yescharacteristic, complemented
162C23—3342no—
163C23—3342no—
164C23—6681no—
165C23—1671no—
166C23—1671no—
167C23—1671no—
168C23C22×D16711yescharacteristic
169Q8—6681no—
170D4—3342no—
171D4—3342no—
172D4—6681no—
173C2×C4—22no—
174C2×C4—3342no—
175C2×C4—6681no—
176C2×C4—6681no—
177C2×C4—6681no—
178C2×C4—1671no—
179C2×C4—1671no—
180C2×C4—1671no—
181C2×C4—1671no—
182C2×C4—1671no—
183C2×C4C22×D16711yescharacteristic
184C2×C4C22×D16711yescharacteristic
185C2×C4C22×D16711yescharacteristic
186C2×C4C22×D16711yescharacteristic
187C2×C4C22×D16711yescharacteristic
188C2×C4C22×D16711yescharacteristic
189C22—21no—
190C22—21no—
191C22—3341no—
192C22—3341no—
193C22—3341no—
194C22—3341no—
195C22—3341no—
196C22—3341no—
197C22—3341no—
198C22—3341no—
199C22—6684no—
200C22—1672no—
201C22#1458195012yes—
202C22C23×D16711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—3341no—
210C4—3341no—
211C4—3341no—
212C4—3341no—
213C4—3341no—
214C2—22no—
215C2—3342no—
216C2—6681no—
217C2—6681no—
218C2#41×D16711yescharacteristic
219C2C2×#1458195011yescharacteristic
220C2#1912887811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D167
commutator, socle, cosocleC22×C167C24
Fitting#117×C167C2
prime radical#19128802C2
hypercenter#117D167
hypocenterC167#551
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #19128827C2semidirect
2. #19128826C2semidirect
3. #19128820C2semidirect
4. #19128817C2semidirect
5. #19128817C2semidirect
6. #19128815C2semidirect
7. #19128810C2semidirect
8. #19128809C2semidirect
10. C4×D668C2semidirect
11. (C22⋊D167)⋊C4C2semidirect
12. #19128789C2semidirect
13. C2×(C2×D167)⋊C4C2semidirect
14. C22⋊C4×D167C2semidirect
15. #117×C167C2semidirect
17. C2×Dic334C22semidirect
24. C2×C4×D167C22semidirect
25. C4⋊Dic167C22semidirect
26. Dic167⋊C4C22semidirect
27. Dic167⋊C4C22semidirect
28. Dic167⋊C4C22semidirect
29. Dic167⋊C4C22semidirect
30. Dic167⋊C4C22semidirect
40. (C2×D167)⋊C4C22semidirect
41. (C2×D167)⋊C4C22semidirect
42. (C2×D167)⋊C4C22semidirect
43. C4×Dic167C22semidirect
45. C4⋊C4×C167C22semidirect
46. C4⋊C4×C167C22semidirect
49. C42×C167C22semidirect
50. C42×C167C22semidirect
110. C167#551semidirect
124. #117D167semidirect
143. C4⋊C4C2×D167semidirect
144. C4⋊C4C2×D167semidirect
160. C42C2×D167semidirect
161. C42C2×D167semidirect