Finite Group Databasebeta

← #34111737

Subgroups — #34111737

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×#19097795C212yesmaximal, complemented
2#23757927C211yesmaximal, characteristic, complemented
3D4×D628C211yesmaximal, characteristic, complemented
4#23757646C211yesmaximal, characteristic, complemented
5#23757520C212yesmaximal, complemented
6#23757478C211yesmaximal, characteristic, complemented
7#23757468C211yesmaximal, characteristic, complemented
8#23757425C212yesmaximal, complemented
9#23757419C212yesmaximal, complemented
10#23757352C211yesmaximal, characteristic, complemented
11C4×D8×C157C211yesmaximal, characteristic, complemented
12C2×#14578953C2212yescomplemented
13C22×D628—21no—
14C2×D4×D157C2212yes—
15#19097795—28no—
16C2×#14578943C2212yescomplemented
17C2×#14578943C2212yescomplemented
18C2×#14578942C2212yes—
19C2×#14578939C2212yescomplemented
20C2×#14578937C2212yescomplemented
21#19097771C2211yescharacteristic, complemented
22#19097768C2211yescharacteristic
23#19097755—22no—
24#19097751C2212yescomplemented
25#19097746C2212yes—
26#19097745—21no—
27#19097745—21no—
28#19097741—22no—
29C4×D628C2211yescharacteristic
30C2×C4⋊Dic157—21no—
31C4⋊#11929434C2211yescharacteristic, complemented
32#19097694C2212yescomplemented
33#19097691C2212yescomplemented
34#19097683C2212yescomplemented
35C2×D8×C157C2211yescharacteristic
36C4×D4×C157C2211yescharacteristic
37C4×D4×C157C2211yescharacteristic, complemented
38#107×C157C2211yescharacteristic, complemented
39#102×C157C2211yescharacteristic
40#102×C157C2211yescharacteristic, complemented
41C4×C8×C157C2211yescharacteristic, complemented
42C23×D157—22no—
43#14578953—24no—
44#14578953—42no—
45C2×Dic314C2312yes—
46C2×D628—22no—
47C2×D628—21no—
48C2×D628—21no—
49C2×D628—44no—
50C2×D628D411yescharacteristic
51C2×D628C2312yes—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54D4×D157—24no—
55D4×D157—42no—
56C22×Dic157—22no—
57C2×C4×D157C2312yes—
58#14578943—24no—
59#14578943—42no—
60#14578942—24no—
61#14578939—24no—
62#14578937—24no—
63C2×#11929434C2312yes—
64C4⋊Dic157—22no—
65C4⋊Dic157—21no—
66C4⋊Dic157—21no—
67C4⋊Dic157D411yescharacteristic
68#14578931—22no—
69#14578931—22no—
70#14578931D412yes—
71C2×D4×C157C2311yescharacteristic
72C2×D4×C157C2311yescharacteristic
73C22×C4×C157—21no—
74C22×C4×C157—21no—
75D8×C157D414yescomplemented
76C2×C8×C157C2311yescharacteristic
77C2×C8×C157C2311yescharacteristic
78C4⋊C4×C157C2311yescharacteristic
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C22⋊C4×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157—22no—
84C22×D157—44no—
85C22×D157—44no—
86C22×D157—82no—
87C22×D157C2×D412yes—
88Dic314—22no—
89Dic314—22no—
90D628—22no—
91D628—22no—
92D628—22no—
93D628—44no—
94D628—42no—
95D628—81no—
96C22⋊D157—44no—
97C22⋊D157—44no—
98C2×Dic157—22no—
99C2×Dic157—44no—
100C2×Dic157C2×D412yes—
101C4×D157—22no—
102C4×D157—22no—
103#11929434—22no—
104C23×C157—21no—
105C23×C157—21no—
106D4×C157—21no—
107D4×C157—21no—
108D4×C157C2×D412yes—
109D4×C157C2×D412yescomplemented
110C2×C4×C157—21no—
111C2×C4×C157—21no—
112C2×C4×C157—42no—
113C2×C4×C157—42no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2411yescharacteristic
117C8×C157—21no—
118C8×C157C2×D412yescomplemented
119C22×C157—22no—
120C22×C157—22no—
121C22×C157—41no—
122C22×C157—41no—
123C22×C157—41no—
124C22×C157—41no—
125C22×C157C22×D411yescharacteristic
126C2×D157—22no—
127C2×D157—22no—
128C2×D157—22no—
129C2×D157—42no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—88no—
133C4×C157—21no—
134C4×C157—41no—
135C4×C157—41no—
136C4×C157AGL(1,ℤ/8)12yescomplemented
137C4×C157C22×D411yescharacteristic
138C4×C157H(5,𝔽2)11yescharacteristic
139Dic157—22no—
140Dic157—42no—
141C2×C157—42no—
142C2×C157—42no—
143C2×C157D4211yescharacteristic
144C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C157#57711yescharacteristic
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×D8D15711yescharacteristic, complemented
160#436—1571no—
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×D8C2×D15711yescharacteristic
170C2×M16—1572no—
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×D4C2×D15711yescharacteristic
181C4×D4C2×D15711yescharacteristic, complemented
182C2×C4⋊C4—3141no—
183#107C2×D15711yescharacteristic, complemented
184C4⋊C8—1571no—
185#103—1572no—
186#102—1572no—
187#102C2×D15711yescharacteristic
188#102C2×D15711yescharacteristic, complemented
189C22⋊C8—1572no—
190C4×C8C2×D15711yescharacteristic, complemented
191C24—3142no—
192#41—3144no—
193#41—6282no—
194C2×Q8—1572no—
195C2×D4—3142no—
196C2×D4—3144no—
197C2×D4—3142no—
198C2×D4—3142no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—6284no—
202C2×D4—6282no—
203C2×D4—1571no—
204C2×D4—1572no—
205C2×D4C22×D15711yescharacteristic
206C2×D4C22×D15711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3142no—
210C22×C4—1572no—
211SD16—3144no—
212SD16—3144no—
213SD16—6282no—
214D8—3144no—
215D8D62814yescomplemented
216M16—3144no—
217C2×C8—1572no—
218C2×C8C22×D15711yescharacteristic
219C2×C8C22×D15711yescharacteristic
220C4⋊C4—3142no—
221C4⋊C4—3141no—
222C4⋊C4—3141no—
223C4⋊C4—1571no—
224C4⋊C4C22×D15711yescharacteristic
225C4⋊C4C22×D15711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—3142no—
229C22⋊C4—3142no—
230C22⋊C4—1572no—
231C42C22×D15711yescharacteristic
232C23—21no—
233C23—21no—
234C23—3142no—
235C23—6284no—
236C23—6284no—
237C23—1572no—
238C23—12562no—
239Q8—3142no—
240Q8—3142no—
241D4—21no—
242D4—21no—
243D4—3142no—
244D4—3142no—
245D4—3142no—
246D4—6284no—
247D4—6284no—
248D4—6284no—
249D4—6282no—
250D4—12561no—
251D4C2×D62812yes—
252D4C2×D62812yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3142no—
258C2×C4—3142no—
259C2×C4—3142no—
260C2×C4—6284no—
261C2×C4—1572no—
262C2×C4D4×D15711yescharacteristic
263C2×C4D4×D15711yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—21no—
266C8—3142no—
267C8C2×D62812yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3142no—
275C22—3142no—
276C22—3142no—
277C22—6282no—
278C22—6282no—
279C22—6282no—
280C22—12568no—
281C22C2×D4×D15711yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—3142no—
286C4—6282no—
287C4#1909779512yescomplemented
288C4C22×D62811yescharacteristic
289C4#1909781711yescharacteristic
290C2—42no—
291C2—42no—
292C2—6282no—
293C2—12562no—
294C2D4×D62811yescharacteristic
295C2C2×#1909779511yescharacteristic
296C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC4×D8×C157C2
socleC22×C157C22×D4
hypercenterC4×D8D157
hypocenterC157#3272
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. #23757927C2semidirect
3. D4×D628C2semidirect
4. #23757646C2semidirect
5. #23757520C2semidirect
6. #23757478C2semidirect
7. #23757468C2semidirect
8. #23757425C2semidirect
9. #23757419C2semidirect
10. #23757352C2semidirect
11. C4×D8×C157C2semidirect
12. C2×#14578953C22semidirect
16. C2×#14578943C22semidirect
17. C2×#14578943C22semidirect
19. C2×#14578939C22semidirect
20. C2×#14578937C22semidirect
21. #19097771C22semidirect
24. #19097751C22semidirect
31. C4⋊#11929434C22semidirect
32. #19097694C22semidirect
33. #19097691C22semidirect
34. #19097683C22semidirect
37. C4×D4×C157C22semidirect
38. #107×C157C22semidirect
40. #102×C157C22semidirect
41. C4×C8×C157C22semidirect
75. D8×C157D4semidirect
109. D4×C157C2×D4semidirect
118. C8×C157C2×D4semidirect
136. C4×C157AGL(1,ℤ/8)semidirect
148. C157#3272semidirect
159. C4×D8D157semidirect
181. C4×D4C2×D157semidirect
183. #107C2×D157semidirect
188. #102C2×D157semidirect
190. C4×C8C2×D157semidirect
215. D8D628semidirect
252. D4C2×D628semidirect
267. C8C2×D628semidirect
287. C4#19097795semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
#237579272
D4×D6282
#237576462
#237575202
#237574782
#237574682
#237574252
#237574192
#237573522
C4×D8×C1572
#3272157