Finite Group Databasebeta

← #34109934

Subgroups — #34109934

Order 20096.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#23757572C212yesmaximal, complemented
4#23757571C212yesmaximal, complemented
5#23757560C211yesmaximal, characteristic, complemented
6#23757554C211yesmaximal, characteristic, complemented
7#23757553C211yesmaximal, characteristic, complemented
8#23757312C212yesmaximal, complemented
9#23757295C211yesmaximal, characteristic, complemented
10#23757276C212yesmaximal
11#476×C157C211yesmaximal, characteristic, complemented
12C2×D4×D157C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578937C2212yes—
15#19097771C2211yescharacteristic, complemented
16#19097762C2212yescomplemented
17#19097751—21no—
18#19097747—21no—
19Q8×Dic157—21no—
20#19097736—21no—
21C4×D628C2211yescharacteristic
22D4×Dic157C2211yescharacteristic
23#19097733C2211yescharacteristic
24C4×C22⋊D157—21no—
25#19097731—21no—
26#19097729C2212yes—
27C4⋊C4×D157C2211yescharacteristic
28C22⋊C4×D157—21no—
29C42×D157C2211yescharacteristic
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097702C2212yescomplemented
33C4⋊#11929434C2211yescharacteristic, complemented
34#19097700C2212yescomplemented
35#19097693C2212yes—
36#19097692C2212yes—
37#19097691C2212yes—
38C4⋊Q8×C157C2211yescharacteristic, complemented
39C4×D4×C157C2211yescharacteristic
40#106×C157C2212yescomplemented
41C4⋊C8×C157C2211yescharacteristic, complemented
42#102×C157C2212yes—
43C23×D157—21no—
44C2×Dic314—21no—
45C2×D628C2311yescharacteristic
46C2×C22⋊D157—21no—
47D4×D157—22no—
48D4×D157Q814yescomplemented
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#14578937—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×C157—21no—
74C2×D4×C157C2311yescharacteristic
75C22×C4×C157—21no—
76C2×C8×C157C2312yes—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C42×C157C2311yescharacteristic
82C22×D157—21no—
83C22×D157—24no—
84C22×D157—42no—
85C22×D157—42no—
86C22×D157C2×D411yescharacteristic
87Dic314—42no—
88D628—21no—
89D628C2×Q812yes—
90C22⋊D157—24no—
91C2×Dic157—21no—
92C2×Dic157—22no—
93C2×Dic157—42no—
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C4×D157—22no—
98C4×D157—41no—
99C4×D157—42no—
100C4×D157—42no—
101C4×D157—42no—
102C4×D157C2×Q812yes—
103C4×D157#4112yes—
104#11929434—22no—
105C23×C157—21no—
106Q8×C157—42no—
107D4×C157—21no—
108D4×C157C2×Q812yes—
109C2×C4×C157—21no—
110C2×C4×C157—22no—
111C2×C4×C157—42no—
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2411yescharacteristic
115C8×C157—22no—
116C22×C157—22no—
117C22×C157—41no—
118C22×C157—41no—
119C22×C157C22×D411yescharacteristic
120C2×D157—21no—
121C2×D157—21no—
122C2×D157—22no—
123C2×D157—48no—
124C2×D157—41no—
125C2×D157—41no—
126C2×D157C22⋊Q812yes—
127C4×C157—21no—
128C4×C157—41no—
129C4×C157—42no—
130C4×C157AGL(1,ℤ/8)12yes—
131C4×C157C22×Q811yescharacteristic
132C4×C157C2×#4111yescharacteristic
133Dic157—21no—
134Dic157—21no—
135Dic157—41no—
136Dic157—42no—
137Dic157C22⋊Q812yes—
138C2×C157—42no—
139C2×C157C2×C22⋊Q811yescharacteristic
140C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C157#57511yescharacteristic
142D157—22no—
143D157—42no—
144C157#307711yescharacteristic, complemented, Sylow
145#3077—1571nomaximal, Sylow
146#533—1571no—
147C2×C4×D4—1571no—
148#478—1572no—
149#478—1572no—
150#476—1571no—
151#476D15711yescharacteristic, complemented
152#474—1571no—
153#474—1571no—
154#428—1572no—
155#423—1571no—
156#418—1572no—
157C22×D4—1571no—
158C23×C4—3141no—
159C2×M16—1572no—
160C4⋊Q8—1571no—
161C4⋊Q8C2×D15711yescharacteristic, complemented
162#125—1572no—
163C22⋊Q8—3141no—
164C22⋊Q8—3141no—
165C4×Q8—3141no—
166C4×Q8—3141no—
167C4×D4—3141no—
168C4×D4—3141no—
169C4×D4—1571no—
170C4×D4—1571no—
171C4×D4—1571no—
172C4×D4C2×D15711yescharacteristic
173#117—1572no—
174C2×C4⋊C4—1571no—
175C2×C22⋊C4—3141no—
176C2×C42—1571no—
177#107—1572no—
178#107—1572no—
179#106—1572no—
180#106C2×D15712yescomplemented
181C4⋊C8—1572no—
182C4⋊C8—1571no—
183C4⋊C8C2×D15711yescharacteristic, complemented
184#102—1572no—
185#102—1572no—
186#102—1572no—
187#102C2×D15712yes—
188C24—3141no—
189C2×Q8—21no—
190C2×Q8—3141no—
191C2×D4—3142no—
192C2×D4—3141no—
193C2×D4—1574no—
194C2×D4—1571no—
195C2×D4C22×D15711yescharacteristic
196C22×C4—21no—
197C22×C4—3141no—
198C22×C4—3141no—
199C22×C4—3141no—
200C22×C4—3141no—
201C22×C4—6284no—
202C22×C4—1571no—
203C22×C4—1571no—
204C22×C4—1571no—
205M16—3144no—
206C2×C8—1572no—
207C2×C8C22×D15712yes—
208C4⋊C4—21no—
209C4⋊C4—3141no—
210C4⋊C4—3142no—
211C4⋊C4—3142no—
212C4⋊C4—3141no—
213C4⋊C4—1572no—
214C4⋊C4—1571no—
215C4⋊C4C22×D15712yes—
216C4⋊C4C22×D15711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—3142no—
219C22⋊C4—3141no—
220C22⋊C4—3141no—
221C22⋊C4—3141no—
222C42—3142no—
223C42—1571no—
224C42—1571no—
225C42—1571no—
226C42C22×D15711yescharacteristic
227C23—21no—
228C23—3144no—
229C23—3141no—
230C23—6282no—
231C23—6282no—
232C23—1571no—
233Q8—42no—
234Q8—6282no—
235D4—21no—
236D4—3144no—
237D4—3141no—
238D4—1572no—
239D4Q8×D15712yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—3142no—
244C2×C4—3142no—
245C2×C4—3141no—
246C2×C4—6281no—
247C2×C4—6282no—
248C2×C4—6282no—
249C2×C4—6282no—
250C2×C4—6282no—
251C2×C4—1571no—
252C2×C4—1571no—
253C2×C4—1571no—
254C2×C4—1572no—
255C2×C4—1572no—
256C2×C4D4×D15711yescharacteristic
257C2×C4D4×D15711yescharacteristic
258C2×C4C23×D15711yescharacteristic
259C8—22no—
260C8—3142no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—3141no—
265C22—3141no—
266C22—3142no—
267C22—6288no—
268C22—6281no—
269C22—6281no—
270C22—1572no—
271C22C2×D4×D15711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—3141no—
276C4—3141no—
277C4—6281no—
278C4—6282no—
279C4—1572no—
280C4#1909779812yes—
281C4C2×Q8×D15711yescharacteristic
282C4#41×D15711yescharacteristic
283C2—42no—
284C2—3142no—
285C2—6282no—
286C2C22⋊Q8×D15711yescharacteristic
287C2C2×#1909779811yescharacteristic
288C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#476×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#476D157
hypocenterC157#3077
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #23757828C2semidirect
3. #23757572C2semidirect
4. #23757571C2semidirect
5. #23757560C2semidirect
6. #23757554C2semidirect
7. #23757553C2semidirect
8. #23757312C2semidirect
9. #23757295C2semidirect
11. #476×C157C2semidirect
15. #19097771C22semidirect
16. #19097762C22semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097702C22semidirect
33. C4⋊#11929434C22semidirect
34. #19097700C22semidirect
38. C4⋊Q8×C157C22semidirect
40. #106×C157C22semidirect
41. C4⋊C8×C157C22semidirect
48. D4×D157Q8semidirect
144. C157#3077semidirect
151. #476D157semidirect
161. C4⋊Q8C2×D157semidirect
180. #106C2×D157semidirect
183. C4⋊C8C2×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
#237575722
#237575712
#237575602
#237575542
#237575532
#237573122
#237572952
#237572762
#476×C1572
#3077157