Finite Group Databasebeta

← #34111900

Subgroups — #34111900

Order 20096.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758084C211yesmaximal, characteristic, complemented
2#23758080C211yesmaximal, characteristic, complemented
3#23758076C211yesmaximal, characteristic, complemented
4#23757661C211yesmaximal, characteristic, complemented
5#23757541C211yesmaximal, characteristic, complemented
6#23757540C211yesmaximal, characteristic, complemented
7#23757536C211yesmaximal, characteristic, complemented
8#23757534C211yesmaximal, characteristic, complemented
9#23757451C211yesmaximal, characteristic, complemented
10#23757444C211yesmaximal, characteristic, complemented
11#23757441C211yesmaximal, characteristic, complemented
12#23757439C211yesmaximal, characteristic, complemented
13#23757395C211yesmaximal, characteristic, complemented
14#23757393C211yesmaximal, characteristic, complemented
15#576×C157C211yesmaximal, characteristic, complemented
16#19097818C2211yescharacteristic, complemented
17#19097817C2211yescharacteristic, complemented
18#19097803—21no—
19#19097801—21no—
20#19097801C2211yescharacteristic, complemented
21#19097801C2211yescharacteristic, complemented
22#19097797—21no—
23#19097797—21no—
24#19097797C2211yescharacteristic, complemented
25#19097797C2211yescharacteristic, complemented
26#19097796—21no—
27#19097796C2211yescharacteristic
28#19097796C2211yescharacteristic, complemented
29#19097795—21no—
30#19097792—21no—
31#19097791—21no—
32#19097791—21no—
33#19097791C2211yescharacteristic
34C2×#14578943C2211yescharacteristic, complemented
35C2×#14578942—21no—
36C2×#14578942C2211yescharacteristic, complemented
37SD16×D157—21no—
38C2×#14578939—21no—
39C2×#14578939C2211yescharacteristic, complemented
40#19097778C2211yescharacteristic, complemented
41#19097778C2211yescharacteristic, complemented
42#19097777C2211yescharacteristic, complemented
43#19097767C2211yescharacteristic, complemented
44#19097758C2211yescharacteristic, complemented
45#19097707C2211yescharacteristic, complemented
46#19097706C2211yescharacteristic, complemented
47#19097698C2211yescharacteristic, complemented
48#19097698C2211yescharacteristic, complemented
49#19097698C2211yescharacteristic, complemented
50#19097697C2211yescharacteristic, complemented
51#19097689C2211yescharacteristic, complemented
52#19097688C2211yescharacteristic, complemented
53#19097688C2211yescharacteristic, complemented
54#19097687C2211yescharacteristic, complemented
55#19097681C2211yescharacteristic, complemented
56H(5,𝔽2)×C157C2211yescharacteristic
57H(5,𝔽2)×C157C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C157—21no—
59AGL(1,ℤ/8)×C157—21no—
60AGL(1,ℤ/8)×C157C2211yescharacteristic
61AGL(1,ℤ/8)×C157C2211yescharacteristic, complemented
62#135×C157—21no—
63#135×C157C2211yescharacteristic
64C2×D8×C157—21no—
65C2×D8×C157C2211yescharacteristic, complemented
66#131×C157C2211yescharacteristic, complemented
67#14578955—21no—
68#14578954—21no—
69#14578954—21no—
70#14578954C2311yescharacteristic
71#14578953—21no—
72#14578953—41no—
73C2×Dic314—21no—
74C2×Dic314C2311yescharacteristic
75Q8×D157—21no—
76C2×D628—21no—
77C2×D628C2311yescharacteristic
78C2×C22⋊D157—41no—
79D4×D157—21no—
80D4×D157—41no—
81#14578944—21no—
82#14578944—21no—
83#14578944D411yescharacteristic, complemented
84#14578943—21no—
85#14578943—21no—
86#14578942—21no—
87#14578942—21no—
88#14578942—21no—
89#14578942—21no—
90#14578942—21no—
91#14578942—21no—
92#14578942D411yescharacteristic, complemented
93#14578941—21no—
94#14578941—21no—
95#14578941D411yescharacteristic, complemented
96#14578939—21no—
97#14578939—21no—
98#14578939—21no—
99#14578939—21no—
100#14578939—21no—
101#14578939—21no—
102#14578939D411yescharacteristic, complemented
103#14578938—21no—
104#14578938—21no—
105#14578938C2311yescharacteristic
106#14578938C2311yescharacteristic
107#14578938C2311yescharacteristic
108#14578937—21no—
109C2×#11929434—21no—
110C2×#11929434—21no—
111C2×#11929434C2311yescharacteristic
112C8×D157—21no—
113C22⋊Dic157—41no—
114C4×Dic157C2311yescharacteristic
115#41×C157—21no—
116#41×C157—21no—
117#41×C157—21no—
118#41×C157—21no—
119#41×C157C2311yescharacteristic
120#41×C157C2311yescharacteristic
121#41×C157C2311yescharacteristic
122C2×D4×C157—21no—
123C2×D4×C157—21no—
124C2×D4×C157—21no—
125C2×D4×C157—21no—
126C2×D4×C157—41no—
127C2×D4×C157—41no—
128C2×D4×C157C2311yescharacteristic
129C2×D4×C157C2311yescharacteristic
130Q16×C157D411yescharacteristic, complemented
131SD16×C157—21no—
132SD16×C157—21no—
133SD16×C157D411yescharacteristic, complemented
134SD16×C157D411yescharacteristic, complemented
135D8×C157—21no—
136D8×C157—21no—
137D8×C157—21no—
138D8×C157—21no—
139D8×C157D411yescharacteristic, complemented
140M16×C157—21no—
141M16×C157C2311yescharacteristic
142C2×C8×C157—21no—
143C2×C8×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157—41no—
146Dic314—21no—
147Dic314—21no—
148Dic314—41no—
149Dic314C2×D411yescharacteristic
150D628—21no—
151D628—21no—
152D628—41no—
153D628C2×D411yescharacteristic
154C22⋊D157—21no—
155C22⋊D157—41no—
156C22⋊D157—41no—
157C22⋊D157—82no—
158C2×Dic157—21no—
159C2×Dic157—21no—
160C2×Dic157—41no—
161C4×D157—21no—
162C4×D157—41no—
163C4×D157—41no—
164#11929434—21no—
165#11929434—21no—
166#11929434—21no—
167#11929434C2×D411yescharacteristic
168#11929434C2×D411yescharacteristic
169C23×C157—21no—
170C23×C157—21no—
171C23×C157—41no—
172C23×C157—41no—
173Q8×C157C2×D411yescharacteristic
174Q8×C157C2×D411yescharacteristic
175Q8×C157C2×D411yescharacteristic
176D4×C157—21no—
177D4×C157—21no—
178D4×C157—21no—
179D4×C157—21no—
180D4×C157—21no—
181D4×C157—21no—
182D4×C157—21no—
183D4×C157—41no—
184D4×C157—41no—
185D4×C157—41no—
186D4×C157—41no—
187D4×C157C2×D411yescharacteristic
188D4×C157C2×D411yescharacteristic
189D4×C157C2×D411yescharacteristic
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157—41no—
194C2×C4×C157—41no—
195C2×C4×C157C2411yescharacteristic
196C8×C157—21no—
197C8×C157C2×D411yescharacteristic
198C8×C157C2×D411yescharacteristic
199C22×C157—21no—
200C22×C157—21no—
201C22×C157—21no—
202C22×C157—41no—
203C22×C157—41no—
204C22×C157—81no—
205C22×C157—81no—
206C22×C157—161no—
207C22×C157—161no—
208C22×C157H(5,𝔽2)11yescharacteristic
209C2×D157—21no—
210C2×D157—41no—
211C2×D157—81no—
212C2×D157—161no—
213C4×C157—21no—
214C4×C157—21no—
215C4×C157—21no—
216C4×C157C22×D411yescharacteristic
217C4×C157C22×D411yescharacteristic
218Dic157—21no—
219Dic157—41no—
220Dic157—41no—
221C2×C157—21no—
222C2×C157—41no—
223C2×C157—41no—
224C2×C157—41no—
225C2×C157—81no—
226C2×C157—81no—
227C2×C157D4211yescharacteristic
228D157—41no—
229D157—81no—
230C157#328811yescharacteristic, complemented, Sylow
231#3288—1571nomaximal, Sylow
232#577—1571no—
233#577—1571no—
234#576—1571no—
235#576D15711yescharacteristic, complemented
236#496—1571no—
237#472—1571no—
238#472—1571no—
239#471—1571no—
240#471—1571no—
241#455—1571no—
242#454—1571no—
243#453—1571no—
244#453—1571no—
245#444—1571no—
246#444—1571no—
2472−1+4—1571no—
248H(5,𝔽2)—1571no—
249H(5,𝔽2)C2×D15711yescharacteristic
250H(5,𝔽2)C2×D15711yescharacteristic, complemented
251#137—3141no—
252#137—3141no—
253#137—1571no—
254#137—1571no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—3141no—
258AGL(1,ℤ/8)—3141no—
259AGL(1,ℤ/8)—3141no—
260AGL(1,ℤ/8)—3141no—
261AGL(1,ℤ/8)—1571no—
262AGL(1,ℤ/8)—1571no—
263AGL(1,ℤ/8)—1571no—
264AGL(1,ℤ/8)—1571no—
265AGL(1,ℤ/8)C2×D15711yescharacteristic
266AGL(1,ℤ/8)C2×D15711yescharacteristic, complemented
267#135—21no—
268#135—3141no—
269#135—3141no—
270#135—3141no—
271#135—1571no—
272#135C2×D15711yescharacteristic
273C2×SD16—3141no—
274C2×SD16—3141no—
275C2×SD16—1571no—
276C2×SD16—1571no—
277C2×D8—21no—
278C2×D8—3141no—
279C2×D8—1571no—
280C2×D8C2×D15711yescharacteristic, complemented
281#131—1571no—
282#131—1571no—
283#131—1571no—
284#131C2×D15711yescharacteristic, complemented
285#127—1571no—
286#124—1571no—
287#108—1571no—
288#108—1571no—
289#104—1571no—
290#104—1571no—
291#104—1571no—
292#104—1571no—
293#101—1571no—
294#100—1571no—
295#100—1571no—
296#100—1571no—
297#97—1571no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—3141no—
303#41—3141no—
304#41—3141no—
305#41—3141no—
306#41—6281no—
307#41—1571no—
308#41C22×D15711yescharacteristic
309#41C22×D15711yescharacteristic
310#41C22×D15711yescharacteristic
311C2×Q8—3141no—
312C2×Q8—3141no—
313C2×Q8—1571no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—3141no—
321C2×D4—3141no—
322C2×D4—6281no—
323C2×D4—6281no—
324C2×D4—1571no—
325C2×D4C22×D15711yescharacteristic
326C2×D4C22×D15711yescharacteristic
327Q16—3141no—
328Q16—3141no—
329Q16—1571no—
330Q16C22⋊D15711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342SD16—3141no—
343SD16—1571no—
344SD16—1571no—
345SD16C22⋊D15711yescharacteristic, complemented
346SD16C22⋊D15711yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—3141no—
352D8—3141no—
353D8—3141no—
354D8—3141no—
355D8—3141no—
356D8—3141no—
357D8—1571no—
358D8C22⋊D15711yescharacteristic, complemented
359M16—21no—
360M16—3141no—
361M16—3141no—
362M16—3141no—
363M16—1571no—
364M16—1571no—
365M16—1571no—
366M16C22×D15711yescharacteristic
367C2×C8—21no—
368C2×C8—3141no—
369C2×C8—3141no—
370C2×C8—3141no—
371C2×C8—1571no—
372C2×C8C22×D15711yescharacteristic
373C22⋊C4—6281no—
374C42—1571no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—3141no—
380C23—6281no—
381Q8—3141no—
382Q8—3141no—
383Q8—6281no—
384Q8—1571no—
385Q8D4×D15711yescharacteristic
386Q8C2×C22⋊D15711yescharacteristic
387Q8C2×C22⋊D15711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—3141no—
400D4—3141no—
401D4—3141no—
402D4—6281no—
403D4—6281no—
404D4—6281no—
405D4—1571no—
406D4—12562no—
407D4D4×D15711yescharacteristic
408D4C2×C22⋊D15711yescharacteristic
409D4C2×C22⋊D15711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—3141no—
416C2×C4—3141no—
417C2×C4—3141no—
418C2×C4—6281no—
419C2×C4—6281no—
420C2×C4—6281no—
421C2×C4C23×D15711yescharacteristic
422C8—21no—
423C8—3141no—
424C8—3141no—
425C8—3141no—
426C8—1571no—
427C8—1571no—
428C8C2×C22⋊D15711yescharacteristic
429C8C2×C22⋊D15711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—3141no—
440C22—6281no—
441C22—25121no—
442C22—12561no—
443C22#1909781611yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—3141no—
448C4—6281no—
449C4—6281no—
450C4C2×D4×D15711yescharacteristic
451C4C22×C22⋊D15711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—6281no—
459C2—12561no—
460C2D4×C22⋊D15711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#576×C157C2
socleC2×C157D42
hypercenter#576D157
hypocenterC157#3288
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #23758084C2semidirect
2. #23758080C2semidirect
3. #23758076C2semidirect
4. #23757661C2semidirect
5. #23757541C2semidirect
6. #23757540C2semidirect
7. #23757536C2semidirect
8. #23757534C2semidirect
9. #23757451C2semidirect
10. #23757444C2semidirect
11. #23757441C2semidirect
12. #23757439C2semidirect
13. #23757395C2semidirect
14. #23757393C2semidirect
15. #576×C157C2semidirect
16. #19097818C22semidirect
17. #19097817C22semidirect
20. #19097801C22semidirect
21. #19097801C22semidirect
24. #19097797C22semidirect
25. #19097797C22semidirect
28. #19097796C22semidirect
34. C2×#14578943C22semidirect
36. C2×#14578942C22semidirect
39. C2×#14578939C22semidirect
40. #19097778C22semidirect
41. #19097778C22semidirect
42. #19097777C22semidirect
43. #19097767C22semidirect
44. #19097758C22semidirect
45. #19097707C22semidirect
46. #19097706C22semidirect
47. #19097698C22semidirect
48. #19097698C22semidirect
49. #19097698C22semidirect
50. #19097697C22semidirect
51. #19097689C22semidirect
52. #19097688C22semidirect
53. #19097688C22semidirect
54. #19097687C22semidirect
55. #19097681C22semidirect
57. H(5,𝔽2)×C157C22semidirect
61. AGL(1,ℤ/8)×C157C22semidirect
65. C2×D8×C157C22semidirect
66. #131×C157C22semidirect
83. #14578944D4semidirect
92. #14578942D4semidirect
95. #14578941D4semidirect
102. #14578939D4semidirect
130. Q16×C157D4semidirect
133. SD16×C157D4semidirect
134. SD16×C157D4semidirect
139. D8×C157D4semidirect
230. C157#3288semidirect
235. #576D157semidirect
250. H(5,𝔽2)C2×D157semidirect
266. AGL(1,ℤ/8)C2×D157semidirect
280. C2×D8C2×D157semidirect
284. #131C2×D157semidirect
330. Q16C22⋊D157semidirect
345. SD16C22⋊D157semidirect
346. SD16C22⋊D157semidirect
358. D8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580842
#237580802
#237580762
#237576612
#237575412
#237575402
#237575362
#237575342
#237574512
#237574442
#237574412
#237574392
#237573952
#237573932
#576×C1572
#3288157