Finite Group Databasebeta

← #34111739

Subgroups — #34111739

Order 20096.

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

normal subgroups only

This group has 546 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#23757926C211yesmaximal, characteristic, complemented
3D4×D628C211yesmaximal, characteristic, complemented
4#23757647C211yesmaximal, characteristic, complemented
5#23757522C212yesmaximal, complemented
6#23757469C211yesmaximal, characteristic, complemented
7#23757422C212yesmaximal, complemented
8#23757421C212yesmaximal, complemented
9C4×#14578939C211yesmaximal, characteristic, complemented
10#23757350C211yesmaximal, characteristic, complemented
11#461×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2212yescomplemented
13C22×D628—21no—
14C2×D4×D157C2212yes—
15#19097798—28no—
16C2×#14578941C2212yescomplemented
17C2×#14578941C2212yescomplemented
18C2×D1256C2212yescomplemented
19C2×#14578939C2211yescharacteristic
20C2×#14578937C2212yescomplemented
21#19097768C2211yescharacteristic
22#19097756—22no—
23#19097747C2212yescomplemented
24#19097746—21no—
25#19097746C2212yes—
26#19097745—21no—
27#19097741—22no—
28C4×D628C2211yescharacteristic
29C4×D628C2211yescharacteristic, complemented
30C4⋊C4×D157—21no—
31#19097704C2211yescharacteristic, complemented
32#19097693C2212yescomplemented
33#19097692C2211yescharacteristic
34#19097691C2211yescharacteristic, complemented
35#19097683C2212yescomplemented
36C4×#11929434C2211yescharacteristic, complemented
37C2×SD16×C157C2212yes—
38C4⋊Q8×C157C2211yescharacteristic, complemented
39C4×D4×C157C2211yescharacteristic
40C4⋊C8×C157C2211yescharacteristic, complemented
41#103×C157C2212yescomplemented
42C23×D157—22no—
43#14578955—24no—
44#14578955—42no—
45C2×D628—22no—
46C2×D628—22no—
47C2×D628—21no—
48C2×D628—21no—
49C2×D628—44no—
50C2×D628D411yescharacteristic
51C2×D628C2312yes—
52C2×D628C2311yescharacteristic
53C2×C22⋊D157—22no—
54D4×D157—24no—
55D4×D157—42no—
56C2×C4×D157—22no—
57C2×C4×D157—21no—
58C2×C4×D157C2312yes—
59#14578941—24no—
60#14578941—42no—
61D1256—24no—
62#14578939D414yescomplemented
63#14578937—24no—
64C2×#11929434C2311yescharacteristic
65C2×#11929434C2311yescharacteristic
66C4⋊Dic157D411yescharacteristic
67C4⋊Dic157C2311yescharacteristic
68#14578933—22no—
69#14578931—22no—
70#14578931—22no—
71#14578931—21no—
72#14578931D412yes—
73C2×Q8×C157C2312yes—
74C2×D4×C157C2311yescharacteristic
75C22×C4×C157—21no—
76SD16×C157—24no—
77C2×C8×C157C2312yes—
78C4⋊C4×C157—21no—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×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—44no—
95D628—42no—
96D628—81no—
97D628C2×D412yescomplemented
98C22⋊D157—44no—
99C2×Dic157—21no—
100C2×Dic157C2×D412yes—
101C4×D157—22no—
102C4×D157—22no—
103C4×D157—44no—
104C4×D157—42no—
105#11929434—21no—
106#11929434C2×D412yescomplemented
107C23×C157—21no—
108Q8×C157—22no—
109Q8×C157—22no—
110D4×C157—21no—
111D4×C157C2×D412yes—
112C2×C4×C157—22no—
113C2×C4×C157—21no—
114C2×C4×C157—42no—
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2411yescharacteristic
118C8×C157—22no—
119C22×C157—22no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157C22×D411yescharacteristic
123C2×D157—22no—
124C2×D157—22no—
125C2×D157—22no—
126C2×D157—22no—
127C2×D157—41no—
128C2×D157—41no—
129C2×D157—42no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—88no—
133C4×C157—21no—
134C4×C157—42no—
135C4×C157—41no—
136C4×C157AGL(1,ℤ/8)12yescomplemented
137C4×C157C22×D411yescharacteristic
138C4×C157H(5,𝔽2)11yescharacteristic
139Dic157—22no—
140Dic157—41no—
141C2×C157—42no—
142C2×C157D4211yescharacteristic
143C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C157#57711yescharacteristic
145D157—42no—
146D157—42no—
147D157—82no—
148C157#327211yescharacteristic, complemented, Sylow
149#3272—1571nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1572no—
151#547—1571no—
152D42—1571no—
153#492—1571no—
154#468—1572no—
155#461D15711yescharacteristic, complemented
156#459—1571no—
157#450—1572no—
158#449—1572no—
159C4×D8—1571no—
160#436—1571no—
161C2×#41—1572no—
162C22×D4—3141no—
163C22×D4—1572no—
164AGL(1,ℤ/8)—3148no—
165C2×SD16—1572no—
166C2×SD16—1572no—
167C2×SD16C2×D15712yes—
168C2×D8—1571no—
169C2×D8—1572no—
170C2×M16—1572no—
171C4⋊Q8C2×D15711yescharacteristic, complemented
172#127—1571no—
173#123—3142no—
174C22⋊Q8—1572no—
175#121—3141no—
176#121—3141no—
177#121—1572no—
178#120—3142no—
179C4×D4—1571no—
180C4×D4—1571no—
181C4×D4C2×D15711yescharacteristic
182C2×C4⋊C4—3141no—
183#107—1571no—
184C4⋊C8C2×D15711yescharacteristic, complemented
185#103C2×D15712yescomplemented
186#102—1571no—
187#102—1572no—
188#102—1571no—
189C22⋊C8—1572no—
190C4×C8—1571no—
191C24—3142no—
192#41—3144no—
193#41—6282no—
194C2×Q8C22×D15712yes—
195C2×D4—3142no—
196C2×D4—3142no—
197C2×D4—3144no—
198C2×D4—3142no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—6284no—
202C2×D4—6282no—
203C2×D4—1571no—
204C2×D4—1572no—
205C2×D4—1571no—
206C2×D4C22×D15711yescharacteristic
207C22×C4—21no—
208C22×C4—3142no—
209C22×C4—3141no—
210C22×C4—1572no—
211SD16—24no—
212SD16—3144no—
213SD16—6282no—
214D8—3144no—
215D8—1574no—
216M16—3144no—
217C2×C8—1571no—
218C2×C8—1571no—
219C2×C8C22×D15712yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3142no—
223C4⋊C4—1571no—
224C4⋊C4—1571no—
225C4⋊C4C22×D15711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3142no—
228C22⋊C4—3142no—
229C22⋊C4—3141no—
230C22⋊C4—1572no—
231C42C22×D15711yescharacteristic
232C23—21no—
233C23—3141no—
234C23—3142no—
235C23—6284no—
236C23—6284no—
237C23—1572no—
238C23—12562no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—3142no—
243D4—3142no—
244D4—3141no—
245D4—3142no—
246D4—6284no—
247D4—6284no—
248D4—6284no—
249D4—6282no—
250D4—1572no—
251D4—12561no—
252D4D4×D15712yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—3142no—
257C2×C4—3142no—
258C2×C4—3141no—
259C2×C4—6284no—
260C2×C4—6282no—
261C2×C4—1572no—
262C2×C4D4×D15711yescharacteristic
263C2×C4D4×D15711yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—22no—
266C8—3141no—
267C8—1572no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—3142no—
272C22—3142no—
273C22—3142no—
274C22—3142no—
275C22—6281no—
276C22—6281no—
277C22—6282no—
278C22—6282no—
279C22—6282no—
280C22—12568no—
281C22C2×D4×D15711yescharacteristic
282C4—21no—
283C4—42no—
284C4—41no—
285C4—3142no—
286C4—6281no—
287C4#1909779812yescomplemented
288C4C2×D4×D15711yescharacteristic
289C4#1909781711yescharacteristic
290C2—42no—
291C2—6282no—
292C2—6282no—
293C2—12562no—
294C2#2375790711yescharacteristic
295C2C2×#1909779811yescharacteristic
296C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#461×C157C2
socleC22×C157C22×D4
hypercenter#461D157
hypocenterC157#3272
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. #23757926C2semidirect
3. D4×D628C2semidirect
4. #23757647C2semidirect
5. #23757522C2semidirect
6. #23757469C2semidirect
7. #23757422C2semidirect
8. #23757421C2semidirect
9. C4×#14578939C2semidirect
10. #23757350C2semidirect
11. #461×C157C2semidirect
12. C2×#14578955C22semidirect
16. C2×#14578941C22semidirect
17. C2×#14578941C22semidirect
18. C2×D1256C22semidirect
20. C2×#14578937C22semidirect
23. #19097747C22semidirect
29. C4×D628C22semidirect
31. #19097704C22semidirect
32. #19097693C22semidirect
34. #19097691C22semidirect
35. #19097683C22semidirect
36. C4×#11929434C22semidirect
38. C4⋊Q8×C157C22semidirect
40. C4⋊C8×C157C22semidirect
41. #103×C157C22semidirect
62. #14578939D4semidirect
97. D628C2×D4semidirect
106. #11929434C2×D4semidirect
136. C4×C157AGL(1,ℤ/8)semidirect
148. C157#3272semidirect
155. #461D157semidirect
171. C4⋊Q8C2×D157semidirect
184. C4⋊C8C2×D157semidirect
185. #103C2×D157semidirect
287. C4#19097798semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
#237579262
D4×D6282
#237576472
#237575222
#237574692
#237574222
#237574212
C4×#145789392
#237573502
#461×C1572
#3272157