Finite Group Databasebeta

← #34111738

Subgroups — #34111738

Order 20096.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19097799C212yesmaximal, complemented
2#23757928C211yesmaximal, characteristic, complemented
3D4×D628C211yesmaximal, characteristic, complemented
4#23757646C211yesmaximal, characteristic, complemented
5#23757519C212yesmaximal, complemented
6#23757479C211yesmaximal, characteristic, complemented
7#23757469C211yesmaximal, characteristic, complemented
8#23757426C212yesmaximal, complemented
9#23757420C212yesmaximal, complemented
10C4×D1256C211yesmaximal, characteristic, complemented
11#436×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2212yescomplemented
13C22×D628C2212yes—
14C2×D4×D157—21no—
15#19097799—28no—
16C2×#14578943C2212yes—
17C2×#14578943C2212yescomplemented
18C2×#14578943C2212yescomplemented
19C2×D1256C2211yescharacteristic
20C2×D1256C2212yescomplemented
21#19097771C2211yescharacteristic, complemented
22#19097768C2211yescharacteristic
23#19097753—22no—
24#19097750C2212yescomplemented
25#19097746—21no—
26#19097745C2212yes—
27#19097742—21no—
28#19097741—22no—
29C4×D628C2211yescharacteristic
30C4×D628C2211yescharacteristic, complemented
31C4⋊C4×D157—21no—
32#19097705C2211yescharacteristic, complemented
33#19097696C2212yescomplemented
34#19097693C2211yescharacteristic
35#19097693C2211yescharacteristic, complemented
36#19097693C2212yescomplemented
37C2×M16×C157C2212yescomplemented
38C4×D4×C157C2211yescharacteristic
39C4⋊C8×C157C2211yescharacteristic, complemented
40C22⋊C8×C157C2212yescomplemented
41C4×C8×C157C2211yescharacteristic, complemented
42C23×D157—22no—
43#14578954—24no—
44#14578954—42no—
45C2×Dic314C2312yes—
46C2×D628—24no—
47C2×D628—22no—
48C2×D628—21no—
49C2×D628—21no—
50C2×D628—42no—
51C2×D628C2312yes—
52C2×D628C2311yescharacteristic
53C2×D628C2311yescharacteristic
54C2×C22⋊D157—22no—
55C2×C22⋊D157—22no—
56D4×D157—44no—
57C2×C4×D157—21no—
58C2×C4×D157—22no—
59C2×C4×D157—21no—
60#14578943—24no—
61#14578943—24no—
62#14578943—42no—
63D1256—24no—
64D1256D414yescomplemented
65C4⋊Dic157—21no—
66C4⋊Dic157—21no—
67C4⋊Dic157C2311yescharacteristic
68C4⋊Dic157C2311yescharacteristic
69#14578933—22no—
70C22⋊Dic157—22no—
71#14578931—21no—
72#14578931—21no—
73#14578931—22no—
74C2×D4×C157D411yescharacteristic
75C22×C4×C157C2312yes—
76M16×C157—24no—
77C2×C8×C157C2311yescharacteristic
78C2×C8×C157C2312yes—
79C2×C8×C157C2311yescharacteristic
80C4⋊C4×C157D411yescharacteristic
81C22⋊C4×C157D412yes—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—21no—
85C22×D157—22no—
86C22×D157—44no—
87C22×D157—44no—
88C22×D157—82no—
89Dic314—22no—
90Dic314—22no—
91D628—22no—
92D628—21no—
93D628—21no—
94D628—22no—
95D628—44no—
96D628—42no—
97D628—81no—
98D628C2×D412yes—
99D628C2×D412yescomplemented
100C22⋊D157—44no—
101C22⋊D157—44no—
102C2×Dic157—21no—
103C2×Dic157—22no—
104C2×Dic157—21no—
105C4×D157—44no—
106C4×D157—42no—
107C4×D157—42no—
108C23×C157C2×D412yes—
109D4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157—22no—
112C2×C4×C157C2×D412yes—
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2411yescharacteristic
116C8×C157—22no—
117C8×C157—21no—
118C8×C157C2×D412yescomplemented
119C22×C157—22no—
120C22×C157—22no—
121C22×C157—22no—
122C22×C157C22×D411yescharacteristic
123C2×D157—22no—
124C2×D157—22no—
125C2×D157—41no—
126C2×D157—41no—
127C2×D157—41no—
128C2×D157—41no—
129C2×D157—42no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—88no—
133C4×C157—22no—
134C4×C157—21no—
135C4×C157AGL(1,ℤ/8)12yescomplemented
136C4×C157C22×D411yescharacteristic
137C4×C157H(5,𝔽2)11yescharacteristic
138Dic157—42no—
139Dic157—41no—
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#461—1571no—
156#459—1571no—
157#450—1572no—
158#449—1572no—
159C4×D8—1571no—
160#436D15711yescharacteristic, complemented
161C2×#41—1572no—
162C22×D4—3141no—
163C22×D4—1572no—
164AGL(1,ℤ/8)—3148no—
165C2×SD16—1572no—
166C2×SD16—1572no—
167C2×SD16—1572no—
168C2×D8—1572no—
169C2×D8—1571no—
170C2×M16C2×D15712yescomplemented
171C4⋊Q8—1571no—
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#103—1572no—
186#102—1571no—
187#102—1571no—
188#102—1572no—
189C22⋊C8C2×D15712yescomplemented
190C4×C8C2×D15711yescharacteristic, complemented
191C24—3142no—
192#41—3144no—
193#41—6282no—
194C2×Q8—1572no—
195C2×D4—3144no—
196C2×D4—3142no—
197C2×D4—3142no—
198C2×D4—3142no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—6282no—
202C2×D4—6284no—
203C2×D4—1572no—
204C2×D4—1571no—
205C2×D4—1571no—
206C2×D4D62811yescharacteristic
207C22×C4—3141no—
208C22×C4—3142no—
209C22×C4—3141no—
210C22×C4C22×D15712yes—
211SD16—3144no—
212SD16—3144no—
213SD16—6282no—
214D8—3144no—
215D8—1574no—
216M16—24no—
217C2×C8C22×D15711yescharacteristic
218C2×C8C22×D15712yes—
219C2×C8C22×D15711yescharacteristic
220C4⋊C4—3141no—
221C4⋊C4—3142no—
222C4⋊C4—3141no—
223C4⋊C4—1571no—
224C4⋊C4—1571no—
225C4⋊C4D62811yescharacteristic
226C22⋊C4—3142no—
227C22⋊C4—3141no—
228C22⋊C4—3141no—
229C22⋊C4—3142no—
230C22⋊C4D62812yes—
231C42C22×D15711yescharacteristic
232C23—3141no—
233C23—3141no—
234C23—3142no—
235C23—6284no—
236C23—6284no—
237C23—12562no—
238C23C2×D62812yes—
239Q8—3142no—
240Q8—3142no—
241D4—22no—
242D4—3142no—
243D4—3141no—
244D4—3141no—
245D4—3142no—
246D4—6284no—
247D4—6284no—
248D4—6284no—
249D4—6282no—
250D4—1572no—
251D4—1572no—
252D4—12561no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3141no—
256C2×C4—3141no—
257C2×C4—3142no—
258C2×C4—6284no—
259C2×C4—6282no—
260C2×C4—6282no—
261C2×C4C2×D62811yescharacteristic
262C2×C4C2×D62812yes—
263C2×C4C2×D62811yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D15712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—3142no—
272C22—3142no—
273C22—6281no—
274C22—6281no—
275C22—6281no—
276C22—6281no—
277C22—6282no—
278C22—6282no—
279C22—6282no—
280C22—12568no—
281C22C22×D62811yescharacteristic
282C4—21no—
283C4—22no—
284C4—6282no—
285C4—6281no—
286C4—6281no—
287C4#1909779912yescomplemented
288C4C2×D4×D15711yescharacteristic
289C4#1909781711yescharacteristic
290C2—42no—
291C2—6282no—
292C2—6282no—
293C2—12562no—
294C2D4×D62811yescharacteristic
295C2C2×#1909779911yescharacteristic
296C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#436×C157C2
socleC22×C157C22×D4
hypercenter#436D157
hypocenterC157#3272
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. #23757928C2semidirect
3. D4×D628C2semidirect
4. #23757646C2semidirect
5. #23757519C2semidirect
6. #23757479C2semidirect
7. #23757469C2semidirect
8. #23757426C2semidirect
9. #23757420C2semidirect
10. C4×D1256C2semidirect
11. #436×C157C2semidirect
12. C2×#14578954C22semidirect
17. C2×#14578943C22semidirect
18. C2×#14578943C22semidirect
20. C2×D1256C22semidirect
21. #19097771C22semidirect
24. #19097750C22semidirect
30. C4×D628C22semidirect
32. #19097705C22semidirect
33. #19097696C22semidirect
35. #19097693C22semidirect
36. #19097693C22semidirect
37. C2×M16×C157C22semidirect
39. C4⋊C8×C157C22semidirect
40. C22⋊C8×C157C22semidirect
41. C4×C8×C157C22semidirect
64. D1256D4semidirect
99. D628C2×D4semidirect
118. C8×C157C2×D4semidirect
135. C4×C157AGL(1,ℤ/8)semidirect
148. C157#3272semidirect
160. #436D157semidirect
170. C2×M16C2×D157semidirect
184. C4⋊C8C2×D157semidirect
189. C22⋊C8C2×D157semidirect
190. C4×C8C2×D157semidirect
267. C8D4×D157semidirect
287. C4#19097799semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
#237579282
D4×D6282
#237576462
#237575192
#237574792
#237574692
#237574262
#237574202
C4×D12562
#436×C1572
#3272157