Finite Group Databasebeta

← #34111903

Subgroups — #34111903

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#23758085C211yesmaximal, characteristic, complemented
2#23758080C211yesmaximal, characteristic, complemented
3#23758078C211yesmaximal, characteristic, complemented
4#23758075C211yesmaximal, characteristic, complemented
5#23757662C211yesmaximal, characteristic, complemented
6#23757544C211yesmaximal, characteristic, complemented
7#23757543C211yesmaximal, characteristic, complemented
8#23757537C211yesmaximal, characteristic, complemented
9#23757536C211yesmaximal, characteristic, complemented
10#23757452C211yesmaximal, characteristic, complemented
11#23757447C211yesmaximal, characteristic, complemented
12#23757442C211yesmaximal, characteristic, complemented
13#23757442C211yesmaximal, characteristic, complemented
14#23757396C211yesmaximal, characteristic, complemented
15#444×C157C211yesmaximal, characteristic, complemented
16#19097818C2211yescharacteristic, complemented
17#19097817C2211yescharacteristic
18#19097817C2211yescharacteristic, complemented
19#19097817C2211yescharacteristic, complemented
20#19097804—21no—
21#19097804C2211yescharacteristic, complemented
22#19097803—21no—
23#19097801C2211yescharacteristic, complemented
24#19097799—21no—
25#19097799—21no—
26#19097799—21no—
27#19097799C2211yescharacteristic
28#19097799C2211yescharacteristic, complemented
29#19097799C2211yescharacteristic, complemented
30#19097798—21no—
31#19097798—21no—
32#19097796C2211yescharacteristic
33#19097796C2211yescharacteristic, complemented
34#19097796C2211yescharacteristic, complemented
35#19097795—21no—
36#19097794—21no—
37#19097793—21no—
38#19097793—21no—
39#19097793C2211yescharacteristic
40#19097792—21no—
41C2×#14578943—21no—
42C2×#14578943C2211yescharacteristic, complemented
43C2×#14578943C2211yescharacteristic, complemented
44SD16×D157—21no—
45C2×D1256—21no—
46C2×D1256C2211yescharacteristic, complemented
47C2×D1256C2211yescharacteristic, complemented
48D8×D157—21no—
49#19097777C2211yescharacteristic, complemented
50#19097777C2211yescharacteristic, complemented
51#19097768C2211yescharacteristic, complemented
52#19097759C2211yescharacteristic, complemented
53#19097707C2211yescharacteristic, complemented
54#19097699C2211yescharacteristic, complemented
55#19097699C2211yescharacteristic, complemented
56#19097689C2211yescharacteristic, complemented
57#19097687C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59#19097687C2211yescharacteristic, complemented
60#135×C157C2211yescharacteristic
61#131×C157C2211yescharacteristic, complemented
62#131×C157C2211yescharacteristic, complemented
63#108×C157C2211yescharacteristic, complemented
64#104×C157C2211yescharacteristic, complemented
65#104×C157C2211yescharacteristic, complemented
66#97×C157C2211yescharacteristic, complemented
67#14578955—21no—
68#14578955—21no—
69#14578955—21no—
70#14578954—21no—
71#14578954—21no—
72#14578954—21no—
73#14578954—21no—
74#14578954C2311yescharacteristic
75#14578954C2311yescharacteristic
76#14578953—21no—
77#14578953—41no—
78C2×Dic314—21no—
79C2×Dic314C2311yescharacteristic
80Q8×D157—21no—
81C2×D628—21no—
82C2×D628—21no—
83C2×D628—21no—
84C2×D628—41no—
85C2×D628C2311yescharacteristic
86C2×D628C2311yescharacteristic
87C2×D628C2311yescharacteristic
88D4×D157—21no—
89D4×D157—21no—
90D4×D157—21no—
91D4×D157—41no—
92D4×D157—41no—
93D4×D157—41no—
94Dic628—21no—
95Dic628D411yescharacteristic, complemented
96#14578944—21no—
97#14578943—21no—
98#14578943—21no—
99#14578943—21no—
100#14578943—21no—
101#14578943—21no—
102#14578943—21no—
103#14578943—21no—
104#14578943—21no—
105#14578943D411yescharacteristic, complemented
106#14578943D411yescharacteristic, complemented
107#14578942—21no—
108#14578941—21no—
109#14578941—21no—
110#14578941—21no—
111D1256—21no—
112D1256—21no—
113D1256—21no—
114D1256—21no—
115D1256—21no—
116D1256—21no—
117D1256—21no—
118D1256D411yescharacteristic, complemented
119#14578939—21no—
120#14578939—21no—
121#14578939—21no—
122#14578938C2311yescharacteristic
123#14578938C2311yescharacteristic
124#14578937—21no—
125#14578937—21no—
126C8×D157—21no—
127C8×D157—21no—
128#14578931—41no—
129#41×C157C2311yescharacteristic
130#41×C157C2311yescharacteristic
131Q16×C157D411yescharacteristic, complemented
132SD16×C157D411yescharacteristic, complemented
133SD16×C157D411yescharacteristic, complemented
134D8×C157D411yescharacteristic, complemented
135M16×C157—21no—
136M16×C157—21no—
137M16×C157C2311yescharacteristic
138M16×C157C2311yescharacteristic
139C2×C8×C157—21no—
140C2×C8×C157—21no—
141C2×C8×C157C2311yescharacteristic
142C2×C8×C157C2311yescharacteristic
143C42×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157—21no—
146C22×D157—21no—
147C22×D157—41no—
148C22×D157—41no—
149C22×D157—41no—
150Dic314—21no—
151Dic314—21no—
152Dic314—41no—
153Dic314C2×D411yescharacteristic
154Dic314C2×D411yescharacteristic
155D628—21no—
156D628—21no—
157D628—21no—
158D628—21no—
159D628—21no—
160D628—21no—
161D628—41no—
162D628—41no—
163D628—41no—
164D628—82no—
165D628C2×D411yescharacteristic
166D628C2×D411yescharacteristic
167C22⋊D157—21no—
168C22⋊D157—21no—
169C22⋊D157—41no—
170C22⋊D157—41no—
171C22⋊D157—41no—
172C22⋊D157—41no—
173C2×Dic157—21no—
174C2×Dic157—41no—
175C4×D157—21no—
176C4×D157—21no—
177C4×D157—41no—
178C4×D157—41no—
179C4×D157—41no—
180C4×D157—41no—
181#11929434—21no—
182#11929434—21no—
183Q8×C157C2×D411yescharacteristic
184Q8×C157C2×D411yescharacteristic
185D4×C157—21no—
186D4×C157—21no—
187D4×C157C2×D411yescharacteristic
188D4×C157C2×D411yescharacteristic
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157C2411yescharacteristic
193C8×C157—21no—
194C8×C157—21no—
195C8×C157C2×D411yescharacteristic
196C8×C157C2×D411yescharacteristic
197C8×C157C2×D411yescharacteristic
198C8×C157C2×D411yescharacteristic
199C22×C157—21no—
200C22×C157—21no—
201C22×C157H(5,𝔽2)11yescharacteristic
202C2×D157—21no—
203C2×D157—21no—
204C2×D157—41no—
205C2×D157—41no—
206C2×D157—41no—
207C2×D157—81no—
208C2×D157—81no—
209C2×D157—81no—
210C2×D157—161no—
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×C157D4211yescharacteristic
225D157—41no—
226D157—41no—
227D157—81no—
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#471—1571no—
241#455—1571no—
242#454—1571no—
243#453—1571no—
244#453—1571no—
245#444—1571no—
246#444D15711yescharacteristic, complemented
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#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)—1571no—
266AGL(1,ℤ/8)—1571no—
267#135—3141no—
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—3141no—
278C2×D8—3141no—
279C2×D8—1571no—
280C2×D8—1571no—
281#131—1571no—
282#131—1571no—
283#131C2×D15711yescharacteristic, complemented
284#131C2×D15711yescharacteristic, complemented
285#127—1571no—
286#124—1571no—
287#108—1571no—
288#108C2×D15711yescharacteristic, complemented
289#104—1571no—
290#104—1571no—
291#104C2×D15711yescharacteristic, complemented
292#104C2×D15711yescharacteristic, complemented
293#101—1571no—
294#100—1571no—
295#100—1571no—
296#100—1571no—
297#97C2×D15711yescharacteristic, complemented
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#41C22×D15711yescharacteristic
310#41C22×D15711yescharacteristic
311C2×Q8—3141no—
312C2×Q8—3141no—
313C2×Q8—1571no—
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×D4—1571no—
327Q16—3141no—
328Q16—3141no—
329Q16—1571no—
330Q16D62811yescharacteristic, complemented
331SD16—3141no—
332SD16—3141no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342SD16—3141no—
343SD16—1571no—
344SD16—1571no—
345SD16D62811yescharacteristic, complemented
346SD16D62811yescharacteristic, complemented
347D8—3141no—
348D8—3141no—
349D8—3141no—
350D8—3141no—
351D8—3141no—
352D8—3141no—
353D8—3141no—
354D8—3141no—
355D8—3141no—
356D8—3141no—
357D8—1571no—
358D8D62811yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—3141no—
362M16—3141no—
363M16—1571no—
364M16—1571no—
365M16C22×D15711yescharacteristic
366M16C22×D15711yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—3141no—
370C2×C8—3141no—
371C2×C8C22×D15711yescharacteristic
372C2×C8C22×D15711yescharacteristic
373C22⋊C4—6281no—
374C42C22×D15711yescharacteristic
375C23—3141no—
376C23—3141no—
377C23—3141no—
378C23—6281no—
379C23—6281no—
380C23—6281no—
381Q8—3141no—
382Q8—3141no—
383Q8—6281no—
384Q8—1571no—
385Q8—1571no—
386Q8C2×D62811yescharacteristic
387Q8C2×D62811yescharacteristic
388D4—21no—
389D4—21no—
390D4—3141no—
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—12562no—
408D4C2×D62811yescharacteristic
409D4C2×D62811yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
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—3141no—
425C8—3141no—
426C8D4×D15711yescharacteristic
427C8D4×D15711yescharacteristic
428C8C2×D62811yescharacteristic
429C8C2×D62811yescharacteristic
430C22—21no—
431C22—21no—
432C22—3141no—
433C22—3141no—
434C22—6281no—
435C22—6281no—
436C22—6281no—
437C22—25121no—
438C22—25121no—
439C22—25121no—
440C22—12561no—
441C22—12561no—
442C22—12561no—
443C22#1909781711yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3141no—
448C4—3141no—
449C4—6281no—
450C4C2×D4×D15711yescharacteristic
451C4C22×D62811yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—6281no—
456C2—6281no—
457C2—12561no—
458C2—12561no—
459C2—12561no—
460C2D4×D62811yescharacteristic
subgroupG/H
centerC2D4×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#444×C157C2
socleC2×C157D42
hypercenter#444D157
hypocenterC157#3288
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #23758085C2semidirect
2. #23758080C2semidirect
3. #23758078C2semidirect
4. #23758075C2semidirect
5. #23757662C2semidirect
6. #23757544C2semidirect
7. #23757543C2semidirect
8. #23757537C2semidirect
9. #23757536C2semidirect
10. #23757452C2semidirect
11. #23757447C2semidirect
12. #23757442C2semidirect
13. #23757442C2semidirect
14. #23757396C2semidirect
15. #444×C157C2semidirect
16. #19097818C22semidirect
18. #19097817C22semidirect
19. #19097817C22semidirect
21. #19097804C22semidirect
23. #19097801C22semidirect
28. #19097799C22semidirect
29. #19097799C22semidirect
33. #19097796C22semidirect
34. #19097796C22semidirect
42. C2×#14578943C22semidirect
43. C2×#14578943C22semidirect
46. C2×D1256C22semidirect
47. C2×D1256C22semidirect
49. #19097777C22semidirect
50. #19097777C22semidirect
51. #19097768C22semidirect
52. #19097759C22semidirect
53. #19097707C22semidirect
54. #19097699C22semidirect
55. #19097699C22semidirect
56. #19097689C22semidirect
57. #19097687C22semidirect
58. #19097687C22semidirect
59. #19097687C22semidirect
61. #131×C157C22semidirect
62. #131×C157C22semidirect
63. #108×C157C22semidirect
64. #104×C157C22semidirect
65. #104×C157C22semidirect
66. #97×C157C22semidirect
95. Dic628D4semidirect
105. #14578943D4semidirect
106. #14578943D4semidirect
118. D1256D4semidirect
131. Q16×C157D4semidirect
132. SD16×C157D4semidirect
133. SD16×C157D4semidirect
134. D8×C157D4semidirect
230. C157#3288semidirect
246. #444D157semidirect
283. #131C2×D157semidirect
284. #131C2×D157semidirect
288. #108C2×D157semidirect
291. #104C2×D157semidirect
292. #104C2×D157semidirect
297. #97C2×D157semidirect
330. Q16D628semidirect
345. SD16D628semidirect
346. SD16D628semidirect
358. D8D628semidirect

Maximal subgroups

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

subgroupindex
#237580852
#237580802
#237580782
#237580752
#237576622
#237575442
#237575432
#237575372
#237575362
#237574522
#237574472
#237574422
#237573962
#444×C1572
#3288157