Finite Group Databasebeta

← #34111656

Subgroups — #34111656

Order 20096.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic, complemented
2C2×#19097792C211yesmaximal, characteristic, complemented
3#23757927C211yesmaximal, characteristic, complemented
4#23757916C211yesmaximal, characteristic, complemented
5#23757668C211yesmaximal, characteristic, complemented
6#23757530C211yesmaximal, characteristic, complemented
7#23757501C211yesmaximal, characteristic, complemented
8#23757479C211yesmaximal, characteristic, complemented
9#23757438C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11#23757425C211yesmaximal, characteristic, complemented
12#23757415C211yesmaximal, characteristic, complemented
13#23757383C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×#14578953C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic
20#19097795—24no—
21#19097792—24no—
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578944C2211yescharacteristic, complemented
24C2×#14578943C2211yescharacteristic
25C2×#14578943C2211yescharacteristic
26C2×#14578942C2211yescharacteristic
27C2×#14578942C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578937C2211yescharacteristic
30C2×C8×D157C2211yescharacteristic, complemented
31#19097771C2211yescharacteristic, complemented
32#19097755—21no—
33#19097755—21no—
34#19097753—21no—
35#19097751C2211yescharacteristic
36#19097751C2211yescharacteristic, complemented
37#19097747C2211yescharacteristic
38#19097745—21no—
39#19097744—21no—
40#19097742C2211yescharacteristic
41#19097741—21no—
42C4×D628C2211yescharacteristic
43C4×D628C2211yescharacteristic
44C2×C4⋊Dic157—21no—
45C22⋊C4×D157—21no—
46#19097702C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097692C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic
51#19097683C2211yescharacteristic
52#19097683C2211yescharacteristic, complemented
53#19097682C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×D8×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×D4×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic
61C23×D157—21no—
62#14578955—22no—
63#14578955—41no—
64#14578953—22no—
65#14578953—22no—
66#14578953—41no—
67#14578953—41no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70C2×D628—21no—
71C2×D628D411yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77D4×D157—22no—
78D4×D157—41no—
79C22×Dic157—21no—
80C22×Dic157—21no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157C2311yescharacteristic
84C2×C4×D157C2311yescharacteristic
85#14578944—22no—
86#14578944—41no—
87#14578943—22no—
88#14578943—22no—
89#14578942—22no—
90#14578942—41no—
91#14578939D414yescomplemented
92#14578937—22no—
93C2×#11929434C2311yescharacteristic
94C2×#11929434C2311yescharacteristic
95C8×D157—22no—
96C4⋊Dic157—21no—
97C4⋊Dic157—21no—
98C4⋊Dic157—21no—
99C4⋊Dic157—21no—
100C4⋊Dic157D411yescharacteristic
101C4⋊Dic157C2311yescharacteristic
102#14578933—21no—
103C22⋊Dic157—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931D411yescharacteristic
109#14578931D411yescharacteristic
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157—21no—
114SD16×C157—22no—
115D8×C157—22no—
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C4⋊C4×C157C2311yescharacteristic
119C22⋊C4×C157—21no—
120C22⋊C4×C157—21no—
121C22⋊C4×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—42no—
125C22×D157—42no—
126C22×D157—81no—
127C22×D157C2×D411yescharacteristic
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—21no—
133D628—21no—
134D628—41no—
135D628—42no—
136D628C2×D412yes—
137C22⋊D157—42no—
138C22⋊D157—42no—
139C22⋊D157—42no—
140C22⋊D157—81no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—21no—
144C2×Dic157—42no—
145C2×Dic157—42no—
146C2×Dic157C2×D411yescharacteristic
147C2×Dic157C2×D411yescharacteristic
148C4×D157—21no—
149C4×D157—21no—
150C4×D157—21no—
151C4×D157—21no—
152C4×D157—42no—
153C4×D157—81no—
154#11929434—21no—
155#11929434C2×D412yes—
156C23×C157—21no—
157C23×C157—21no—
158Q8×C157—21no—
159Q8×C157—21no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157C2×D412yes—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—42no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157—21no—
172C22×C157—22no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157C22×D411yescharacteristic
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—22no—
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—21no—
185C2×D157—21no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C4×C157—21no—
190C4×C157—21no—
191C4×C157—41no—
192C4×C157—41no—
193C4×C157C22×D411yescharacteristic
194C4×C157H(5,𝔽2)11yescharacteristic
195Dic157—21no—
196Dic157—21no—
197Dic157—41no—
198Dic157—41no—
199Dic157—41no—
200C2×C157—42no—
201C2×C157—81no—
202C2×C157D4211yescharacteristic
203C2×C157#57511yescharacteristic
204C2×C157#57711yescharacteristic
205D157—41no—
206D157—41no—
207D157—42no—
208C157#326711yescharacteristic, complemented, Sylow
209#3267—1571nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1571no—
211C2×#135—1571no—
212#547—1571no—
213#546—1571no—
214#497—1571no—
215#470—1571no—
216#465—1571no—
217#463D15711yescharacteristic, complemented
218#461—1571no—
219#452—1571no—
220#452—1571no—
221#450—1571no—
222#449—1571no—
223#442—1571no—
224#435—1571no—
225C2×#41—1571no—
226C2×#41—1571no—
227C2×#41—1571no—
228C22×D4—1571no—
229AGL(1,ℤ/8)—3144no—
230#135—3144no—
231C2×Q16—1571no—
232C2×Q16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×SD16C2×D15711yescharacteristic, complemented
238C2×D8—1571no—
239C2×D8C2×D15711yescharacteristic
240C2×M16—1571no—
241C22×C8—1571no—
242C4⋊Q8—1571no—
243#124C2×D15711yescharacteristic
244#123—3141no—
245#123—3141no—
246#123—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8—1571no—
249C22⋊Q8—1571no—
250#121—3141no—
251#121—3141no—
252#121—1571no—
253#120—3141no—
254C4×D4—1571no—
255C4×D4—1571no—
256C4×D4C2×D15711yescharacteristic, complemented
257C2×C4⋊C4—3141no—
258C2×C22⋊C4—3141no—
259#106—1571no—
260C4⋊C8C2×D15711yescharacteristic, complemented
261#103—1571no—
262#103—1571no—
263#103C2×D15711yescharacteristic, complemented
264#102—1571no—
265#102—1571no—
266#102C2×D15711yescharacteristic
267C22⋊C8—1571no—
268C22⋊C8—1571no—
269#97—1571no—
270C24—3141no—
271#41—3142no—
272#41—3142no—
273#41—3142no—
274#41—6281no—
275#41—6281no—
276#41—6281no—
277C2×Q8—1571no—
278C2×Q8—1571no—
279C2×Q8C22×D15711yescharacteristic
280C2×D4—3141no—
281C2×D4—3142no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—6281no—
287C2×D4—1571no—
288C2×D4—1571no—
289C2×D4C22×D15711yescharacteristic
290C2×D4C22×D15711yescharacteristic
291C22×C4—21no—
292C22×C4—3141no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—1571no—
297C22×C4—1571no—
298Q16—3142no—
299Q16—6281no—
300SD16—22no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—6281no—
305D8—22no—
306D8—1574no—
307M16—3142no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—3141no—
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—3141no—
318C4⋊C4—1571no—
319C4⋊C4—1571no—
320C4⋊C4C22×D15711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4—1571no—
330C22⋊C4—1571no—
331C42C22×D15711yescharacteristic
332C23—21no—
333C23—21no—
334C23—3141no—
335C23—6282no—
336C23—6282no—
337C23—1571no—
338C23—1571no—
339C23—12561no—
340Q8—21no—
341Q8—21no—
342Q8—3141no—
343Q8—3141no—
344Q8—3141no—
345Q8—3141no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—3141no—
350D4—6281no—
351D4—6282no—
352D4—6282no—
353D4—6282no—
354D4—6282no—
355D4—1572no—
356D4—12561no—
357D4D4×D15712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—3141no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4—12561no—
374C2×C4D4×D15711yescharacteristic
375C2×C4D4×D15711yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—21no—
378C8—21no—
379C8—3141no—
380C8—1572no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3141no—
388C22—3141no—
389C22—3141no—
390C22—3141no—
391C22—3142no—
392C22—3141no—
393C22—3141no—
394C22—6281no—
395C22—6281no—
396C22—12564no—
397C22C2×D4×D15711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3141no—
403C4—3141no—
404C4—6281no—
405C4—6281no—
406C4—6281no—
407C4C2×D4×D15711yescharacteristic
408C4#1909781711yescharacteristic
409C2—42no—
410C2—81no—
411C2—6281no—
412C2—6281no—
413C2—6282no—
414C2#2375790711yescharacteristic
415C2#2375806911yescharacteristic
416C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
hypercenter#463D157
hypocenterC157#3267
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×#19097792C2semidirect
3. #23757927C2semidirect
4. #23757916C2semidirect
5. #23757668C2semidirect
6. #23757530C2semidirect
7. #23757501C2semidirect
8. #23757479C2semidirect
9. #23757438C2semidirect
10. #23757432C2semidirect
11. #23757425C2semidirect
12. #23757415C2semidirect
13. #23757383C2semidirect
14. #23757343C2semidirect
15. #463×C157C2semidirect
18. C2×#14578953C22semidirect
22. C2×#14578944C22semidirect
23. C2×#14578944C22semidirect
30. C2×C8×D157C22semidirect
31. #19097771C22semidirect
36. #19097751C22semidirect
46. #19097702C22semidirect
47. #19097696C22semidirect
48. #19097696C22semidirect
49. #19097692C22semidirect
52. #19097683C22semidirect
53. #19097682C22semidirect
54. C2×SD16×C157C22semidirect
57. C4×D4×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
91. #14578939D4semidirect
208. C157#3267semidirect
217. #463D157semidirect
237. C2×SD16C2×D157semidirect
256. C4×D4C2×D157semidirect
260. C4⋊C8C2×D157semidirect
263. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
C2×#190977922
#237579272
#237579162
#237576682
#237575302
#237575012
#237574792
#237574382
#237574322
#237574252
#237574152
#237573832
#237573432
#463×C1572
#3267157