Finite Group Databasebeta

← #34111901

Subgroups — #34111901

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#23758082C211yesmaximal, characteristic, complemented
2#23758079C211yesmaximal, characteristic, complemented
3#23758078C211yesmaximal, characteristic, complemented
4#23758077C211yesmaximal, characteristic, complemented
5#23757659C211yesmaximal, characteristic, complemented
6#23757544C211yesmaximal, characteristic, complemented
7#23757542C211yesmaximal, characteristic, complemented
8#23757539C211yesmaximal, characteristic, complemented
9#23757449C211yesmaximal, characteristic, complemented
10#23757448C211yesmaximal, characteristic, complemented
11#23757443C211yesmaximal, characteristic, complemented
12#23757439C211yesmaximal, characteristic, complemented
13#23757394C211yesmaximal, characteristic, complemented
14#23757393C211yesmaximal, characteristic, complemented
15#471×C157C211yesmaximal, characteristic, complemented
16#19097819C2211yescharacteristic, complemented
17#19097817C2211yescharacteristic
18#19097817C2211yescharacteristic, complemented
19#19097816C2211yescharacteristic, complemented
20#19097803—21no—
21#19097802C2211yescharacteristic, complemented
22#19097800—21no—
23#19097799—21no—
24#19097799—21no—
25#19097799C2211yescharacteristic
26#19097799C2211yescharacteristic, complemented
27#19097799C2211yescharacteristic, complemented
28#19097799C2211yescharacteristic, complemented
29#19097798—21no—
30#19097798—21no—
31#19097798—21no—
32#19097797C2211yescharacteristic, complemented
33#19097795—21no—
34#19097794—21no—
35#19097793—21no—
36#19097793C2211yescharacteristic
37#19097792—21no—
38#19097792—21no—
39#19097791C2211yescharacteristic
40C2×#14578941C2211yescharacteristic, complemented
41SD16×D157—21no—
42SD16×D157—21no—
43C2×D1256—21no—
44C2×D1256C2211yescharacteristic, complemented
45C2×#14578939C2211yescharacteristic, complemented
46D8×D157—21no—
47#19097779C2211yescharacteristic, complemented
48#19097779C2211yescharacteristic, complemented
49#19097777C2211yescharacteristic, complemented
50#19097767C2211yescharacteristic, complemented
51#19097760C2211yescharacteristic, complemented
52#19097706C2211yescharacteristic, complemented
53#19097698C2211yescharacteristic, complemented
54#19097697C2211yescharacteristic, complemented
55#19097697C2211yescharacteristic, complemented
56#19097697C2211yescharacteristic, complemented
57#19097687C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59#19097681C2211yescharacteristic, complemented
60#137×C157C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C157C2211yescharacteristic
62C2×SD16×C157C2211yescharacteristic, complemented
63#131×C157C2211yescharacteristic, complemented
64#108×C157C2211yescharacteristic, complemented
65#101×C157C2211yescharacteristic, complemented
66#100×C157C2211yescharacteristic, complemented
67#14578955—21no—
68#14578955—21no—
69#14578955—21no—
70#14578955—21no—
71#14578954—21no—
72#14578954—21no—
73#14578954—41no—
74#14578954C2311yescharacteristic
75#14578954C2311yescharacteristic
76#14578954C2311yescharacteristic
77#14578953—21no—
78#14578953—21no—
79Q8×D157—21no—
80Q8×D157—21no—
81C2×D628—21no—
82C2×D628—21no—
83C2×D628C2311yescharacteristic
84C2×D628C2311yescharacteristic
85C2×C22⋊D157—41no—
86C2×C22⋊D157—41no—
87D4×D157—21no—
88D4×D157—21no—
89D4×D157—21no—
90D4×D157—21no—
91D4×D157—41no—
92D4×D157—41no—
93Dic628D411yescharacteristic, complemented
94#14578944—21no—
95#14578944D411yescharacteristic, complemented
96#14578943—21no—
97#14578943—21no—
98#14578943—21no—
99#14578943—21no—
100#14578943D411yescharacteristic, complemented
101#14578943D411yescharacteristic, complemented
102#14578942—21no—
103#14578942D411yescharacteristic, complemented
104#14578941—21no—
105#14578941—21no—
106#14578941—21no—
107#14578941D411yescharacteristic, complemented
108D1256—21no—
109D1256—21no—
110D1256—21no—
111D1256—21no—
112D1256—21no—
113D1256—21no—
114D1256D411yescharacteristic, complemented
115#14578939—21no—
116#14578939—21no—
117#14578939—21no—
118#14578939D411yescharacteristic, complemented
119#14578938C2311yescharacteristic
120#14578937—21no—
121#14578937—21no—
122#14578937—21no—
123C2×#11929434C2311yescharacteristic
124C8×D157—21no—
125C8×D157—21no—
126C8×D157—21no—
127#14578931—41no—
128C4×Dic157C2311yescharacteristic
129#41×C157C2311yescharacteristic
130C2×Q8×C157C2311yescharacteristic
131C2×D4×C157C2311yescharacteristic
132Q16×C157—21no—
133SD16×C157—21no—
134SD16×C157—21no—
135SD16×C157—21no—
136SD16×C157—21no—
137D8×C157—21no—
138M16×C157—21no—
139M16×C157C2311yescharacteristic
140M16×C157C2311yescharacteristic
141M16×C157C2311yescharacteristic
142C2×C8×C157—21no—
143C2×C8×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157—21no—
146C22×D157—41no—
147C22×D157—41no—
148C22×D157—41no—
149Dic314—41no—
150Dic314C2×D411yescharacteristic
151Dic314C2×D411yescharacteristic
152Dic314C2×D411yescharacteristic
153D628—21no—
154D628—21no—
155D628—21no—
156D628—21no—
157D628—41no—
158D628—41no—
159D628—41no—
160D628C2×D411yescharacteristic
161D628C2×D411yescharacteristic
162D628C2×D411yescharacteristic
163C22⋊D157—21no—
164C22⋊D157—21no—
165C22⋊D157—21no—
166C22⋊D157—41no—
167C22⋊D157—41no—
168C22⋊D157—41no—
169C22⋊D157—41no—
170C22⋊D157—82no—
171C2×Dic157—21no—
172C2×Dic157—41no—
173C4×D157—21no—
174C4×D157—21no—
175C4×D157—21no—
176C4×D157—41no—
177C4×D157—41no—
178C4×D157—41no—
179C4×D157—41no—
180#11929434—21no—
181#11929434C2×D411yescharacteristic
182#11929434C2×D411yescharacteristic
183C23×C157—21no—
184Q8×C157—21no—
185Q8×C157—21no—
186Q8×C157C2×D411yescharacteristic
187D4×C157—21no—
188D4×C157—21no—
189D4×C157—21no—
190D4×C157C2×D411yescharacteristic
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157C2411yescharacteristic
194C8×C157—21no—
195C8×C157—21no—
196C8×C157—21no—
197C8×C157C2×D411yescharacteristic
198C8×C157C2×D411yescharacteristic
199C22×C157—21no—
200C22×C157—41no—
201C22×C157—81no—
202C22×C157H(5,𝔽2)11yescharacteristic
203C2×D157—21no—
204C2×D157—21no—
205C2×D157—21no—
206C2×D157—41no—
207C2×D157—41no—
208C2×D157—81no—
209C2×D157—81no—
210C2×D157—161no—
211C2×D157—161no—
212C2×D157—161no—
213C4×C157—21no—
214C4×C157—41no—
215C4×C157C22×D411yescharacteristic
216C4×C157C22×D411yescharacteristic
217Dic157—21no—
218Dic157—21no—
219Dic157—21no—
220Dic157—41no—
221C2×C157—21no—
222C2×C157—41no—
223C2×C157—81no—
224C2×C157D4211yescharacteristic
225D157—41no—
226D157—41no—
227D157—41no—
228D157—81no—
229D157—81no—
230C157#328811yescharacteristic, complemented, Sylow
231#3288—1571nomaximal, Sylow
232#577—1571no—
233#577—1571no—
234#576—1571no—
235#576—1571no—
236#496—1571no—
237#472—1571no—
238#472—1571no—
239#471—1571no—
240#471D15711yescharacteristic, complemented
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)—1571no—
250H(5,𝔽2)—1571no—
251#137—3141no—
252#137—3141no—
253#137—1571no—
254#137C2×D15711yescharacteristic, complemented
255AGL(1,ℤ/8)—3141no—
256AGL(1,ℤ/8)—3141no—
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)—1571no—
266AGL(1,ℤ/8)C2×D15711yescharacteristic
267#135—3141no—
268#135—3141no—
269#135—3141no—
270#135—3141no—
271#135—1571no—
272#135—1571no—
273C2×SD16—3141no—
274C2×SD16—3141no—
275C2×SD16—1571no—
276C2×SD16C2×D15711yescharacteristic, complemented
277C2×D8—3141no—
278C2×D8—3141no—
279C2×D8—1571no—
280C2×D8—1571no—
281#131—1571no—
282#131—1571no—
283#131—1571no—
284#131C2×D15711yescharacteristic, complemented
285#127—1571no—
286#124—1571no—
287#108—1571no—
288#108C2×D15711yescharacteristic, complemented
289#104—1571no—
290#104—1571no—
291#104—1571no—
292#104—1571no—
293#101C2×D15711yescharacteristic, complemented
294#100—1571no—
295#100—1571no—
296#100C2×D15711yescharacteristic, complemented
297#97—1571no—
298#41—3141no—
299#41—3141no—
300#41—3141no—
301#41—3141no—
302#41—3141no—
303#41—3141no—
304#41—3141no—
305#41—3141no—
306#41—6281no—
307#41—1571no—
308#41—1571no—
309#41—1571no—
310#41C22×D15711yescharacteristic
311C2×Q8—3141no—
312C2×Q8—3141no—
313C2×Q8C22×D15711yescharacteristic
314C2×D4—3141no—
315C2×D4—3141no—
316C2×D4—3141no—
317C2×D4—3141no—
318C2×D4—3141no—
319C2×D4—3141no—
320C2×D4—6281no—
321C2×D4—6281no—
322C2×D4—6281no—
323C2×D4—6281no—
324C2×D4—1571no—
325C2×D4—1571no—
326C2×D4C22×D15711yescharacteristic
327Q16—21no—
328Q16—3141no—
329Q16—1571no—
330Q16—1571no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342SD16—3141no—
343SD16—1571no—
344SD16—1571no—
345SD16—1571no—
346SD16—1571no—
347D8—21no—
348D8—3141no—
349D8—3141no—
350D8—3141no—
351D8—3141no—
352D8—3141no—
353D8—3141no—
354D8—3141no—
355D8—3141no—
356D8—3141no—
357D8—1571no—
358D8—1571no—
359M16—21no—
360M16—3141no—
361M16—3141no—
362M16—3141no—
363M16—1571no—
364M16C22×D15711yescharacteristic
365M16C22×D15711yescharacteristic
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—3141no—
377C23—3141no—
378C23—6281no—
379C23—6281no—
380C23—6281no—
381Q8—21no—
382Q8—21no—
383Q8—6281no—
384Q8—1571no—
385Q8—1571no—
386Q8—1571no—
387Q8D4×D15711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—3141no—
392D4—3141no—
393D4—3141no—
394D4—3141no—
395D4—3141no—
396D4—3141no—
397D4—3141no—
398D4—6281no—
399D4—6281no—
400D4—6281no—
401D4—6281no—
402D4—6281no—
403D4—6281no—
404D4—6281no—
405D4—1571no—
406D4—1571no—
407D4—1571no—
408D4—12562no—
409D4D4×D15711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—3141no—
413C2×C4—3141no—
414C2×C4—3141no—
415C2×C4—3141no—
416C2×C4—6281no—
417C2×C4—6281no—
418C2×C4—6281no—
419C2×C4—6281no—
420C2×C4—6281no—
421C2×C4C23×D15711yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—3141no—
426C8—1571no—
427C8—1571no—
428C8D4×D15711yescharacteristic
429C8D4×D15711yescharacteristic
430C22—21no—
431C22—41no—
432C22—81no—
433C22—3141no—
434C22—3141no—
435C22—3141no—
436C22—6281no—
437C22—6281no—
438C22—25121no—
439C22—25121no—
440C22—25121no—
441C22—12561no—
442C22—12561no—
443C22#1909781711yescharacteristic
444C4—21no—
445C4—41no—
446C4—3141no—
447C4—3141no—
448C4—3141no—
449C4—6281no—
450C4C2×D4×D15711yescharacteristic
451C4C2×D4×D15711yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—6281no—
456C2—6281no—
457C2—6281no—
458C2—12561no—
459C2—12561no—
460C2#2375790711yescharacteristic
subgroupG/H
centerC2#23757907
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#471×C157C2
socleC2×C157D42
hypercenter#471D157
hypocenterC157#3288
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #23758082C2semidirect
2. #23758079C2semidirect
3. #23758078C2semidirect
4. #23758077C2semidirect
5. #23757659C2semidirect
6. #23757544C2semidirect
7. #23757542C2semidirect
8. #23757539C2semidirect
9. #23757449C2semidirect
10. #23757448C2semidirect
11. #23757443C2semidirect
12. #23757439C2semidirect
13. #23757394C2semidirect
14. #23757393C2semidirect
15. #471×C157C2semidirect
16. #19097819C22semidirect
18. #19097817C22semidirect
19. #19097816C22semidirect
21. #19097802C22semidirect
26. #19097799C22semidirect
27. #19097799C22semidirect
28. #19097799C22semidirect
32. #19097797C22semidirect
40. C2×#14578941C22semidirect
44. C2×D1256C22semidirect
45. C2×#14578939C22semidirect
47. #19097779C22semidirect
48. #19097779C22semidirect
49. #19097777C22semidirect
50. #19097767C22semidirect
51. #19097760C22semidirect
52. #19097706C22semidirect
53. #19097698C22semidirect
54. #19097697C22semidirect
55. #19097697C22semidirect
56. #19097697C22semidirect
57. #19097687C22semidirect
58. #19097687C22semidirect
59. #19097681C22semidirect
60. #137×C157C22semidirect
62. C2×SD16×C157C22semidirect
63. #131×C157C22semidirect
64. #108×C157C22semidirect
65. #101×C157C22semidirect
66. #100×C157C22semidirect
93. Dic628D4semidirect
95. #14578944D4semidirect
100. #14578943D4semidirect
101. #14578943D4semidirect
103. #14578942D4semidirect
107. #14578941D4semidirect
114. D1256D4semidirect
118. #14578939D4semidirect
230. C157#3288semidirect
240. #471D157semidirect
254. #137C2×D157semidirect
276. C2×SD16C2×D157semidirect
284. #131C2×D157semidirect
288. #108C2×D157semidirect
293. #101C2×D157semidirect
296. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580822
#237580792
#237580782
#237580772
#237576592
#237575442
#237575422
#237575392
#237574492
#237574482
#237574432
#237574392
#237573942
#237573932
#471×C1572
#3288157