Finite Group Databasebeta

← #34110226

Subgroups — #34110226

Order 20096.

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

normal subgroups only

This group has 488 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#23757821C211yesmaximal, characteristic, complemented
2C4×D4×D157C211yesmaximal, characteristic, complemented
3#23757590C212yesmaximal, complemented
4#23757564C211yesmaximal, characteristic, complemented
5#23757563C211yesmaximal, characteristic, complemented
6#23757465C211yesmaximal, characteristic, complemented
7#23757405C212yesmaximal, complemented
8#23757385C212yesmaximal, complemented
9#23757382C212yesmaximal, complemented
10#23757329C211yesmaximal, characteristic, complemented
11#459×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—21no—
13C2×D4×D157C2211yescharacteristic
14C22×C4×D157—21no—
15C2×#14578939C2212yes—
16#19097771C2211yescharacteristic, complemented
17#19097758—21no—
18#19097742—21no—
19#19097742C2212yes—
20C4×D628C2211yescharacteristic
21D4×Dic157—21no—
22D4×Dic157C2211yescharacteristic
23D4×Dic157C2212yescomplemented
24#19097733C2211yescharacteristic
25C4×C22⋊D157—21no—
26#19097731—21no—
27C4⋊C4×D157C2211yescharacteristic
28C22⋊C4×D157—21no—
29C42×D157C2211yescharacteristic
30#19097703C2212yescomplemented
31#19097702C2212yescomplemented
32C4⋊#11929434C2211yescharacteristic, complemented
33#19097693C2212yes—
34#19097692C2212yes—
35#19097692C2212yescomplemented
36#19097683C2212yes—
37#19097681C2212yescomplemented
38C2×D8×C157C2212yes—
39#127×C157C2211yescharacteristic, complemented
40C4×D4×C157C2211yescharacteristic
41C4⋊C8×C157C2211yescharacteristic, complemented
42#102×C157C2212yescomplemented
43C23×D157—21no—
44#14578953—44no—
45C2×Dic314—21no—
46C2×D628C2311yescharacteristic
47C2×C22⋊D157—22no—
48C2×C22⋊D157—21no—
49D4×D157—22no—
50D4×D157—22no—
51C22×Dic157—21no—
52C22×Dic157—22no—
53C2×C4×D157—21no—
54C2×C4×D157—21no—
55C2×C4×D157—21no—
56C2×C4×D157—44no—
57C2×C4×D157D411yescharacteristic
58C2×C4×D157D411yescharacteristic
59C2×C4×D157C2311yescharacteristic
60#14578939—24no—
61C2×#11929434C2312yes—
62C4⋊Dic157—21no—
63C4⋊Dic157C2312yes—
64C4⋊Dic157C2311yescharacteristic
65#14578933—21no—
66C22⋊Dic157—21no—
67C22⋊Dic157—22no—
68C22⋊Dic157—22no—
69#14578931—21no—
70#14578931—21no—
71C4×Dic157D411yescharacteristic
72C4×Dic157D411yescharacteristic
73C4×Dic157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157—21no—
76C2×D4×C157C2312yes—
77C2×D4×C157C2311yescharacteristic
78C22×C4×C157—21no—
79D8×C157—24no—
80C2×C8×C157C2312yes—
81C4⋊C4×C157C2311yescharacteristic
82C22⋊C4×C157—21no—
83C42×C157C2311yescharacteristic
84C22×D157—24no—
85C22×D157—21no—
86C22×D157—42no—
87C22×D157—42no—
88C22×D157C2×D411yescharacteristic
89Dic314—42no—
90D628—21no—
91D628#4112yescomplemented
92C22⋊D157—24no—
93C22⋊D157—44no—
94C2×Dic157—21no—
95C2×Dic157—22no—
96C2×Dic157—42no—
97C2×Dic157—44no—
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D411yescharacteristic
100C2×Dic157C2×D411yescharacteristic
101C4×D157—22no—
102C4×D157—22no—
103C4×D157—21no—
104C4×D157—21no—
105C4×D157—42no—
106C4×D157—42no—
107C4×D157—42no—
108#11929434—22no—
109C23×C157—21no—
110C23×C157—22no—
111D4×C157—21no—
112D4×C157—22no—
113D4×C157—22no—
114D4×C157—42no—
115D4×C157—42no—
116D4×C157#4112yescomplemented
117C2×C4×C157—21no—
118C2×C4×C157—42no—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2411yescharacteristic
122C8×C157—22no—
123C22×C157—22no—
124C22×C157—41no—
125C22×C157—41no—
126C22×C157—42no—
127C22×C157—42no—
128C22×C157—42no—
129C22×C157C22×D411yescharacteristic
130C2×D157—21no—
131C2×D157—21no—
132C2×D157—22no—
133C2×D157—48no—
134C2×D157—41no—
135C2×D157—41no—
136C2×D157AGL(1,ℤ/8)12yes—
137C4×C157—21no—
138C4×C157—41no—
139C4×C157AGL(1,ℤ/8)12yes—
140C4×C157C2×#4111yescharacteristic
141C4×C157C2×#4111yescharacteristic
142Dic157—21no—
143Dic157—21no—
144Dic157—21no—
145Dic157—41no—
146Dic157—42no—
147C2×C157—42no—
148C2×C157—82no—
149C2×C157#52511yescharacteristic
150C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
152D157—22no—
153D157—42no—
154C157#310911yescharacteristic, complemented, Sylow
155#3109—1571nomaximal, Sylow
156#532—1571no—
157C2×C4×D4—1571no—
158#481—1572no—
159#476—1571no—
160#476—1571no—
161#459—1571no—
162#459D15711yescharacteristic, complemented
163#447—1572no—
164#442—1572no—
165#442—1572no—
166#432—1571no—
167C2×#41—3141no—
168C22×D4—1571no—
169C23×C4—3141no—
170C2×D8—1572no—
171C2×D8C2×D15712yes—
172C4⋊Q8—1571no—
173#127C2×D15711yescharacteristic, complemented
174#124—3141no—
175#121—3141no—
176#121—1572no—
177C4×D4—3141no—
178C4×D4—3141no—
179C4×D4—3141no—
180C4×D4—1571no—
181C4×D4—1572no—
182C4×D4—1571no—
183C4×D4—1571no—
184C4×D4C2×D15711yescharacteristic
185C2×C4⋊C4—1571no—
186C2×C22⋊C4—3141no—
187C2×C42—1571no—
188#106—1572no—
189#106—1572no—
190C4⋊C8—1571no—
191C4⋊C8C2×D15711yescharacteristic, complemented
192#102—1572no—
193#102—1572no—
194#102—1572no—
195#102C2×D15712yescomplemented
196C22⋊C8—1572no—
197#97—1572no—
198C24—3141no—
199#41—6284no—
200C2×Q8—3141no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—3142no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—1571no—
208C2×D4C22×D15712yes—
209C2×D4C22×D15711yescharacteristic
210C22×C4—21no—
211C22×C4—3141no—
212C22×C4—3141no—
213C22×C4—3141no—
214C22×C4—3142no—
215C22×C4—3141no—
216C22×C4—6284no—
217C22×C4—1571no—
218C22×C4—1571no—
219C22×C4—1571no—
220D8—24no—
221D8—3144no—
222C2×C8—1572no—
223C2×C8C22×D15712yes—
224C4⋊C4—3141no—
225C4⋊C4—3141no—
226C4⋊C4—1571no—
227C4⋊C4—1572no—
228C4⋊C4C22×D15711yescharacteristic
229C22⋊C4—21no—
230C22⋊C4—3141no—
231C22⋊C4—3141no—
232C22⋊C4—3142no—
233C22⋊C4—3142no—
234C22⋊C4—3141no—
235C42—1571no—
236C42—1571no—
237C42—1571no—
238C42C22×D15711yescharacteristic
239C23—21no—
240C23—22no—
241C23—3144no—
242C23—3141no—
243C23—6282no—
244C23—6282no—
245C23—1571no—
246Q8—6282no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—3144no—
253D4—3141no—
254D4—6284no—
255D4—1572no—
256D4#1457895312yescomplemented
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—3141no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—3142no—
265C2×C4—6282no—
266C2×C4—6282no—
267C2×C4—6282no—
268C2×C4—6284no—
269C2×C4—6282no—
270C2×C4—1571no—
271C2×C4—1571no—
272C2×C4—1571no—
273C2×C4D4×D15711yescharacteristic
274C2×C4D4×D15711yescharacteristic
275C2×C4C23×D15711yescharacteristic
276C8—22no—
277C8—3142no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—3141no—
285C22—3141no—
286C22—3142no—
287C22—6288no—
288C22—6281no—
289C22—6281no—
290C22—1572no—
291C22C2×D4×D15711yescharacteristic
292C4—21no—
293C4—41no—
294C4—3141no—
295C4—3141no—
296C4—3141no—
297C4—6282no—
298C4—6281no—
299C4#1909779512yes—
300C4#41×D15711yescharacteristic
301C4C2×#1457895311yescharacteristic
302C2—42no—
303C2—82no—
304C2—3142no—
305C2—6282no—
306C2#2375778311yescharacteristic
307C2AGL(1,ℤ/8)×D15711yescharacteristic
308C2C2×#1909779511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3109
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #23757821C2semidirect
2. C4×D4×D157C2semidirect
3. #23757590C2semidirect
4. #23757564C2semidirect
5. #23757563C2semidirect
6. #23757465C2semidirect
7. #23757405C2semidirect
8. #23757385C2semidirect
9. #23757382C2semidirect
10. #23757329C2semidirect
11. #459×C157C2semidirect
16. #19097771C22semidirect
23. D4×Dic157C22semidirect
30. #19097703C22semidirect
31. #19097702C22semidirect
32. C4⋊#11929434C22semidirect
35. #19097692C22semidirect
37. #19097681C22semidirect
39. #127×C157C22semidirect
41. C4⋊C8×C157C22semidirect
42. #102×C157C22semidirect
91. D628#41semidirect
116. D4×C157#41semidirect
154. C157#3109semidirect
162. #459D157semidirect
173. #127C2×D157semidirect
191. C4⋊C8C2×D157semidirect
195. #102C2×D157semidirect
256. D4#14578953semidirect

Maximal subgroups

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

subgroupindex
#237578212
C4×D4×D1572
#237575902
#237575642
#237575632
#237574652
#237574052
#237573852
#237573822
#237573292
#459×C1572
#3109157