Finite Group Databasebeta

← #34111751

Subgroups — #34111751

Order 20096.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19097798C212yesmaximal, complemented
2#23757945C211yesmaximal, characteristic, complemented
3#23757908C211yesmaximal, characteristic, complemented
4#23757649C211yesmaximal, characteristic, complemented
5#23757525C212yesmaximal, complemented
6#23757469C211yesmaximal, characteristic, complemented
7#23757417C212yesmaximal, complemented
8#23757408C212yesmaximal, complemented
9C4×#14578941C211yesmaximal, characteristic, complemented
10#23757350C211yesmaximal, characteristic, complemented
11#460×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2212yescomplemented
13C2×D4×D157—21no—
14C2×D4×D157C2212yes—
15#19097798—28no—
16C2×#14578941C2211yescharacteristic
17C2×D1256C2212yes—
18C2×#14578939C2212yescomplemented
19C2×#14578939C2212yescomplemented
20C2×#14578937C2212yescomplemented
21#19097768—21no—
22#19097768C2211yescharacteristic, complemented
23#19097746—22no—
24#19097746C2212yescomplemented
25#19097742—21no—
26#19097742C2212yes—
27#19097739—22no—
28C4×D628—21no—
29C4×D628C2211yescharacteristic
30C4×D628C2211yescharacteristic
31#19097702C2211yescharacteristic, complemented
32#19097695C2211yescharacteristic, complemented
33#19097693C2212yescomplemented
34#19097691C2211yescharacteristic
35#19097683C2212yescomplemented
36C4×#11929434C2211yescharacteristic, complemented
37C2×SD16×C157C2212yescomplemented
38#127×C157C2211yescharacteristic
39C4×Q8×C157C2211yescharacteristic, complemented
40C4⋊C8×C157C2211yescharacteristic, complemented
41#102×C157C2212yescomplemented
42C23×D157—22no—
43#14578955—24no—
44#14578955—42no—
45C2×D628—22no—
46C2×D628—22no—
47C2×D628—21no—
48C2×D628—21no—
49C2×D628D411yescharacteristic
50C2×D628C2312yes—
51C2×D628C2311yescharacteristic
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54D4×D157—24no—
55D4×D157—42no—
56D4×D157—44no—
57C2×C4×D157—22no—
58C2×C4×D157—21no—
59C2×C4×D157C2312yes—
60#14578941D414yescomplemented
61D1256—24no—
62#14578939—24no—
63#14578939—42no—
64#14578937—24no—
65C2×#11929434C2311yescharacteristic
66C2×#11929434C2311yescharacteristic
67C4⋊Dic157D411yescharacteristic
68C4⋊Dic157C2311yescharacteristic
69C22⋊Dic157—22no—
70#14578931—22no—
71#14578931—21no—
72#14578931D412yes—
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157—21no—
76C2×D4×C157C2312yes—
77SD16×C157—24no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C42×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—44no—
86C22×D157—44no—
87C22×D157—82no—
88C22×D157C2×D412yes—
89D628—22no—
90D628—22no—
91D628—21no—
92D628—22no—
93D628—44no—
94D628—42no—
95D628—81no—
96D628—82no—
97D628C2×D412yes—
98C22⋊D157—44no—
99C22⋊D157—44no—
100C2×Dic157—21no—
101C2×Dic157C2×D412yes—
102C4×D157—22no—
103C4×D157—22no—
104C4×D157—44no—
105C4×D157—42no—
106#11929434—21no—
107#11929434C2×D412yescomplemented
108C23×C157—22no—
109Q8×C157—21no—
110Q8×C157C2×D412yescomplemented
111D4×C157—22no—
112D4×C157—22no—
113D4×C157—42no—
114D4×C157—81no—
115C2×C4×C157—21no—
116C2×C4×C157—21no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2411yescharacteristic
120C8×C157—22no—
121C22×C157—42no—
122C22×C157—42no—
123C22×C157—42no—
124C22×C157C22×D411yescharacteristic
125C2×D157—22no—
126C2×D157—22no—
127C2×D157—22no—
128C2×D157—22no—
129C2×D157—41no—
130C2×D157—41no—
131C2×D157—42no—
132C2×D157—42no—
133C2×D157—42no—
134C2×D157—88no—
135C4×C157—22no—
136C4×C157—21no—
137C4×C157—41no—
138C4×C157AGL(1,ℤ/8)12yescomplemented
139C4×C157C22×D411yescharacteristic
140C4×C157H(5,𝔽2)11yescharacteristic
141Dic157—22no—
142Dic157—41no—
143C2×C157—82no—
144C2×C157D4211yescharacteristic
145C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C157#57611yescharacteristic
147D157—42no—
148D157—42no—
149D157—82no—
150C157#327411yescharacteristic, complemented, Sylow
151#3274—1571nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1572no—
153#550—1571no—
154D42—1571no—
155#493—1571no—
156#469—1572no—
157#460D15711yescharacteristic, complemented
158#459—1571no—
159#449—1572no—
160#447—1572no—
161C4×SD16—1571no—
162#436—1571no—
163C2×#41—1572no—
164C22×D4—3141no—
165C22×D4—1572no—
166AGL(1,ℤ/8)—3148no—
167C2×SD16—1571no—
168C2×SD16C2×D15712yescomplemented
169C2×D8—1572no—
170C2×D8—1572no—
171C2×D8—1572no—
172C2×M16—1572no—
173#127—3141no—
174#127—1571no—
175#127C2×D15711yescharacteristic
176#121—3141no—
177#121—3142no—
178#121—1572no—
179#121—1572no—
180#120—3142no—
181C4×Q8C2×D15711yescharacteristic, complemented
182C4×D4—3141no—
183C4×D4—1571no—
184C4×D4—1571no—
185#106—1571no—
186C4⋊C8C2×D15711yescharacteristic, complemented
187#103—1571no—
188#102—1572no—
189#102—1571no—
190#102C2×D15712yescomplemented
191C22⋊C8—1572no—
192C4×C8—1571no—
193C24—3142no—
194#41—3144no—
195#41—6282no—
196C2×Q8C22×D15711yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—3142no—
200C2×D4—3142no—
201C2×D4—3144no—
202C2×D4—3142no—
203C2×D4—3142no—
204C2×D4—3141no—
205C2×D4—3141no—
206C2×D4—6282no—
207C2×D4—6284no—
208C2×D4—1571no—
209C2×D4—1572no—
210C2×D4—1571no—
211C2×D4C22×D15712yes—
212C22×C4—3142no—
213C22×C4—3141no—
214C22×C4—1572no—
215SD16—24no—
216SD16—1574no—
217D8—3144no—
218D8—3144no—
219D8—6282no—
220M16—3144no—
221C2×C8—1571no—
222C2×C8—1571no—
223C2×C8C22×D15712yes—
224C4⋊C4—21no—
225C4⋊C4—1571no—
226C4⋊C4—1571no—
227C4⋊C4C22×D15711yescharacteristic
228C22⋊C4—3142no—
229C22⋊C4—3142no—
230C22⋊C4—3141no—
231C22⋊C4—1572no—
232C42—21no—
233C42C22×D15711yescharacteristic
234C23—22no—
235C23—3141no—
236C23—3142no—
237C23—6284no—
238C23—6284no—
239C23—1572no—
240C23—12562no—
241Q8—21no—
242Q8D4×D15712yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—3142no—
248D4—3142no—
249D4—3141no—
250D4—3142no—
251D4—6284no—
252D4—6284no—
253D4—6284no—
254D4—6282no—
255D4—1572no—
256D4—12561no—
257D4—12562no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—6284no—
264C2×C4—6282no—
265C2×C4—1572no—
266C2×C4D4×D15711yescharacteristic
267C2×C4D4×D15711yescharacteristic
268C2×C4C23×D15711yescharacteristic
269C8—22no—
270C8—3141no—
271C8—1572no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3142no—
276C22—3142no—
277C22—3142no—
278C22—3142no—
279C22—6281no—
280C22—6281no—
281C22—6282no—
282C22—6282no—
283C22—6282no—
284C22—12568no—
285C22C2×D4×D15711yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—3142no—
290C4—6281no—
291C4#1909779812yescomplemented
292C4C2×D4×D15711yescharacteristic
293C4#1909781711yescharacteristic
294C2—82no—
295C2—6282no—
296C2—6282no—
297C2—12562no—
298C2#2375790711yescharacteristic
299C2C2×#1909779811yescharacteristic
300C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#460×C157C2
socleC22×C157C22×D4
hypercenter#460D157
hypocenterC157#3274
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. #23757945C2semidirect
3. #23757908C2semidirect
4. #23757649C2semidirect
5. #23757525C2semidirect
6. #23757469C2semidirect
7. #23757417C2semidirect
8. #23757408C2semidirect
9. C4×#14578941C2semidirect
10. #23757350C2semidirect
11. #460×C157C2semidirect
12. C2×#14578955C22semidirect
18. C2×#14578939C22semidirect
19. C2×#14578939C22semidirect
20. C2×#14578937C22semidirect
22. #19097768C22semidirect
24. #19097746C22semidirect
31. #19097702C22semidirect
32. #19097695C22semidirect
33. #19097693C22semidirect
35. #19097683C22semidirect
36. C4×#11929434C22semidirect
37. C2×SD16×C157C22semidirect
39. C4×Q8×C157C22semidirect
40. C4⋊C8×C157C22semidirect
41. #102×C157C22semidirect
60. #14578941D4semidirect
107. #11929434C2×D4semidirect
110. Q8×C157C2×D4semidirect
138. C4×C157AGL(1,ℤ/8)semidirect
150. C157#3274semidirect
157. #460D157semidirect
168. C2×SD16C2×D157semidirect
181. C4×Q8C2×D157semidirect
186. C4⋊C8C2×D157semidirect
190. #102C2×D157semidirect
242. Q8D4×D157semidirect
291. C4#19097798semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
#237579452
#237579082
#237576492
#237575252
#237574692
#237574172
#237574082
C4×#145789412
#237573502
#460×C1572
#3274157