Finite Group Databasebeta

← #34111897

Subgroups — #34111897

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#23758083C211yesmaximal, characteristic, complemented
3#23758077C211yesmaximal, characteristic, complemented
4#23758076C211yesmaximal, characteristic, complemented
5#23757660C211yesmaximal, characteristic, complemented
6#23757542C211yesmaximal, characteristic, complemented
7#23757541C211yesmaximal, characteristic, complemented
8#23757535C211yesmaximal, characteristic, complemented
9#23757534C211yesmaximal, characteristic, complemented
10#23757452C211yesmaximal, characteristic, complemented
11#23757447C211yesmaximal, characteristic, complemented
12#23757440C211yesmaximal, characteristic, complemented
13#23757392C211yesmaximal, characteristic, complemented
14#23757392C211yesmaximal, characteristic, complemented
15#453×C157C211yesmaximal, characteristic, complemented
16#19097818C2211yescharacteristic, complemented
17#19097817C2211yescharacteristic
18#19097816C2211yescharacteristic, complemented
19#19097804C2211yescharacteristic, complemented
20#19097801—21no—
21#19097801C2211yescharacteristic, complemented
22#19097800—21no—
23#19097799C2211yescharacteristic, complemented
24#19097798—21no—
25#19097797—21no—
26#19097797—21no—
27#19097797C2211yescharacteristic
28#19097797C2211yescharacteristic, complemented
29#19097796—21no—
30#19097796C2211yescharacteristic, complemented
31#19097795—21no—
32#19097795—21no—
33#19097792—21no—
34#19097791—21no—
35#19097791—21no—
36#19097791C2211yescharacteristic
37#19097791C2211yescharacteristic
38#19097790—21no—
39C2×#14578942—21no—
40C2×#14578942C2211yescharacteristic, complemented
41C2×#14578942C2211yescharacteristic, complemented
42SD16×D157—21no—
43C2×#14578939—21no—
44C2×#14578939C2211yescharacteristic, complemented
45C2×#14578939C2211yescharacteristic, complemented
46D8×D157—21no—
47#19097779C2211yescharacteristic, complemented
48#19097779C2211yescharacteristic, complemented
49#19097778C2211yescharacteristic, complemented
50#19097778C2211yescharacteristic, complemented
51#19097759C2211yescharacteristic, complemented
52#19097706C2211yescharacteristic, complemented
53#19097706C2211yescharacteristic, complemented
54#19097699C2211yescharacteristic, complemented
55#19097699C2211yescharacteristic, complemented
56#19097689C2211yescharacteristic, complemented
57#19097688C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59#19097682C2211yescharacteristic, complemented
60H(5,𝔽2)×C157C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C157C2211yescharacteristic
62AGL(1,ℤ/8)×C157C2211yescharacteristic, complemented
63#127×C157C2211yescharacteristic, complemented
64#104×C157C2211yescharacteristic, complemented
65#104×C157C2211yescharacteristic, complemented
66#100×C157C2211yescharacteristic, complemented
67#14578955—21no—
68#14578954—21no—
69#14578954—21no—
70#14578954C2311yescharacteristic
71#14578954C2311yescharacteristic
72#14578953—21no—
73#14578953—21no—
74#14578953—21no—
75#14578953—41no—
76C2×Dic314—21no—
77C2×Dic314C2311yescharacteristic
78Q8×D157—21no—
79C2×D628—21no—
80C2×D628C2311yescharacteristic
81C2×C22⋊D157—41no—
82D4×D157—21no—
83D4×D157—21no—
84D4×D157—21no—
85D4×D157—41no—
86Dic628—21no—
87#14578944—21no—
88#14578944D411yescharacteristic, complemented
89#14578944D411yescharacteristic, complemented
90#14578943—21no—
91#14578943—21no—
92#14578942—21no—
93#14578942—21no—
94#14578942—21no—
95#14578942—21no—
96#14578942—21no—
97#14578942—21no—
98#14578942—21no—
99#14578942D411yescharacteristic, complemented
100#14578942D411yescharacteristic, complemented
101#14578941—21no—
102#14578941D411yescharacteristic, complemented
103#14578941D411yescharacteristic, complemented
104D1256—21no—
105#14578939—21no—
106#14578939—21no—
107#14578939—21no—
108#14578939—21no—
109#14578939—21no—
110#14578939—21no—
111#14578939—21no—
112#14578939D411yescharacteristic, complemented
113#14578939D411yescharacteristic, complemented
114#14578938—21no—
115#14578938—21no—
116#14578938C2311yescharacteristic
117#14578938C2311yescharacteristic
118#14578937—21no—
119#14578937—21no—
120C2×#11929434—21no—
121C2×#11929434—21no—
122C2×#11929434C2311yescharacteristic
123C2×#11929434C2311yescharacteristic
124C8×D157—21no—
125C8×D157—21no—
126#14578931—41no—
127#41×C157—21no—
128#41×C157—21no—
129#41×C157C2311yescharacteristic
130#41×C157C2311yescharacteristic
131C2×D4×C157—21no—
132C2×D4×C157—21no—
133C2×D4×C157—41no—
134C2×D4×C157—41no—
135C2×D4×C157C2311yescharacteristic
136C2×D4×C157C2311yescharacteristic
137SD16×C157—21no—
138SD16×C157—21no—
139D8×C157—21no—
140D8×C157—21no—
141M16×C157C2311yescharacteristic
142M16×C157C2311yescharacteristic
143C42×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157—41no—
146C22×D157—41no—
147Dic314—21no—
148Dic314—21no—
149Dic314—41no—
150Dic314C2×D411yescharacteristic
151Dic314C2×D411yescharacteristic
152D628—21no—
153D628—21no—
154D628—41no—
155D628C2×D411yescharacteristic
156D628C2×D411yescharacteristic
157C22⋊D157—21no—
158C22⋊D157—21no—
159C22⋊D157—41no—
160C22⋊D157—41no—
161C22⋊D157—41no—
162C22⋊D157—41no—
163C2×Dic157—21no—
164C2×Dic157—41no—
165C2×Dic157—41no—
166C4×D157—21no—
167C4×D157—21no—
168C4×D157—41no—
169C4×D157—41no—
170#11929434—21no—
171#11929434—21no—
172#11929434C2×D411yescharacteristic
173#11929434C2×D411yescharacteristic
174#11929434C2×D411yescharacteristic
175#11929434C2×D411yescharacteristic
176C23×C157—21no—
177C23×C157—21no—
178C23×C157—41no—
179Q8×C157C2×D411yescharacteristic
180Q8×C157C2×D411yescharacteristic
181D4×C157—21no—
182D4×C157—21no—
183D4×C157—21no—
184D4×C157—21no—
185D4×C157—21no—
186D4×C157—21no—
187D4×C157—41no—
188D4×C157—41no—
189D4×C157—82no—
190D4×C157C2×D411yescharacteristic
191D4×C157C2×D411yescharacteristic
192C2×C4×C157—21no—
193C2×C4×C157—21no—
194C2×C4×C157—21no—
195C2×C4×C157—41no—
196C2×C4×C157C2411yescharacteristic
197C8×C157—21no—
198C8×C157—21no—
199C22×C157—21no—
200C22×C157—21no—
201C22×C157—41no—
202C22×C157—41no—
203C22×C157—81no—
204C22×C157—81no—
205C22×C157—161no—
206C22×C157H(5,𝔽2)11yescharacteristic
207C2×D157—21no—
208C2×D157—21no—
209C2×D157—41no—
210C2×D157—81no—
211C2×D157—161no—
212C2×D157—161no—
213C4×C157—21no—
214C4×C157—21no—
215C4×C157—41no—
216C4×C157C22×D411yescharacteristic
217C4×C157C22×D411yescharacteristic
218Dic157—21no—
219Dic157—21no—
220Dic157—41no—
221C2×C157—21no—
222C2×C157—41no—
223C2×C157—41no—
224C2×C157—81no—
225C2×C157—81no—
226C2×C157D4211yescharacteristic
227D157—41no—
228D157—41no—
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#471—1571no—
241#455—1571no—
242#454—1571no—
243#453—1571no—
244#453D15711yescharacteristic, complemented
245#444—1571no—
246#444—1571no—
2472−1+4—1571no—
248H(5,𝔽2)—1571no—
249H(5,𝔽2)—1571no—
250H(5,𝔽2)C2×D15711yescharacteristic, complemented
251#137—3141no—
252#137—3141no—
253#137—1571no—
254#137—1571no—
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)C2×D15711yescharacteristic
266AGL(1,ℤ/8)C2×D15711yescharacteristic, complemented
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×SD16—1571no—
277C2×D8—3141no—
278C2×D8—3141no—
279C2×D8—1571no—
280C2×D8—1571no—
281#131—1571no—
282#131—1571no—
283#131—1571no—
284#131—1571no—
285#127C2×D15711yescharacteristic, complemented
286#124—1571no—
287#108—1571no—
288#108—1571no—
289#104—1571no—
290#104—1571no—
291#104C2×D15711yescharacteristic, complemented
292#104C2×D15711yescharacteristic, complemented
293#101—1571no—
294#100—1571no—
295#100—1571no—
296#100C2×D15711yescharacteristic, complemented
297#97—1571no—
298#41—21no—
299#41—21no—
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#41C22×D15711yescharacteristic
310#41C22×D15711yescharacteristic
311C2×Q8—3141no—
312C2×Q8—3141no—
313C2×Q8—1571no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—3141no—
319C2×D4—3141no—
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—
330Q16—1571no—
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—
345SD16—1571no—
346SD16—1571no—
347D8—21no—
348D8—21no—
349D8—3141no—
350D8—3141no—
351D8—3141no—
352D8—3141no—
353D8—3141no—
354D8—3141no—
355D8—3141no—
356D8—3141no—
357D8—1571no—
358D8—1571no—
359M16—3141no—
360M16—3141no—
361M16—3141no—
362M16—3141no—
363M16—1571no—
364M16—1571no—
365M16C22×D15711yescharacteristic
366M16C22×D15711yescharacteristic
367C2×C8—3141no—
368C2×C8—3141no—
369C2×C8—3141no—
370C2×C8—3141no—
371C2×C8—1571no—
372C2×C8—1571no—
373C22⋊C4—6281no—
374C42C22×D15711yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—3141no—
379C23—6281no—
380C23—6281no—
381Q8—3141no—
382Q8—3141no—
383Q8—6281no—
384Q8—1571no—
385Q8—1571no—
386Q8D4×D15711yescharacteristic
387Q8D4×D15711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—3141no—
398D4—3141no—
399D4—3141no—
400D4—3141no—
401D4—6281no—
402D4—6281no—
403D4—6281no—
404D4—6281no—
405D4—6281no—
406D4—1571no—
407D4—1571no—
408D4D4×D15711yescharacteristic
409D4D4×D15711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—3141no—
415C2×C4—3141no—
416C2×C4—3141no—
417C2×C4—6281no—
418C2×C4—6281no—
419C2×C4—6281no—
420C2×C4—6281no—
421C2×C4C23×D15711yescharacteristic
422C8—21no—
423C8—21no—
424C8—3141no—
425C8—3141no—
426C8—1571no—
427C8—1571no—
428C8—1571no—
429C8—1571no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—3141no—
438C22—3141no—
439C22—6281no—
440C22—25121no—
441C22—25121no—
442C22—12561no—
443C22#1909781611yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3141no—
448C4—3141no—
449C4—6281no—
450C4C2×D4×D15711yescharacteristic
451C4C2×D4×D15711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—6281no—
458C2—6281no—
459C2—12561no—
460C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#453×C157C2
socleC2×C157D42
hypercenter#453D157
hypocenterC157#3288
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23758084C2semidirect
2. #23758083C2semidirect
3. #23758077C2semidirect
4. #23758076C2semidirect
5. #23757660C2semidirect
6. #23757542C2semidirect
7. #23757541C2semidirect
8. #23757535C2semidirect
9. #23757534C2semidirect
10. #23757452C2semidirect
11. #23757447C2semidirect
12. #23757440C2semidirect
13. #23757392C2semidirect
14. #23757392C2semidirect
15. #453×C157C2semidirect
16. #19097818C22semidirect
18. #19097816C22semidirect
19. #19097804C22semidirect
21. #19097801C22semidirect
23. #19097799C22semidirect
28. #19097797C22semidirect
30. #19097796C22semidirect
40. C2×#14578942C22semidirect
41. C2×#14578942C22semidirect
44. C2×#14578939C22semidirect
45. C2×#14578939C22semidirect
47. #19097779C22semidirect
48. #19097779C22semidirect
49. #19097778C22semidirect
50. #19097778C22semidirect
51. #19097759C22semidirect
52. #19097706C22semidirect
53. #19097706C22semidirect
54. #19097699C22semidirect
55. #19097699C22semidirect
56. #19097689C22semidirect
57. #19097688C22semidirect
58. #19097687C22semidirect
59. #19097682C22semidirect
60. H(5,𝔽2)×C157C22semidirect
62. AGL(1,ℤ/8)×C157C22semidirect
63. #127×C157C22semidirect
64. #104×C157C22semidirect
65. #104×C157C22semidirect
66. #100×C157C22semidirect
88. #14578944D4semidirect
89. #14578944D4semidirect
99. #14578942D4semidirect
100. #14578942D4semidirect
102. #14578941D4semidirect
103. #14578941D4semidirect
112. #14578939D4semidirect
113. #14578939D4semidirect
230. C157#3288semidirect
244. #453D157semidirect
250. H(5,𝔽2)C2×D157semidirect
266. AGL(1,ℤ/8)C2×D157semidirect
285. #127C2×D157semidirect
291. #104C2×D157semidirect
292. #104C2×D157semidirect
296. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580842
#237580832
#237580772
#237580762
#237576602
#237575422
#237575412
#237575352
#237575342
#237574522
#237574472
#237574402
#237573922
#453×C1572
#3288157