Finite Group Databasebeta

← #34111980

Subgroups — #34111980

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757919C211yesmaximal, characteristic, complemented
2#23757630C211yesmaximal, characteristic, complemented
3#23757608C211yesmaximal, characteristic, complemented
4#23757597C211yesmaximal, characteristic, complemented
5#23757570C211yesmaximal, characteristic, complemented
6#23757507C211yesmaximal, characteristic, complemented
7#23757500C211yesmaximal, characteristic, complemented
8#23757488C211yesmaximal, characteristic, complemented
9#23757416C211yesmaximal, characteristic, complemented
10#23757406C211yesmaximal, characteristic, complemented
11C4×#14578939C211yesmaximal, characteristic, complemented
12D4×#11929434C211yesmaximal, characteristic, complemented
13#23757267C211yesmaximal, characteristic, complemented
14C2×#19097691C211yesmaximal, characteristic, complemented
15#553×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×#14578942C2211yescharacteristic, complemented
19C2×#14578939C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21C22×#11929434C2211yescharacteristic
22C22×#11929434C2211yescharacteristic, complemented
23#19097759C2211yescharacteristic
24#19097756—21no—
25#19097750C2211yescharacteristic
26#19097745C2211yescharacteristic
27#19097744—21no—
28#19097739—21no—
29C4×D628C2211yescharacteristic
30C22⋊C4×D157—21no—
31#19097704C2211yescharacteristic, complemented
32#19097704C2211yescharacteristic, complemented
33#19097702C2211yescharacteristic, complemented
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097694C2211yescharacteristic, complemented
36#19097694C2211yescharacteristic, complemented
37#19097694C2211yescharacteristic, complemented
38#19097692C2211yescharacteristic, complemented
39#19097691C2211yescharacteristic
40#19097691C2211yescharacteristic
41#19097691C2211yescharacteristic
42#19097691C2211yescharacteristic
43#19097691C2211yescharacteristic
44#19097691C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic, complemented
47C4×#11929434C2211yescharacteristic, complemented
48#126×C157—21no—
49#125×C157C2211yescharacteristic, complemented
50#123×C157—21no—
51#121×C157—21no—
52#121×C157C2211yescharacteristic
53#121×C157C2211yescharacteristic, complemented
54C4×D4×C157—21no—
55C4×D4×C157C2211yescharacteristic
56C4×D4×C157C2211yescharacteristic, complemented
57#117×C157C2211yescharacteristic, complemented
58C2×C4⋊C4×C157C2211yescharacteristic
59C23×D157—21no—
60#14578954—22no—
61#14578954—41no—
62C2×Dic314C2311yescharacteristic
63C2×D628—22no—
64C2×D628—21no—
65C2×D628—41no—
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72#14578942—41no—
73#14578939—22no—
74#14578939—41no—
75C2×#11929434—41no—
76C2×#11929434—41no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88C2×D4×C157—21no—
89C2×D4×C157—21no—
90C2×D4×C157D411yescharacteristic
91C2×D4×C157C2311yescharacteristic
92C22×C4×C157—21no—
93C22×C4×C157—21no—
94C22×C4×C157C2311yescharacteristic
95C22×C4×C157C2311yescharacteristic
96C4⋊C4×C157—21no—
97C4⋊C4×C157—21no—
98C4⋊C4×C157—21no—
99C4⋊C4×C157D411yescharacteristic
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C22⋊C4×C157—21no—
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C22⋊C4×C157D411yescharacteristic
108C22⋊C4×C157D411yescharacteristic
109C42×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—42no—
114C22×D157—42no—
115C22×D157—81no—
116Dic314—21no—
117Dic314—21no—
118D628—21no—
119D628—21no—
120D628—21no—
121D628—42no—
122D628#4112yes—
123C22⋊D157—42no—
124C22⋊D157—81no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C4×D157—42no—
128C4×D157—81no—
129#11929434—21no—
130#11929434—21no—
131#11929434—21no—
132#11929434—21no—
133C23×C157—21no—
134C23×C157C2×D411yescharacteristic
135C23×C157C2×D411yescharacteristic
136D4×C157—22no—
137D4×C157—21no—
138D4×C157—21no—
139D4×C157—42no—
140D4×C157—81no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—81no—
149C2×C4×C157—81no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2×D411yescharacteristic
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2411yescharacteristic
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157#13512yes—
164C22×C157C22×D411yescharacteristic
165C2×D157—22no—
166C2×D157—41no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—84no—
172C4×C157—21no—
173C4×C157—21no—
174C4×C157—21no—
175C4×C157—21no—
176C4×C157—41no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157C2×#4111yescharacteristic
180C4×C157H(5,𝔽2)11yescharacteristic
181Dic157—41no—
182Dic157—41no—
183C2×C157—22no—
184C2×C157—41no—
185C2×C157—81no—
186C2×C157#54611yescharacteristic
187C2×C157C2×#13511yescharacteristic
188C2×C157#57611yescharacteristic
189D157—42no—
190D157—81no—
191C157#329711yescharacteristic, complemented, Sylow
192#3297—1571nomaximal, Sylow
193#553D15711yescharacteristic, complemented
194#546—1571no—
195#488—1571no—
196#484—1571no—
197#482—1571no—
198#478—1571no—
199#466—1571no—
200#465—1571no—
201#463—1571no—
202#449—1571no—
203#447—1571no—
204C4×D8—1571no—
205C8×D4—1571no—
206#416—1571no—
207C2×#102—1571no—
208C2×#41—1571no—
209C22×D4—1571no—
210C2×SD16—1571no—
211C2×D8—1571no—
212C2×D8—1571no—
213C22×C8—1571no—
214C22×C8—1571no—
215#126—21no—
216#125C2×D15711yescharacteristic, complemented
217#124—1571no—
218#123—21no—
219#123—3141no—
220C22⋊Q8—1571no—
221#121—21no—
222#121—3141no—
223#121—1571no—
224#121C2×D15711yescharacteristic
225#121C2×D15711yescharacteristic, complemented
226#120—3141no—
227C4×D4—21no—
228C4×D4—1571no—
229C4×D4C2×D15711yescharacteristic
230C4×D4C2×D15711yescharacteristic, complemented
231#117C2×D15711yescharacteristic, complemented
232C2×C4⋊C4C2×D15711yescharacteristic
233C2×C22⋊C4—3141no—
234#107—1571no—
235#107—1571no—
236#106—1571no—
237C4⋊C8—1571no—
238#103—1571no—
239#103—1571no—
240#103—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248C22⋊C8—1571no—
249C22⋊C8—1571no—
250C4×C8—1571no—
251C24—3141no—
252#41—3142no—
253#41—6281no—
254C2×Q8—1571no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3142no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—3141no—
261C2×D4—6281no—
262C2×D4—1571no—
263C2×D4—1571no—
264C2×D4C22⋊D15711yescharacteristic
265C2×D4C22×D15711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4C22×D15711yescharacteristic
271C22×C4C22×D15711yescharacteristic
272SD16—6281no—
273D8—3142no—
274D8—6281no—
275C2×C8—6281no—
276C2×C8—6281no—
277C2×C8—1571no—
278C2×C8—1571no—
279C2×C8—1571no—
280C2×C8—1571no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—3141no—
285C4⋊C4—1571no—
286C4⋊C4C22⋊D15711yescharacteristic
287C4⋊C4C22×D15711yescharacteristic
288C4⋊C4C22×D15711yescharacteristic
289C4⋊C4C22×D15711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4C22⋊D15711yescharacteristic
300C22⋊C4C22⋊D15711yescharacteristic
301C42—21no—
302C42C22×D15711yescharacteristic
303C23—21no—
304C23—3141no—
305C23—3141no—
306C23—6282no—
307C23—6282no—
308C23—12561no—
309C23C2×C22⋊D15711yescharacteristic
310C23C2×C22⋊D15711yescharacteristic
311Q8—3141no—
312Q8—3141no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—3141no—
319D4—3141no—
320D4—3141no—
321D4—6282no—
322D4—6282no—
323D4—1572no—
324D4—12561no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—3141no—
335C2×C4—3141no—
336C2×C4—6282no—
337C2×C4—12561no—
338C2×C4C2×C22⋊D15711yescharacteristic
339C2×C4C2×C22⋊D15711yescharacteristic
340C2×C4C2×C22⋊D15711yescharacteristic
341C2×C4C2×C22⋊D15711yescharacteristic
342C2×C4C23×D15711yescharacteristic
343C8—3141no—
344C8—3141no—
345C8—3141no—
346C8—3141no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3142no—
356C22—6281no—
357C22—6281no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—12564no—
362C22#1909779112yes—
363C22C22×C22⋊D15711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—6281no—
372C4—6281no—
373C4#41×D15711yescharacteristic
374C4#1909781711yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—6282no—
379C2—12561no—
380C2#2375791111yescharacteristic
381C2C2×#1909779111yescharacteristic
382C2#2375807611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#553×C157C2
socleC22×C157C22×D4
hypercenter#553D157
hypocenterC157#3297
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #23757919C2semidirect
2. #23757630C2semidirect
3. #23757608C2semidirect
4. #23757597C2semidirect
5. #23757570C2semidirect
6. #23757507C2semidirect
7. #23757500C2semidirect
8. #23757488C2semidirect
9. #23757416C2semidirect
10. #23757406C2semidirect
11. C4×#14578939C2semidirect
12. D4×#11929434C2semidirect
13. #23757267C2semidirect
14. C2×#19097691C2semidirect
15. #553×C157C2semidirect
18. C2×#14578942C22semidirect
22. C22×#11929434C22semidirect
31. #19097704C22semidirect
32. #19097704C22semidirect
33. #19097702C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097694C22semidirect
36. #19097694C22semidirect
37. #19097694C22semidirect
38. #19097692C22semidirect
46. C22⋊#11929434C22semidirect
47. C4×#11929434C22semidirect
49. #125×C157C22semidirect
53. #121×C157C22semidirect
56. C4×D4×C157C22semidirect
57. #117×C157C22semidirect
191. C157#3297semidirect
193. #553D157semidirect
216. #125C2×D157semidirect
225. #121C2×D157semidirect
230. C4×D4C2×D157semidirect
231. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579192
#237576302
#237576082
#237575972
#237575702
#237575072
#237575002
#237574882
#237574162
#237574062
C4×#145789392
D4×#119294342
#237572672
C2×#190976912
#553×C1572
#3297157