Finite Group Databasebeta

← #34113463

Subgroups — #34113463

Order 20096.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D157C212yesmaximal, complemented
2#23757850C214yesmaximal, complemented
3#23757848C212yesmaximal, complemented
4#23757843C212yesmaximal, complemented
5#23757841C212yesmaximal, complemented
6#23757836C212yesmaximal, complemented
7#23757831C214yesmaximal, complemented
8#23757823C211yesmaximal, characteristic, complemented
9#23757819C212yesmaximal, complemented
10#23757782C218yesmaximal, complemented
11C2×#19097728C211yesmaximal, characteristic, complemented
12#532×C157C211yesmaximal, characteristic, complemented
13C2×#14578955C2212yes—
14C2×#14578954C2214yes—
15C2×#14578954C2212yescomplemented
16C2×#14578953C2212yes—
17C2×#14578953C2214yescomplemented
18#41×D157—216no—
19C2×Q8×D157C2212yescomplemented
20C2×D4×D157C2214yes—
21C2×D4×D157C2212yescomplemented
22C22×C4×D157C2211yescharacteristic
23C22×C4×D157C2214yescomplemented
24#19097771C2211yescharacteristic, complemented
25#19097769C2212yescomplemented
26#19097768C2211yescharacteristic, complemented
27#19097767C2212yescomplemented
28#19097760C2212yescomplemented
29#19097759C2212yescomplemented
30#19097758C2212yescomplemented
31#19097756C2218yescomplemented
32#19097753C2218yescomplemented
33#19097749C2218yescomplemented
34#19097748C2218yescomplemented
35#19097745C2214yescomplemented
36#19097744C2218yescomplemented
37#19097743C2218yescomplemented
38#19097743C2218yescomplemented
39#19097741C2218yescomplemented
40#19097739C2218yescomplemented
41C4×D628C2214yescomplemented
42C4×C22⋊D157C2218yescomplemented
43#19097728C2212yescomplemented
44#19097728C2212yescomplemented
45#19097728C2214yescomplemented
46C2×#14578933C2212yescomplemented
47C2×#14578931C2212yes—
48C2×C4×Dic157C2211yescharacteristic, complemented
49C2×#41×C157C2212yescomplemented
50C4⋊Q8×C157C2211yescharacteristic, complemented
51#127×C157C2211yescharacteristic, complemented
52#124×C157C2212yescomplemented
53#121×C157C2214yescomplemented
54C4×D4×C157C2214yescomplemented
55C2×C42×C157C2211yescharacteristic, complemented
56C23×D157C2314yes—
57C23×D157C2311yescharacteristic
58#14578955—28no—
59#14578954—216no—
60#14578954—28no—
61#14578953—216no—
62#14578953—28no—
63C2×Dic314C2314yes—
64C2×Dic314C2312yes—
65Q8×D157—28no—
66C2×D628C2314yes—
67C2×D628C2312yes—
68C2×D628C2314yes—
69C2×C22⋊D157C2314yes—
70C2×C22⋊D157C2314yes—
71C2×C22⋊D157C2314yes—
72C2×C22⋊D157C2318yes—
73D4×D157—216no—
74D4×D157—28no—
75C22×Dic157C2314yes—
76C22×Dic157C2312yes—
77C22×Dic157C2311yescharacteristic
78C2×C4×D157—216no—
79C2×C4×D157D418yescomplemented
80C2×C4×D157C2314yes—
81C2×C4×D157C2314yes—
82C2×C4×D157C2314yes—
83C2×C4×D157C2312yes—
84C2×C4×D157C2312yes—
85C4⋊Dic157C2314yes—
86#14578933C2314yes—
87#14578933C2314yes—
88#14578933C2318yes—
89C22⋊Dic157C2318yes—
90#14578931C2314yes—
91#14578931C2314yes—
92#14578931C2318yes—
93#14578931C2318yes—
94C4×Dic157C2312yes—
95C4×Dic157C2312yes—
96#41×C157—28no—
97C2×Q8×C157C2312yes—
98C2×D4×C157C2314yes—
99C2×D4×C157C2312yes—
100C2×D4×C157C2314yes—
101C22×C4×C157C2314yes—
102C22×C4×C157C2312yes—
103C22×C4×C157C2311yescharacteristic
104C4⋊C4×C157C2314yes—
105C22⋊C4×C157C2318yes—
106C42×C157C2312yes—
107C42×C157C2311yescharacteristic
108C42×C157C2311yescharacteristic
109C22×D157—28no—
110C22×D157—42no—
111C22×D157—48no—
112C22×D157C2×D414yes—
113C22×D157C2414yes—
114C22×D157C2412yes—
115Dic314—24no—
116Dic314—28no—
117D628—28no—
118D628—24no—
119D628—44no—
120C22⋊D157—28no—
121C22⋊D157—28no—
122C22⋊D157—28no—
123C22⋊D157—48no—
124C2×Dic157—28no—
125C2×Dic157—42no—
126C2×Dic157C2×D414yes—
127C2×Dic157C2414yes—
128C2×Dic157C2414yes—
129C2×Dic157C2412yes—
130C4×D157—28no—
131C4×D157—28no—
132C4×D157—28no—
133C4×D157C2×D418yes—
134C4×D157C2×D418yes—
135C23×C157C2414yes—
136C23×C157C2411yescharacteristic
137Q8×C157—24no—
138D4×C157—28no—
139D4×C157—24no—
140D4×C157—44no—
141C2×C4×C157—28no—
142C2×C4×C157—42no—
143C2×C4×C157C2×D414yes—
144C2×C4×C157C2414yes—
145C2×C4×C157C2412yes—
146C2×C4×C157C2412yes—
147C2×C4×C157C2411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C22×C157—22no—
150C22×C157—24no—
151C22×C157—28no—
152C22×C157C22×D412yes—
153C22×C157C2511yescharacteristic
154C2×D157—24no—
155C2×D157—24no—
156C2×D157—28no—
157C2×D157—44no—
158C2×D157—416no—
159C2×D157C22×D414yes—
160C4×C157—24no—
161C4×C157—22no—
162C4×C157—22no—
163C4×C157C22×D412yes—
164C4×C157C22×D412yes—
165Dic157—24no—
166Dic157—24no—
167Dic157C22×D414yes—
168C2×C157—22no—
169C2×C157—44no—
170C2×C157C23×D411yescharacteristic
171C2×C157#58512yes—
172D157—24no—
173D157—44no—
174C157#344911yescharacteristic, complemented, Sylow
175#3449—1571nomaximal, Sylow
176C22×#41—1572no—
177#537—1574no—
178#536—1572no—
179#536—1572no—
180#535—1572no—
181#535—1572no—
182#534—1574no—
183#532—1572no—
184#532—1571no—
185#532D15711yescharacteristic, complemented
186#525—1578no—
187C2×#117—1571no—
188C2×#41—31416no—
189C2×#41—1572no—
190C2×#41—1572no—
191C2×#41—1574no—
192C2×#41—1574no—
193C2×#41—1572no—
194C2×#41C2×D15712yescomplemented
195C22×Q8—1572no—
196C22×D4—1574no—
197C22×D4—1572no—
198C23×C4—1574no—
199C23×C4—1571no—
200C4⋊Q8—1572no—
201C4⋊Q8—1571no—
202C4⋊Q8C2×D15711yescharacteristic, complemented
203#127—1572no—
204#127—1571no—
205#127C2×D15711yescharacteristic, complemented
206#124—1572no—
207#124—1572no—
208#124—1572no—
209#124C2×D15712yescomplemented
210#123—1578no—
211#123—1578no—
212C22⋊Q8—1578no—
213C22⋊Q8—1578no—
214#121—1578no—
215#121—1578no—
216#121—1574no—
217#121—1578no—
218#121C2×D15714yescomplemented
219#120—1578no—
220#120—1578no—
221C4×D4—1578no—
222C4×D4—1574no—
223C4×D4C2×D15714yescomplemented
224#117—1574no—
225#117—1572no—
226#117—1572no—
227C2×C4⋊C4—1572no—
228C2×C22⋊C4—1572no—
229C2×C42—1571no—
230C2×C42C2×D15711yescharacteristic, complemented
231C24—1574no—
232C24—1571no—
233#41—28no—
234#41—31416no—
235#41—31416no—
236#41—3148no—
237#41—3148no—
238#41—3148no—
239C2×Q8—3148no—
240C2×Q8—1574no—
241C2×Q8—1572no—
242C2×Q8C22×D15712yes—
243C2×D4—31416no—
244C2×D4—3148no—
245C2×D4—1574no—
246C2×D4—1574no—
247C2×D4—1574no—
248C2×D4—1578no—
249C2×D4—1574no—
250C2×D4—1572no—
251C2×D4—1574no—
252C2×D4C22×D15714yes—
253C2×D4C22×D15712yes—
254C2×D4C22×D15714yes—
255C22×C4—31416no—
256C22×C4—1574no—
257C22×C4—1574no—
258C22×C4—1574no—
259C22×C4—1574no—
260C22×C4—1572no—
261C22×C4—1571no—
262C22×C4—1578no—
263C22×C4—1572no—
264C22×C4—1572no—
265C22×C4C22×D15714yes—
266C22×C4C22×D15712yes—
267C22×C4C22×D15711yescharacteristic
268C4⋊C4—1574no—
269C4⋊C4—1574no—
270C4⋊C4—1578no—
271C4⋊C4—1574no—
272C4⋊C4C22×D15714yes—
273C22⋊C4—1578no—
274C22⋊C4—1574no—
275C22⋊C4—1574no—
276C22⋊C4—1578no—
277C22⋊C4—1578no—
278C22⋊C4C22×D15718yes—
279C42—1572no—
280C42—1572no—
281C42C22×D15711yescharacteristic
282C42C22×D15711yescharacteristic
283C42C22×D15712yes—
284C23—3148no—
285C23—6282no—
286C23—6288no—
287C23—1574no—
288C23—1572no—
289C23—1574no—
290C23C23×D15714yes—
291C23C23×D15711yescharacteristic
292Q8—24no—
293Q8—3148no—
294Q8—3144no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—3148no—
299D4—3148no—
300D4—3148no—
301D4—3148no—
302D4—3144no—
303D4—6284no—
304D4—6288no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—3148no—
308C2×C4—3148no—
309C2×C4—3148no—
310C2×C4—3148no—
311C2×C4—6282no—
312C2×C4—1574no—
313C2×C4—1574no—
314C2×C4—1574no—
315C2×C4—1572no—
316C2×C4—1578no—
317C2×C4—1578no—
318C2×C4D4×D15714yes—
319C2×C4C23×D15712yes—
320C2×C4C23×D15712yes—
321C2×C4C23×D15714yes—
322C2×C4C23×D15711yescharacteristic
323C2×C4C23×D15711yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—3144no—
328C22—3144no—
329C22—3148no—
330C22—6284no—
331C22—62816no—
332C22—1574no—
333C22C2×D4×D15712yes—
334C22C24×D15711yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—3144no—
339C4—3144no—
340C4—1574no—
341C4C2×D4×D15712yes—
342C4C2×D4×D15712yes—
343C2—22no—
344C2—44no—
345C2—3144no—
346C2—6284no—
347C2C22×D4×D15711yescharacteristic
348C2#2375810912yes—
subgroupG/H
centerC2×C4C23×D157
commutator, socle, cosocleC22×C157C25
FrattiniC22C24×D157
Fitting#532×C157C2
hypercenter#532D157
hypocenterC157#3449
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D157C2semidirect
2. #23757850C2semidirect
3. #23757848C2semidirect
4. #23757843C2semidirect
5. #23757841C2semidirect
6. #23757836C2semidirect
7. #23757831C2semidirect
8. #23757823C2semidirect
9. #23757819C2semidirect
10. #23757782C2semidirect
11. C2×#19097728C2semidirect
12. #532×C157C2semidirect
15. C2×#14578954C22semidirect
17. C2×#14578953C22semidirect
19. C2×Q8×D157C22semidirect
21. C2×D4×D157C22semidirect
23. C22×C4×D157C22semidirect
24. #19097771C22semidirect
25. #19097769C22semidirect
26. #19097768C22semidirect
27. #19097767C22semidirect
28. #19097760C22semidirect
29. #19097759C22semidirect
30. #19097758C22semidirect
31. #19097756C22semidirect
32. #19097753C22semidirect
33. #19097749C22semidirect
34. #19097748C22semidirect
35. #19097745C22semidirect
36. #19097744C22semidirect
37. #19097743C22semidirect
38. #19097743C22semidirect
39. #19097741C22semidirect
40. #19097739C22semidirect
41. C4×D628C22semidirect
42. C4×C22⋊D157C22semidirect
43. #19097728C22semidirect
44. #19097728C22semidirect
45. #19097728C22semidirect
46. C2×#14578933C22semidirect
48. C2×C4×Dic157C22semidirect
49. C2×#41×C157C22semidirect
50. C4⋊Q8×C157C22semidirect
51. #127×C157C22semidirect
52. #124×C157C22semidirect
53. #121×C157C22semidirect
54. C4×D4×C157C22semidirect
55. C2×C42×C157C22semidirect
79. C2×C4×D157D4semidirect
174. C157#3449semidirect
185. #532D157semidirect
194. C2×#41C2×D157semidirect
202. C4⋊Q8C2×D157semidirect
205. #127C2×D157semidirect
209. #124C2×D157semidirect
218. #121C2×D157semidirect
223. C4×D4C2×D157semidirect
230. C2×C42C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1572
#237578502
#237578482
#237578432
#237578412
#237578362
#237578312
#237578232
#237578192
#237577822
C2×#190977282
#532×C1572
#3449157