Finite Group Databasebeta

← #23757912

Subgroups — #23757912

Order 10048.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#14578953C211yesmaximal, characteristic, complemented
2C2×D4×D157C211yesmaximal, characteristic, complemented
3#19097757C211yesmaximal, characteristic, complemented
4#19097756C211yesmaximal, characteristic, complemented
5#19097752C211yesmaximal, characteristic, complemented
6#19097749C211yesmaximal, characteristic, complemented
7#19097744C211yesmaximal, characteristic, complemented
8#19097743C211yesmaximal, characteristic, complemented
9#19097739C211yesmaximal, characteristic, complemented
10#19097739C211yesmaximal, characteristic, complemented
11C4×C22⋊D157C211yesmaximal, characteristic, complemented
12#19097731C211yesmaximal, characteristic, complemented
13C2×#14578931C211yesmaximal, characteristic, complemented
14C22⋊C4×D157C211yesmaximal, characteristic, complemented
15#121×C157C211yesmaximal, characteristic, complemented
16C23×D157C2211yescharacteristic
17C23×D157C2211yescharacteristic
18#14578953—24no—
19C2×Dic314C2211yescharacteristic, complemented
20C2×D628C2211yescharacteristic
21C2×C22⋊D157C2211yescharacteristic
22C2×C22⋊D157C2211yescharacteristic
23C2×C22⋊D157C2211yescharacteristic
24C2×C22⋊D157C2211yescharacteristic, complemented
25C2×C22⋊D157C2211yescharacteristic, complemented
26D4×D157—24no—
27C22×Dic157C2211yescharacteristic, complemented
28C22×Dic157C2211yescharacteristic, complemented
29C2×C4×D157C2211yescharacteristic
30C2×C4×D157C2211yescharacteristic, complemented
31C2×C4×D157C2211yescharacteristic, complemented
32#14578933C2211yescharacteristic, complemented
33#14578933C2211yescharacteristic, complemented
34#14578933C2211yescharacteristic, complemented
35C22⋊Dic157C2211yescharacteristic, complemented
36C22⋊Dic157C2211yescharacteristic, complemented
37C22⋊Dic157C2211yescharacteristic, complemented
38#14578931C2211yescharacteristic
39#14578931C2211yescharacteristic
40#14578931C2211yescharacteristic
41#14578931C2211yescharacteristic, complemented
42#14578931C2211yescharacteristic, complemented
43#14578931C2211yescharacteristic, complemented
44#14578931C2211yescharacteristic, complemented
45C4×Dic157C2211yescharacteristic, complemented
46C2×D4×C157C2211yescharacteristic
47C2×D4×C157C2211yescharacteristic, complemented
48C2×D4×C157C2211yescharacteristic, complemented
49C22×C4×C157C2211yescharacteristic, complemented
50C4⋊C4×C157C2211yescharacteristic, complemented
51C22⋊C4×C157C2211yescharacteristic, complemented
52C22⋊C4×C157C2211yescharacteristic, complemented
53C22×D157—22no—
54C22×D157—22no—
55C22×D157—22no—
56C22×D157—22no—
57C22×D157—41no—
58C22×D157—41no—
59C22×D157C2311yescharacteristic
60C22×D157C2311yescharacteristic
61C22×D157C2311yescharacteristic
62Dic314—22no—
63D628—22no—
64C22⋊D157—22no—
65C22⋊D157—22no—
66C22⋊D157—22no—
67C22⋊D157—41no—
68C22⋊D157D414yescomplemented
69C2×Dic157—22no—
70C2×Dic157—22no—
71C2×Dic157C2311yescharacteristic
72C2×Dic157C2311yescharacteristic
73C2×Dic157C2311yescharacteristic
74C2×Dic157C2311yescharacteristic
75C2×Dic157C2311yescharacteristic
76C4×D157—22no—
77C4×D157—22no—
78C4×D157—41no—
79C23×C157C2311yescharacteristic
80C23×C157C2311yescharacteristic
81C23×C157C2311yescharacteristic
82D4×C157—22no—
83D4×C157—22no—
84D4×C157—41no—
85C2×C4×C157—41no—
86C2×C4×C157C2311yescharacteristic
87C2×C4×C157C2311yescharacteristic
88C2×C4×C157C2311yescharacteristic
89C2×C4×C157C2311yescharacteristic
90C22×C157—21no—
91C22×C157—21no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—21no—
95C22×C157—21no—
96C22×C157—21no—
97C22×C157—21no—
98C22×C157C2×D412yes—
99C22×C157C2411yescharacteristic
100C2×D157—21no—
101C2×D157—21no—
102C2×D157—21no—
103C2×D157—21no—
104C2×D157—21no—
105C2×D157—21no—
106C2×D157—21no—
107C2×D157—44no—
108C2×D157—44no—
109C2×D157C2×D412yes—
110C2×D157#4112yescomplemented
111C4×C157—21no—
112C4×C157—21no—
113C4×C157—21no—
114C4×C157—21no—
115Dic157—21no—
116Dic157—21no—
117Dic157—21no—
118Dic157—21no—
119Dic157C2×D412yes—
120C2×C157—22no—
121C2×C157—41no—
122C2×C157—41no—
123C2×C157C22×D411yescharacteristic
124C2×C157C2×#4111yescharacteristic
125C2×C157H(5,𝔽2)11yescharacteristic
126D157—22no—
127D157—22no—
128D157—41no—
129C157#54611yescharacteristic, complemented, Sylow
130#546—1571nomaximal, Sylow
131C2×#41—1571no—
132C22×D4—1571no—
133#124—1571no—
134#123—1571no—
135#123—1571no—
136C22⋊Q8—1571no—
137#121—1571no—
138#121—1571no—
139#121D15711yescharacteristic, complemented
140#120—1571no—
141#120—1571no—
142C4×D4—1571no—
143C4×D4—1571no—
144C2×C22⋊C4—1571no—
145C2×C22⋊C4—1571no—
146C24—1571no—
147C24—1571no—
148#41—3144no—
149C2×Q8—1571no—
150C2×D4—3144no—
151C2×D4—1571no—
152C2×D4—1571no—
153C2×D4—1571no—
154C2×D4—1571no—
155C2×D4—1571no—
156C2×D4—1571no—
157C2×D4C2×D15711yescharacteristic
158C2×D4C2×D15711yescharacteristic, complemented
159C2×D4C2×D15711yescharacteristic, complemented
160C22×C4—1571no—
161C22×C4—1571no—
162C22×C4—1571no—
163C22×C4—1571no—
164C22×C4—1571no—
165C22×C4C2×D15711yescharacteristic, complemented
166C4⋊C4—1571no—
167C4⋊C4—1571no—
168C4⋊C4—1571no—
169C4⋊C4C2×D15711yescharacteristic, complemented
170C22⋊C4—1571no—
171C22⋊C4—1571no—
172C22⋊C4—1571no—
173C22⋊C4—1571no—
174C22⋊C4—1571no—
175C22⋊C4—1571no—
176C22⋊C4—1571no—
177C22⋊C4—1571no—
178C22⋊C4—1571no—
179C22⋊C4—1571no—
180C22⋊C4C2×D15711yescharacteristic, complemented
181C22⋊C4C2×D15711yescharacteristic, complemented
182C42—1571no—
183C23—3142no—
184C23—3142no—
185C23—3142no—
186C23—3142no—
187C23—6281no—
188C23—6281no—
189C23—1571no—
190C23—1571no—
191C23—1571no—
192C23C22×D15711yescharacteristic
193C23C22×D15711yescharacteristic
194C23C22×D15711yescharacteristic
195Q8—3142no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—3142no—
200D4—3142no—
201D4—3142no—
202D4—3142no—
203D4—6281no—
204D4—1574no—
205C2×C4—41no—
206C2×C4—3142no—
207C2×C4—3142no—
208C2×C4—3142no—
209C2×C4—3142no—
210C2×C4—6281no—
211C2×C4—1571no—
212C2×C4—1571no—
213C2×C4—1571no—
214C2×C4—1571no—
215C2×C4—1571no—
216C2×C4C22×D15711yescharacteristic
217C2×C4C22×D15711yescharacteristic
218C2×C4C22×D15711yescharacteristic
219C2×C4C22×D15711yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—3141no—
229C22—3141no—
230C22—3141no—
231C22—3141no—
232C22—3141no—
233C22—3141no—
234C22—3141no—
235C22—6284no—
236C22—6284no—
237C22—1572no—
238C22—1572no—
239C22D4×D15712yes—
240C22C23×D15711yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—3141no—
246C4—3141no—
247C4—3141no—
248C4—3141no—
249C4—1572no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—3142no—
254C2—3142no—
255C2—6281no—
256C2C2×D4×D15711yescharacteristic
257C2#41×D15711yescharacteristic
258C2#1909781611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D157
commutator, socle, cosocleC22×C157C24
Fitting#121×C157C2
hypercenter#121D157
hypocenterC157#546
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#14578953C2semidirect
2. C2×D4×D157C2semidirect
3. #19097757C2semidirect
4. #19097756C2semidirect
5. #19097752C2semidirect
6. #19097749C2semidirect
7. #19097744C2semidirect
8. #19097743C2semidirect
9. #19097739C2semidirect
10. #19097739C2semidirect
11. C4×C22⋊D157C2semidirect
12. #19097731C2semidirect
13. C2×#14578931C2semidirect
14. C22⋊C4×D157C2semidirect
15. #121×C157C2semidirect
19. C2×Dic314C22semidirect
24. C2×C22⋊D157C22semidirect
25. C2×C22⋊D157C22semidirect
27. C22×Dic157C22semidirect
28. C22×Dic157C22semidirect
30. C2×C4×D157C22semidirect
31. C2×C4×D157C22semidirect
32. #14578933C22semidirect
33. #14578933C22semidirect
34. #14578933C22semidirect
35. C22⋊Dic157C22semidirect
36. C22⋊Dic157C22semidirect
37. C22⋊Dic157C22semidirect
41. #14578931C22semidirect
42. #14578931C22semidirect
43. #14578931C22semidirect
44. #14578931C22semidirect
45. C4×Dic157C22semidirect
47. C2×D4×C157C22semidirect
48. C2×D4×C157C22semidirect
49. C22×C4×C157C22semidirect
50. C4⋊C4×C157C22semidirect
51. C22⋊C4×C157C22semidirect
52. C22⋊C4×C157C22semidirect
68. C22⋊D157D4semidirect
110. C2×D157#41semidirect
129. C157#546semidirect
139. #121D157semidirect
158. C2×D4C2×D157semidirect
159. C2×D4C2×D157semidirect
165. C22×C4C2×D157semidirect
169. C4⋊C4C2×D157semidirect
180. C22⋊C4C2×D157semidirect
181. C22⋊C4C2×D157semidirect