Finite Group Databasebeta

← #541×D157 #34113498

Subgroups — #541×D157 #34113498

Order 20096.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757886C211yesmaximal, characteristic, complemented
2#23757885C211yesmaximal, characteristic, complemented
3#23757884C211yesmaximal, characteristic, complemented
4#23757883C212yesmaximal, complemented
5#23757882C214yesmaximal, complemented
6#23757881C212yesmaximal, complemented
7#23757880C214yesmaximal, complemented
8C4⋊Q8×D157C211yesmaximal, characteristic, complemented
9#126×D157C214yesmaximal
10#125×D157C214yesmaximal, complemented
11#125×D157C211yesmaximal, characteristic, complemented
12C22⋊Q8×D157C212yesmaximal
13C4×Q8×D157C212yesmaximal, complemented
14#117×D157C211yesmaximal, characteristic
15#541×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2212yes—
17C22×C4×D157C2211yescharacteristic
18#19097771C2211yescharacteristic, complemented
19#19097770C2214yescomplemented
20#19097769C2212yescomplemented
21#19097766C2214yes—
22#19097766C2214yes—
23#19097766C2214yes—
24#19097765C2214yes—
25#19097764C2214yes—
26#19097764C2214yes—
27#19097764C2214yes—
28#19097763C2214yescomplemented
29#19097763C2211yescharacteristic, complemented
30#19097762C2214yescomplemented
31#19097762C2212yescomplemented
32#19097762C2214yescomplemented
33#19097761C2214yescomplemented
34#19097761C2214yescomplemented
35#19097761C2214yescomplemented
36#19097761C2214yescomplemented
37#19097761C2214yescomplemented
38#19097751C2212yes—
39#19097750C2212yes—
40#19097749C2214yes—
41#19097748C2214yes—
42#19097747C2212yes—
43Q8×Dic157C2212yescomplemented
44C4×Dic314C2214yescomplemented
45C4×Dic314C2212yescomplemented
46#19097736C2214yescomplemented
47#19097736C2212yescomplemented
48#19097730C2211yescharacteristic
49#19097729C2212yes—
50#19097728C2211yescharacteristic
51#19097728C2211yescharacteristic
52#19097727C2212yes—
53C4⋊C4×D157C2212yes—
54C4⋊C4×D157C2212yes—
55C4⋊C4×D157C2214yes—
56C4⋊C4×D157C2214yes—
57C4⋊C4×D157C2214yes—
58C4⋊C4×D157C2214yes—
59C22⋊C4×D157C2212yes—
60C22⋊C4×D157C2214yes—
61C42×D157C2211yescharacteristic
62C42×D157C2211yescharacteristic
63C42×D157C2214yes—
64C4⋊Q8×C157C2211yescharacteristic, complemented
65#126×C157C2214yes—
66#125×C157C2214yescomplemented
67#125×C157C2211yescharacteristic, complemented
68C22⋊Q8×C157C2212yes—
69C4×Q8×C157C2212yescomplemented
70#117×C157C2211yescharacteristic
71C23×D157C2311yescharacteristic
72C2×Dic314C2314yes—
73C2×Dic314C2312yes—
74Q8×D157—28no—
75C22×Dic157C2311yescharacteristic
76C2×C4×D157—24no—
77C2×C4×D157C2312yes—
78C2×C4×D157C2312yes—
79C2×C4×D157C2314yes—
80C2×C4×D157C2314yes—
81C2×C4×D157C2311yescharacteristic
82C2×C4×D157C2311yescharacteristic
83C2×C4×D157C2311yescharacteristic
84C2×C4×D157C2311yescharacteristic
85C4⋊Dic157C2312yes—
86C4⋊Dic157C2312yes—
87C4⋊Dic157C2314yes—
88C4⋊Dic157C2314yes—
89C4⋊Dic157C2314yes—
90C4⋊Dic157C2314yes—
91#14578933C2312yes—
92#14578933C2312yes—
93#14578933C2314yes—
94#14578933C2314yes—
95#14578933C2314yes—
96#14578933C2314yes—
97#14578933C2314yes—
98#14578933C2314yes—
99#14578933C2314yes—
100#14578933C2314yes—
101#14578933C2314yes—
102C22⋊Dic157C2312yes—
103C22⋊Dic157C2314yes—
104#14578931C2312yes—
105#14578931C2312yes—
106#14578931C2314yes—
107#14578931C2314yes—
108C4×Dic157C2312yes—
109C4×Dic157C2312yes—
110C4×Dic157C2311yescharacteristic
111C4×Dic157C2311yescharacteristic
112C4×Dic157C2314yes—
113C4×Dic157C2314yes—
114C4×Dic157C2314yes—
115C2×Q8×C157C2312yes—
116C22×C4×C157C2311yescharacteristic
117C4⋊C4×C157C2312yes—
118C4⋊C4×C157C2312yes—
119C4⋊C4×C157C2314yes—
120C4⋊C4×C157C2314yes—
121C4⋊C4×C157C2314yes—
122C4⋊C4×C157C2314yes—
123C22⋊C4×C157C2312yes—
124C22⋊C4×C157C2314yes—
125C42×C157C2311yescharacteristic
126C42×C157C2311yescharacteristic
127C42×C157C2314yes—
128C22×D157—22no—
129C22×D157—24no—
130C22×D157C2411yescharacteristic
131C22×D157C2411yescharacteristic
132Dic314—24no—
133Dic314—28no—
134C2×Dic157—22no—
135C2×Dic157C2411yescharacteristic
136C2×Dic157C2414yes—
137C2×Dic157C2412yes—
138C2×Dic157C2412yes—
139C2×Dic157C2414yes—
140C2×Dic157C2411yescharacteristic
141C4×D157—22no—
142C4×D157—28no—
143C4×D157—24no—
144C4×D157—24no—
145C4×D157—28no—
146C4×D157—22no—
147C4×D157—22no—
148C4×D157#4114yes—
149C23×C157C2411yescharacteristic
150Q8×C157—24no—
151C2×C4×C157—22no—
152C2×C4×C157C2411yescharacteristic
153C2×C4×C157C2414yes—
154C2×C4×C157C2412yes—
155C2×C4×C157C2412yes—
156C2×C4×C157C2414yes—
157C2×C4×C157C2411yescharacteristic
158C22×C157—21no—
159C22×C157—22no—
160C22×C157C2511yescharacteristic
161C2×D157—21no—
162C2×D157—22no—
163C2×D157—44no—
164C2×D157C2×#4112yes—
165C2×D1572−1+414yes—
166C4×C157—21no—
167C4×C157—24no—
168C4×C157—22no—
169C4×C157—22no—
170C4×C157—24no—
171C4×C157C2×#4112yes—
172Dic157—21no—
173Dic157—24no—
174Dic157—22no—
175Dic157—22no—
176Dic157—24no—
177Dic157C2×#4112yes—
178C2×C157—41no—
179C2×C157C22×#4111yescharacteristic
180C2×C157C2×2−1+412yes—
181D157—41no—
182D157#54114yesdirect product factor
183C157C2×#54111yescharacteristic, complemented, Sylow
184C2×#541—1571nomaximal, Sylow
185#541—1571no—
186#541—1574no—
187#541—1574no—
188#541—1571no—
189#541—1572no—
190#541—1572no—
191#541—1571no—
192#541D15711yescharacteristic, direct product factor
193C2×C4⋊Q8—1571no—
194C2×#126—1574no—
195C2×#125—1571no—
196C2×#125—1574no—
197C2×C22⋊Q8—1572no—
198C2×C4×Q8—1572no—
199C2×#117—1571no—
200C22×Q8—1572no—
201C23×C4—1571no—
202C4⋊Q8—1572no—
203C4⋊Q8—1574no—
204C4⋊Q8—1571no—
205C4⋊Q8C2×D15711yescharacteristic, complemented
206#126—1574no—
207#126—1574no—
208#126—1574no—
209#126—1574no—
210#126—1574no—
211#126—1574no—
212#126—1574no—
213#126C2×D15714yes—
214#125—1574no—
215#125—1574no—
216#125—1574no—
217#125—1574no—
218#125—1574no—
219#125—1572no—
220#125—1574no—
221#125—1574no—
222#125—1571no—
223#125—1574no—
224#125C2×D15711yescharacteristic, complemented
225#125C2×D15714yescomplemented
226C22⋊Q8—1572no—
227C22⋊Q8—1572no—
228C22⋊Q8—1574no—
229C22⋊Q8—1572no—
230C22⋊Q8—1574no—
231C22⋊Q8C2×D15712yes—
232C4×Q8—1572no—
233C4×Q8—1574no—
234C4×Q8—1572no—
235C4×Q8—1574no—
236C4×Q8—1572no—
237C4×Q8C2×D15712yescomplemented
238#117—1571no—
239#117—1572no—
240#117—1572no—
241#117—1571no—
242#117—1571no—
243#117C2×D15711yescharacteristic
244C2×C4⋊C4—1572no—
245C2×C4⋊C4—1572no—
246C2×C4⋊C4—1574no—
247C2×C4⋊C4—1574no—
248C2×C4⋊C4—1574no—
249C2×C4⋊C4—1574no—
250C2×C22⋊C4—1572no—
251C2×C22⋊C4—1574no—
252C2×C42—1571no—
253C2×C42—1571no—
254C2×C42—1574no—
255C24—1571no—
256C2×Q8—3148no—
257C2×Q8—1572no—
258C2×Q8—1574no—
259C2×Q8C22×D15712yes—
260C22×C4—3144no—
261C22×C4—1571no—
262C22×C4—1572no—
263C22×C4—1572no—
264C22×C4—1574no—
265C22×C4—1574no—
266C22×C4—1571no—
267C22×C4—1571no—
268C22×C4—1571no—
269C22×C4—1571no—
270C22×C4C22×D15711yescharacteristic
271C4⋊C4—1572no—
272C4⋊C4—1572no—
273C4⋊C4—1572no—
274C4⋊C4—1574no—
275C4⋊C4—1572no—
276C4⋊C4—1574no—
277C4⋊C4—1574no—
278C4⋊C4—1574no—
279C4⋊C4—1574no—
280C4⋊C4—1574no—
281C4⋊C4—1574no—
282C4⋊C4—1574no—
283C4⋊C4—1574no—
284C4⋊C4—1574no—
285C4⋊C4—1574no—
286C4⋊C4—1574no—
287C4⋊C4—1574no—
288C4⋊C4C22×D15712yes—
289C4⋊C4C22×D15712yes—
290C4⋊C4C22×D15714yes—
291C4⋊C4C22×D15714yes—
292C4⋊C4C22×D15714yes—
293C4⋊C4C22×D15714yes—
294C22⋊C4—1572no—
295C22⋊C4—1574no—
296C22⋊C4—1572no—
297C22⋊C4—1572no—
298C22⋊C4—1574no—
299C22⋊C4—1574no—
300C22⋊C4C22×D15712yes—
301C22⋊C4C22×D15714yes—
302C42—1571no—
303C42—1572no—
304C42—1571no—
305C42—1572no—
306C42—1574no—
307C42—1574no—
308C42—1574no—
309C42C22×D15711yescharacteristic
310C42C22×D15711yescharacteristic
311C42C22×D15714yes—
312C23—3142no—
313C23—3144no—
314C23—1571no—
315C23—1571no—
316C23C23×D15711yescharacteristic
317Q8—24no—
318Q8—3148no—
319Q8—3144no—
320C2×C4—22no—
321C2×C4—3142no—
322C2×C4—3148no—
323C2×C4—3144no—
324C2×C4—3144no—
325C2×C4—3148no—
326C2×C4—3142no—
327C2×C4—3142no—
328C2×C4—3142no—
329C2×C4—1572no—
330C2×C4—1572no—
331C2×C4—1574no—
332C2×C4—1574no—
333C2×C4—1571no—
334C2×C4—1571no—
335C2×C4—1574no—
336C2×C4C23×D15712yes—
337C2×C4C23×D15712yes—
338C2×C4C23×D15714yes—
339C2×C4C23×D15714yes—
340C2×C4C23×D15711yescharacteristic
341C2×C4C23×D15711yescharacteristic
342C22—21no—
343C22—22no—
344C22—3141no—
345C22—3142no—
346C22—6284no—
347C22—1572no—
348C22—1574no—
349C22C24×D15711yescharacteristic
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—24no—
355C4—3141no—
356C4—3144no—
357C4—3142no—
358C4—3142no—
359C4—3144no—
360C4—1572no—
361C4#41×D15712yes—
362C2—41no—
363C2—6281no—
364C2—1574no—
365C2C2×#41×D15711yescharacteristic
366C22−1+4×D15712yes—
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#541×C157C2
prime radicalC23×D157C23
hypercenter#541D157
hypocenterC157C2×#541
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #23757886C2semidirect
2. #23757885C2semidirect
3. #23757884C2semidirect
4. #23757883C2semidirect
5. #23757882C2semidirect
6. #23757881C2semidirect
7. #23757880C2semidirect
8. C4⋊Q8×D157C2semidirect
10. #125×D157C2semidirect
11. #125×D157C2semidirect
13. C4×Q8×D157C2semidirect
15. #541×C157C2semidirect
18. #19097771C22semidirect
19. #19097770C22semidirect
20. #19097769C22semidirect
28. #19097763C22semidirect
29. #19097763C22semidirect
30. #19097762C22semidirect
31. #19097762C22semidirect
32. #19097762C22semidirect
33. #19097761C22semidirect
34. #19097761C22semidirect
35. #19097761C22semidirect
36. #19097761C22semidirect
37. #19097761C22semidirect
43. Q8×Dic157C22semidirect
44. C4×Dic314C22semidirect
45. C4×Dic314C22semidirect
46. #19097736C22semidirect
47. #19097736C22semidirect
64. C4⋊Q8×C157C22semidirect
66. #125×C157C22semidirect
67. #125×C157C22semidirect
69. C4×Q8×C157C22semidirect
182. D157#541direct (N × H)
183. C157C2×#541semidirect
192. #541D157direct (N × H)
205. C4⋊Q8C2×D157semidirect
224. #125C2×D157semidirect
225. #125C2×D157semidirect
237. C4×Q8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578862
#237578852
#237578842
#237578832
#237578822
#237578812
#237578802
C4⋊Q8×D1572
#126×D1572
#125×D1572
C22⋊Q8×D1572
C4×Q8×D1572
#117×D1572
#541×C1572
C2×#541157