Finite Group Databasebeta

← #34111668

Subgroups — #34111668

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×#19097799C211yesmaximal, characteristic, complemented
2C2×#19097794C211yesmaximal, characteristic, complemented
3#23757925C211yesmaximal, characteristic, complemented
4#23757914C211yesmaximal, characteristic, complemented
5#23757666C211yesmaximal, characteristic, complemented
6#23757503C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757477C211yesmaximal, characteristic, complemented
9#23757438C211yesmaximal, characteristic, complemented
10#23757433C211yesmaximal, characteristic, complemented
11#23757423C211yesmaximal, characteristic, complemented
12#23757421C211yesmaximal, characteristic, complemented
13#23757384C211yesmaximal, characteristic, complemented
14#23757342C211yesmaximal, characteristic, complemented
15#470×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578955C2211yescharacteristic, complemented
18C2×#14578954C2211yescharacteristic
19C22×D628C2211yescharacteristic
20#19097799—24no—
21#19097794—24no—
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic
24C2×#14578943C2211yescharacteristic
25C2×#14578941C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578941C2211yescharacteristic, complemented
28C2×D1256C2211yescharacteristic
29C2×D1256C2211yescharacteristic
30C2×C8×D157C2211yescharacteristic, complemented
31#19097769C2211yescharacteristic, complemented
32#19097760C2211yescharacteristic
33#19097756—21no—
34#19097756—21no—
35#19097753—21no—
36#19097751C2211yescharacteristic, complemented
37#19097746—21no—
38#19097746C2211yescharacteristic
39#19097744—21no—
40#19097741—21no—
41#19097733C2211yescharacteristic
42#19097733C2211yescharacteristic, complemented
43C4×C22⋊D157C2211yescharacteristic
44C4⋊C4×D157—21no—
45C22⋊C4×D157—21no—
46#19097700C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097691C2211yescharacteristic, complemented
51C22⋊#11929434C2211yescharacteristic
52#19097683C2211yescharacteristic, complemented
53#19097681C2211yescharacteristic, complemented
54C2×Q16×C157C2211yescharacteristic, complemented
55C2×M16×C157C2211yescharacteristic
56C22⋊Q8×C157C2211yescharacteristic
57C22⋊Q8×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#103×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578955—22no—
63#14578955—22no—
64#14578955—41no—
65#14578955—41no—
66#14578954—22no—
67#14578954—41no—
68C2×Dic314C2311yescharacteristic
69C2×D628—22no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628—21no—
74C2×D628—41no—
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×D628C2311yescharacteristic
78C2×C22⋊D157—21no—
79C2×C22⋊D157D411yescharacteristic
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85C2×C4×D157C2311yescharacteristic
86#14578944—41no—
87#14578943—22no—
88#14578943—22no—
89#14578941—22no—
90#14578941—22no—
91#14578941—41no—
92D1256—22no—
93D1256D414yescomplemented
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C8×D157—22no—
97C4⋊Dic157—21no—
98#14578933—21no—
99#14578933—21no—
100#14578933D411yescharacteristic
101C22⋊Dic157D411yescharacteristic
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931D411yescharacteristic
109C4×Dic157C2311yescharacteristic
110C2×Q8×C157C2311yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C22×C4×C157C2311yescharacteristic
113Q16×C157—22no—
114M16×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C4⋊C4×C157—21no—
118C4⋊C4×C157—21no—
119C4⋊C4×C157C2311yescharacteristic
120C4⋊C4×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
122C22⋊C4×C157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—81no—
129C22×D157C2×D411yescharacteristic
130Dic314—21no—
131Dic314—21no—
132D628—21no—
133D628—21no—
134D628—21no—
135D628—21no—
136D628—42no—
137D628—42no—
138D628—42no—
139D628—81no—
140D628C2×D412yes—
141D628C2×D412yes—
142C22⋊D157—41no—
143C22⋊D157—42no—
144C2×Dic157—21no—
145C2×Dic157C2×D411yescharacteristic
146C2×Dic157C2×D411yescharacteristic
147C2×Dic157C2×D411yescharacteristic
148C4×D157—21no—
149C4×D157—21no—
150C4×D157—42no—
151C4×D157—42no—
152C4×D157—42no—
153C4×D157—42no—
154C4×D157—81no—
155#11929434—21no—
156#11929434—21no—
157C23×C157C2×D411yescharacteristic
158Q8×C157—21no—
159Q8×C157—21no—
160Q8×C157—21no—
161Q8×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—21no—
175C22×C157C22×D411yescharacteristic
176C2×D157—21no—
177C2×D157—22no—
178C2×D157—22no—
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C4×C157—21no—
190C4×C157—41no—
191C4×C157—41no—
192C4×C157—41no—
193C4×C157—41no—
194C4×C157C22×D411yescharacteristic
195C4×C157H(5,𝔽2)11yescharacteristic
196Dic157—21no—
197Dic157—21no—
198Dic157—21no—
199Dic157—41no—
200C2×C157—41no—
201C2×C157D4211yescharacteristic
202C2×C157#57511yescharacteristic
203C2×C157#57711yescharacteristic
204D157—41no—
205D157—42no—
206D157—42no—
207D157—81no—
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#470D15711yescharacteristic, complemented
216#465—1571no—
217#463—1571no—
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×Q16C2×D15711yescharacteristic, complemented
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×SD16—1571no—
238C2×D8—1571no—
239C2×D8—1571no—
240C2×M16C2×D15711yescharacteristic
241C22×C8—1571no—
242C4⋊Q8—1571no—
243#124—1571no—
244#123—3141no—
245#123—3141no—
246#123—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8C2×D15711yescharacteristic
249C22⋊Q8C2×D15711yescharacteristic
250#121—3141no—
251#121—3141no—
252#121—1571no—
253#120—3141no—
254C4×D4—1571no—
255C4×D4—1571no—
256C4×D4—1571no—
257C2×C4⋊C4—3141no—
258C2×C22⋊C4—3141no—
259#106C2×D15711yescharacteristic, complemented
260C4⋊C8—1571no—
261#103—1571no—
262#103C2×D15711yescharacteristic, complemented
263#103C2×D15711yescharacteristic, complemented
264#102—1571no—
265#102—1571no—
266#102—1571no—
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×Q8C22×D15711yescharacteristic
279C2×Q8C22×D15711yescharacteristic
280C2×D4—3142no—
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—6281no—
287C2×D4—1571no—
288C2×D4—1571no—
289C2×D4—1571no—
290C2×D4—1571no—
291C22×C4—3141no—
292C22×C4—3141no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—1571no—
297C22×C4C22×D15711yescharacteristic
298Q16—22no—
299Q16—6281no—
300SD16—3142no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—6281no—
305D8—3142no—
306D8—1574no—
307M16—22no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—3141no—
318C4⋊C4—1571no—
319C4⋊C4C22×D15711yescharacteristic
320C4⋊C4C22×D15711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4—1571no—
330C22⋊C4—1571no—
331C42—1571no—
332C23—3141no—
333C23—3141no—
334C23—3141no—
335C23—6282no—
336C23—6282no—
337C23—1571no—
338C23—12561no—
339C23D4×D15711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3141no—
345Q8—3141no—
346D4—3141no—
347D4—3141no—
348D4—3141no—
349D4—3141no—
350D4—6282no—
351D4—6282no—
352D4—6282no—
353D4—6281no—
354D4—6282no—
355D4—1572no—
356D4—1572no—
357D4—12561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—6282no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4—1571no—
374C2×C4—12561no—
375C2×C4D4×D15711yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—21no—
378C8—3141no—
379C8—3141no—
380C8D4×D15712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—3141no—
385C22—3141no—
386C22—3142no—
387C22—3142no—
388C22—3141no—
389C22—6281no—
390C22—6281no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22—6281no—
396C22—12564no—
397C22C2×D4×D15711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3141no—
404C4—3141no—
405C4—3141no—
406C4—6281no—
407C4C2×D4×D15711yescharacteristic
408C4#1909781711yescharacteristic
409C2—41no—
410C2—6281no—
411C2—6282no—
412C2—6282no—
413C2—12561no—
414C2#2375790711yescharacteristic
415C2#2375807211yescharacteristic
416C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#470×C157C2
socleC22×C157C22×D4
hypercenter#470D157
hypocenterC157#3267
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×#19097794C2semidirect
3. #23757925C2semidirect
4. #23757914C2semidirect
5. #23757666C2semidirect
6. #23757503C2semidirect
7. #23757491C2semidirect
8. #23757477C2semidirect
9. #23757438C2semidirect
10. #23757433C2semidirect
11. #23757423C2semidirect
12. #23757421C2semidirect
13. #23757384C2semidirect
14. #23757342C2semidirect
15. #470×C157C2semidirect
17. C2×#14578955C22semidirect
22. C2×#14578944C22semidirect
27. C2×#14578941C22semidirect
30. C2×C8×D157C22semidirect
31. #19097769C22semidirect
36. #19097751C22semidirect
42. #19097733C22semidirect
46. #19097700C22semidirect
47. #19097694C22semidirect
50. #19097691C22semidirect
52. #19097683C22semidirect
53. #19097681C22semidirect
54. C2×Q16×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
60. #103×C157C22semidirect
93. D1256D4semidirect
208. C157#3267semidirect
215. #470D157semidirect
232. C2×Q16C2×D157semidirect
259. #106C2×D157semidirect
262. #103C2×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×#190977992
C2×#190977942
#237579252
#237579142
#237576662
#237575032
#237574912
#237574772
#237574382
#237574332
#237574232
#237574212
#237573842
#237573422
#470×C1572
#3267157