Finite Group Databasebeta

← #34110234

Subgroups — #34110234

Order 20096.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#23757828C211yesmaximal, characteristic, complemented
2C4×D4×D157C211yesmaximal, characteristic
3#23757607C212yesmaximal, complemented
4#23757554C211yesmaximal, characteristic, complemented
5#23757553C211yesmaximal, characteristic, complemented
6#23757475C211yesmaximal, characteristic, complemented
7#23757422C212yesmaximal
8#23757376C212yesmaximal, complemented
9#23757374C212yesmaximal, complemented
10#23757329C211yesmaximal, characteristic, complemented
11#461×C157C211yesmaximal, characteristic, complemented
12C2×D4×D157C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578941C2212yes—
15#19097771C2211yescharacteristic, complemented
16#19097762—21no—
17#19097751—21no—
18#19097747C2212yes—
19Q8×Dic157C2212yescomplemented
20#19097736—21no—
21C4×D628C2211yescharacteristic
22D4×Dic157C2211yescharacteristic
23#19097733C2211yescharacteristic
24C4×C22⋊D157—21no—
25#19097731—21no—
26#19097729—21no—
27C4⋊C4×D157C2211yescharacteristic
28C22⋊C4×D157—21no—
29C42×D157C2211yescharacteristic
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32C4⋊#11929434C2211yescharacteristic, complemented
33#19097695C2212yescomplemented
34#19097693C2212yes—
35#19097692C2212yes—
36#19097683C2212yes—
37#19097681C2212yescomplemented
38C2×SD16×C157C2212yes—
39C4⋊Q8×C157C2211yescharacteristic, complemented
40C4×D4×C157C2211yescharacteristic
41C4⋊C8×C157C2211yescharacteristic, complemented
42#103×C157C2212yescomplemented
43C23×D157—21no—
44C2×Dic314—21no—
45C2×D628C2311yescharacteristic
46C2×C22⋊D157—21no—
47D4×D157—22no—
48D4×D157—22no—
49C22×Dic157—21no—
50C2×C4×D157—21no—
51C2×C4×D157—21no—
52C2×C4×D157—21no—
53C2×C4×D157—44no—
54C2×C4×D157D411yescharacteristic
55C2×C4×D157D411yescharacteristic
56C2×C4×D157C2311yescharacteristic
57#14578941—24no—
58C2×#11929434C2312yes—
59C4⋊Dic157—22no—
60C4⋊Dic157—21no—
61C4⋊Dic157C2312yes—
62C4⋊Dic157C2311yescharacteristic
63#14578933—21no—
64#14578933—22no—
65C22⋊Dic157—21no—
66#14578931—22no—
67#14578931—21no—
68#14578931—21no—
69C4×Dic157—22no—
70C4×Dic157D411yescharacteristic
71C4×Dic157D411yescharacteristic
72C4×Dic157C2311yescharacteristic
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—24no—
84C22×D157—21no—
85C22×D157—42no—
86C22×D157—42no—
87C22×D157C2×D411yescharacteristic
88Dic314—42no—
89D628—21no—
90D628#4112yes—
91C22⋊D157—24no—
92C2×Dic157—21no—
93C2×Dic157—22no—
94C2×Dic157—42no—
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C4×D157—22no—
99C4×D157—22no—
100C4×D157—21no—
101C4×D157—21no—
102C4×D157—42no—
103C4×D157—42no—
104C4×D157—42no—
105#11929434—22no—
106C23×C157—21no—
107Q8×C157—22no—
108Q8×C157—22no—
109D4×C157—21no—
110D4×C157#4112yes—
111C2×C4×C157—21no—
112C2×C4×C157—22no—
113C2×C4×C157—42no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2411yescharacteristic
117C8×C157—22no—
118C22×C157—22no—
119C22×C157—41no—
120C22×C157—41no—
121C22×C157C22×D411yescharacteristic
122C2×D157—21no—
123C2×D157—21no—
124C2×D157—22no—
125C2×D157—48no—
126C2×D157—41no—
127C2×D157—41no—
128C2×D157#13712yes—
129C4×C157—21no—
130C4×C157—42no—
131C4×C157—41no—
132C4×C157AGL(1,ℤ/8)12yes—
133C4×C157C2×#4111yescharacteristic
134C4×C157C2×#4111yescharacteristic
135Dic157—21no—
136Dic157—21no—
137Dic157—21no—
138Dic157—41no—
139Dic157—42no—
140C2×C157—42no—
141C2×C157#52511yescharacteristic
142C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C157C2×#13711yescharacteristic
144D157—22no—
145D157—42no—
146C157#311011yescharacteristic, complemented, Sylow
147#3110—1571nomaximal, Sylow
148#533—1571no—
149C2×C4×D4—1571no—
150#484—1572no—
151#474—1571no—
152#474—1571no—
153#461—1571no—
154#461D15711yescharacteristic, complemented
155#450—1572no—
156#440—1572no—
157#440—1572no—
158#432—1571no—
159C22×D4—1571no—
160C23×C4—3141no—
161C2×SD16—1572no—
162C2×SD16C2×D15712yes—
163C4⋊Q8—1571no—
164C4⋊Q8C2×D15711yescharacteristic, complemented
165#125—3141no—
166C22⋊Q8—3141no—
167C22⋊Q8—1572no—
168C4×Q8—3141no—
169C4×Q8—1572no—
170C4×D4—3141no—
171C4×D4—3141no—
172C4×D4—1571no—
173C4×D4—1571no—
174C4×D4—1571no—
175C4×D4C2×D15711yescharacteristic
176#117—3141no—
177C2×C4⋊C4—1571no—
178C2×C22⋊C4—3141no—
179C2×C42—1571no—
180#107—1572no—
181#107—1572no—
182C4⋊C8—1571no—
183C4⋊C8C2×D15711yescharacteristic, complemented
184#103—1572no—
185#103C2×D15712yescomplemented
186#102—1572no—
187#102—1572no—
188C22⋊C8—1572no—
189#97—1572no—
190C24—3141no—
191C2×Q8—3141no—
192C2×Q8C22×D15712yes—
193C2×D4—3142no—
194C2×D4—3142no—
195C2×D4—3141no—
196C2×D4—1571no—
197C2×D4C22×D15711yescharacteristic
198C22×C4—21no—
199C22×C4—3141no—
200C22×C4—3141no—
201C22×C4—3141no—
202C22×C4—3141no—
203C22×C4—6284no—
204C22×C4—1571no—
205C22×C4—1571no—
206C22×C4—1571no—
207SD16—24no—
208SD16—3144no—
209C2×C8—1572no—
210C2×C8C22×D15712yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—3141no—
214C4⋊C4—3142no—
215C4⋊C4—3142no—
216C4⋊C4—3141no—
217C4⋊C4—1572no—
218C4⋊C4—1571no—
219C4⋊C4C22×D15711yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—3142no—
222C22⋊C4—3141no—
223C22⋊C4—3141no—
224C22⋊C4—3141no—
225C42—3142no—
226C42—1571no—
227C42—1571no—
228C42—1571no—
229C42C22×D15711yescharacteristic
230C23—21no—
231C23—3144no—
232C23—3141no—
233C23—6282no—
234C23—6282no—
235C23—1571no—
236Q8—22no—
237Q8—22no—
238Q8—6282no—
239D4—21no—
240D4—3144no—
241D4—3141no—
242D4—1572no—
243D4#1457895312yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—3142no—
248C2×C4—3142no—
249C2×C4—3141no—
250C2×C4—3141no—
251C2×C4—3142no—
252C2×C4—3141no—
253C2×C4—6282no—
254C2×C4—6282no—
255C2×C4—6282no—
256C2×C4—6282no—
257C2×C4—1571no—
258C2×C4—1571no—
259C2×C4—1571no—
260C2×C4D4×D15711yescharacteristic
261C2×C4D4×D15711yescharacteristic
262C2×C4C23×D15711yescharacteristic
263C8—22no—
264C8—3142no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—3141no—
269C22—3141no—
270C22—3142no—
271C22—6288no—
272C22—6281no—
273C22—6281no—
274C22—1572no—
275C22C2×D4×D15711yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—3141no—
280C4—3141no—
281C4—3141no—
282C4—6281no—
283C4—6282no—
284C4#1909779812yes—
285C4#41×D15711yescharacteristic
286C4C2×#1457895311yescharacteristic
287C2—42no—
288C2—3142no—
289C2—6282no—
290C2#2375778311yescharacteristic
291C2C2×#1909779811yescharacteristic
292C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#461×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#461D157
hypocenterC157#3110
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. #23757828C2semidirect
3. #23757607C2semidirect
4. #23757554C2semidirect
5. #23757553C2semidirect
6. #23757475C2semidirect
8. #23757376C2semidirect
9. #23757374C2semidirect
10. #23757329C2semidirect
11. #461×C157C2semidirect
15. #19097771C22semidirect
19. Q8×Dic157C22semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. C4⋊#11929434C22semidirect
33. #19097695C22semidirect
37. #19097681C22semidirect
39. C4⋊Q8×C157C22semidirect
41. C4⋊C8×C157C22semidirect
42. #103×C157C22semidirect
146. C157#3110semidirect
154. #461D157semidirect
164. C4⋊Q8C2×D157semidirect
183. C4⋊C8C2×D157semidirect
185. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578282
C4×D4×D1572
#237576072
#237575542
#237575532
#237574752
#237574222
#237573762
#237573742
#237573292
#461×C1572
#3110157