Finite Group Databasebeta

← #34111928

Subgroups — #34111928

Order 20096.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758085C211yesmaximal, characteristic, complemented
2#23758083C211yesmaximal, characteristic, complemented
3#23758082C211yesmaximal, characteristic, complemented
4#23758079C211yesmaximal, characteristic, complemented
5#23757644C211yesmaximal, characteristic, complemented
6#23757539C211yesmaximal, characteristic, complemented
7#23757537C211yesmaximal, characteristic, complemented
8#23757535C211yesmaximal, characteristic, complemented
9#23757458C211yesmaximal, characteristic, complemented
10#23757454C211yesmaximal, characteristic, complemented
11#23757443C211yesmaximal, characteristic, complemented
12#23757439C211yesmaximal, characteristic, complemented
13#23757389C211yesmaximal, characteristic, complemented
14#23757388C211yesmaximal, characteristic, complemented
15#471×C157C211yesmaximal, characteristic, complemented
16#19097819C2211yescharacteristic
17#19097818C2211yescharacteristic
18#19097817C2211yescharacteristic, complemented
19#19097816C2211yescharacteristic, complemented
20#19097804—21no—
21#19097804C2211yescharacteristic, complemented
22#19097804C2211yescharacteristic, complemented
23#19097803—21no—
24#19097802C2211yescharacteristic, complemented
25#19097800—21no—
26#19097800—21no—
27#19097799—21no—
28#19097799C2211yescharacteristic
29#19097799C2211yescharacteristic
30#19097798—21no—
31#19097798—21no—
32#19097797C2211yescharacteristic
33#19097795—21no—
34#19097794—21no—
35#19097793—21no—
36#19097793C2211yescharacteristic
37#19097792—21no—
38#19097791C2211yescharacteristic
39#19097790—21no—
40C2×#14578943—21no—
41C2×#14578943C2211yescharacteristic, complemented
42C2×#14578942C2211yescharacteristic, complemented
43C2×#14578941C2211yescharacteristic, complemented
44SD16×D157—21no—
45SD16×D157—21no—
46SD16×D157—21no—
47#19097779C2211yescharacteristic, complemented
48#19097779C2211yescharacteristic, complemented
49#19097777C2211yescharacteristic, complemented
50#19097769C2211yescharacteristic, complemented
51#19097767C2211yescharacteristic
52#19097706C2211yescharacteristic, complemented
53#19097698C2211yescharacteristic, complemented
54#19097697C2211yescharacteristic, complemented
55#19097697C2211yescharacteristic, complemented
56#19097697C2211yescharacteristic, complemented
57#19097689C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59C8×Dic157C2211yescharacteristic, 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#14578954—21no—
71#14578954—21no—
72#14578954—41no—
73#14578954C2311yescharacteristic
74#14578954C2311yescharacteristic
75#14578954C2311yescharacteristic
76#14578953—21no—
77#14578953—21no—
78#14578953—21no—
79#14578953—41no—
80C2×Dic314—21no—
81C2×Dic314C2311yescharacteristic
82Q8×D157—21no—
83Q8×D157—21no—
84Q8×D157—21no—
85C2×D628—21no—
86C2×D628C2311yescharacteristic
87C2×C22⋊D157—41no—
88C2×C22⋊D157—41no—
89D4×D157—21no—
90D4×D157—21no—
91D4×D157—21no—
92D4×D157—41no—
93Dic628—21no—
94Dic628—21no—
95Dic628D411yescharacteristic, complemented
96#14578944—21no—
97#14578944D411yescharacteristic, complemented
98#14578943—21no—
99#14578943—21no—
100#14578943—21no—
101#14578943—21no—
102#14578943—21no—
103#14578943—21no—
104#14578943D411yescharacteristic, complemented
105#14578943D411yescharacteristic, complemented
106#14578942—21no—
107#14578942—21no—
108#14578942—21no—
109#14578942D411yescharacteristic, complemented
110#14578941—21no—
111#14578941—21no—
112#14578941—21no—
113#14578941D411yescharacteristic, complemented
114D1256—21no—
115D1256—21no—
116D1256D411yescharacteristic, complemented
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#14578933—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—41no—
146C22×D157—41no—
147Dic314—21no—
148Dic314—21no—
149Dic314—41no—
150Dic314—41no—
151Dic314C2×D411yescharacteristic
152Dic314C2×D411yescharacteristic
153Dic314C2×D411yescharacteristic
154D628—21no—
155D628—21no—
156D628—41no—
157D628—41no—
158D628C2×D411yescharacteristic
159D628C2×D411yescharacteristic
160D628C2×D411yescharacteristic
161C22⋊D157—21no—
162C22⋊D157—21no—
163C22⋊D157—21no—
164C22⋊D157—41no—
165C22⋊D157—41no—
166C22⋊D157—41no—
167C22⋊D157—41no—
168C22⋊D157—82no—
169C2×Dic157—21no—
170C2×Dic157—21no—
171C2×Dic157—41no—
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—81no—
208C2×D157—161no—
209C2×D157—161no—
210C4×C157—21no—
211C4×C157—41no—
212C4×C157C22×D411yescharacteristic
213C4×C157C22×D411yescharacteristic
214Dic157—22no—
215Dic157—21no—
216Dic157—21no—
217Dic157—21no—
218Dic157—41no—
219C2×C157—21no—
220C2×C157—41no—
221C2×C157—81no—
222C2×C157D4211yescharacteristic
223D157—41no—
224D157—41no—
225D157—41no—
226D157—81no—
227C157#329111yescharacteristic, complemented, Sylow
228#3291—1571nomaximal, Sylow
229#577—1571no—
230#577—1571no—
231#577—1571no—
232#577—1571no—
233#492—1571no—
234#471—1571no—
235#471—1571no—
236#471—1571no—
237#471D15711yescharacteristic, complemented
238#456—1571no—
239#456—1571no—
240#453—1571no—
241#453—1571no—
242#443—1571no—
243#443—1571no—
2442−1+4—1571no—
2452−1+4—1571no—
246H(5,𝔽2)—1571no—
247H(5,𝔽2)—1571no—
248#137—3141no—
249#137—3141no—
250#137—3141no—
251#137—3141no—
252#137—1571no—
253#137—1571no—
254#137—1571no—
255#137C2×D15711yescharacteristic, complemented
256AGL(1,ℤ/8)—3141no—
257AGL(1,ℤ/8)—3141no—
258AGL(1,ℤ/8)—3141no—
259AGL(1,ℤ/8)—3141no—
260AGL(1,ℤ/8)—1571no—
261AGL(1,ℤ/8)—1571no—
262AGL(1,ℤ/8)—1571no—
263AGL(1,ℤ/8)C2×D15711yescharacteristic
264#135—3141no—
265#135—3141no—
266#135—3141no—
267#135—3141no—
268#135—1571no—
269#135—1571no—
270C2×SD16—3141no—
271C2×SD16—3141no—
272C2×SD16—3141no—
273C2×SD16—3141no—
274C2×SD16—1571no—
275C2×SD16—1571no—
276C2×SD16—1571no—
277C2×SD16C2×D15711yescharacteristic, complemented
278#131—1571no—
279#131—1571no—
280#131—1571no—
281#131C2×D15711yescharacteristic, complemented
282C4⋊Q8—1571no—
283#127—1571no—
284#108—1571no—
285#108C2×D15711yescharacteristic, complemented
286#104—1571no—
287#104—1571no—
288#104—1571no—
289#104—1571no—
290#101—1571no—
291#101C2×D15711yescharacteristic, complemented
292#100—1571no—
293#100C2×D15711yescharacteristic, complemented
294C4×C8—1571no—
295#41—3141no—
296#41—3141no—
297#41—3141no—
298#41—3141no—
299#41—3141no—
300#41—3141no—
301#41—3141no—
302#41—3141no—
303#41—6281no—
304#41—6281no—
305#41—1571no—
306#41—1571no—
307#41—1571no—
308#41C22×D15711yescharacteristic
309C2×Q8—3141no—
310C2×Q8—3141no—
311C2×Q8—3141no—
312C2×Q8—3141no—
313C2×Q8—1571no—
314C2×Q8C22×D15711yescharacteristic
315C2×D4—3141no—
316C2×D4—3141no—
317C2×D4—3141no—
318C2×D4—3141no—
319C2×D4—6281no—
320C2×D4—6281no—
321C2×D4—6281no—
322C2×D4—1571no—
323C2×D4C22×D15711yescharacteristic
324Q16—21no—
325Q16—3141no—
326Q16—3141no—
327Q16—3141no—
328Q16—1571no—
329Q16—1571no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342SD16—3141no—
343SD16—3141no—
344SD16—3141no—
345SD16—3141no—
346SD16—1571no—
347SD16—1571no—
348SD16—1571no—
349SD16—1571no—
350D8—21no—
351D8—3141no—
352D8—3141no—
353D8—3141no—
354D8—1571no—
355D8—1571no—
356M16—21no—
357M16—3141no—
358M16—3141no—
359M16—3141no—
360M16—1571no—
361M16C22×D15711yescharacteristic
362M16C22×D15711yescharacteristic
363M16C22×D15711yescharacteristic
364C2×C8—21no—
365C2×C8—3141no—
366C2×C8—3141no—
367C2×C8—3141no—
368C2×C8—1571no—
369C2×C8C22×D15711yescharacteristic
370C4⋊C4—6281no—
371C42—1571no—
372C23—21no—
373C23—3141no—
374C23—6281no—
375C23—6281no—
376Q8—21no—
377Q8—21no—
378Q8—3141no—
379Q8—3141no—
380Q8—6281no—
381Q8—6281no—
382Q8—1571no—
383Q8—1571no—
384Q8—1571no—
385Q8D4×D15711yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—3141no—
390D4—3141no—
391D4—3141no—
392D4—3141no—
393D4—3141no—
394D4—6281no—
395D4—6281no—
396D4—6281no—
397D4—6281no—
398D4—6281no—
399D4—6281no—
400D4—1571no—
401D4—1571no—
402D4—1571no—
403D4—12562no—
404D4D4×D15711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3141no—
408C2×C4—3141no—
409C2×C4—3141no—
410C2×C4—3141no—
411C2×C4—3141no—
412C2×C4—6281no—
413C2×C4—6281no—
414C2×C4—6281no—
415C2×C4—6281no—
416C2×C4—6281no—
417C2×C4—6281no—
418C2×C4C23×D15711yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—3141no—
423C8—1571no—
424C8—1571no—
425C8D4×D15711yescharacteristic
426C8D4×D15711yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—3141no—
431C22—3141no—
432C22—3141no—
433C22—6281no—
434C22—25121no—
435C22—25121no—
436C22—12561no—
437C22#1909781711yescharacteristic
438C4—21no—
439C4—41no—
440C4—3142no—
441C4—3141no—
442C4—3141no—
443C4—3141no—
444C4—6281no—
445C4C2×D4×D15711yescharacteristic
446C4C2×D4×D15711yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—6281no—
451C2—6281no—
452C2—6281no—
453C2—12561no—
454C2#2375790711yescharacteristic
subgroupG/H
centerC2#23757907
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#471×C157C2
socleC2×C157D42
hypercenter#471D157
hypocenterC157#3291
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. #23758085C2semidirect
2. #23758083C2semidirect
3. #23758082C2semidirect
4. #23758079C2semidirect
5. #23757644C2semidirect
6. #23757539C2semidirect
7. #23757537C2semidirect
8. #23757535C2semidirect
9. #23757458C2semidirect
10. #23757454C2semidirect
11. #23757443C2semidirect
12. #23757439C2semidirect
13. #23757389C2semidirect
14. #23757388C2semidirect
15. #471×C157C2semidirect
18. #19097817C22semidirect
19. #19097816C22semidirect
21. #19097804C22semidirect
22. #19097804C22semidirect
24. #19097802C22semidirect
41. C2×#14578943C22semidirect
42. C2×#14578942C22semidirect
43. C2×#14578941C22semidirect
47. #19097779C22semidirect
48. #19097779C22semidirect
49. #19097777C22semidirect
50. #19097769C22semidirect
52. #19097706C22semidirect
53. #19097698C22semidirect
54. #19097697C22semidirect
55. #19097697C22semidirect
56. #19097697C22semidirect
57. #19097689C22semidirect
58. #19097687C22semidirect
59. C8×Dic157C22semidirect
60. #137×C157C22semidirect
62. C2×SD16×C157C22semidirect
63. #131×C157C22semidirect
64. #108×C157C22semidirect
65. #101×C157C22semidirect
66. #100×C157C22semidirect
95. Dic628D4semidirect
97. #14578944D4semidirect
104. #14578943D4semidirect
105. #14578943D4semidirect
109. #14578942D4semidirect
113. #14578941D4semidirect
116. D1256D4semidirect
118. #14578939D4semidirect
227. C157#3291semidirect
237. #471D157semidirect
255. #137C2×D157semidirect
277. C2×SD16C2×D157semidirect
281. #131C2×D157semidirect
285. #108C2×D157semidirect
291. #101C2×D157semidirect
293. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580852
#237580832
#237580822
#237580792
#237576442
#237575392
#237575372
#237575352
#237574582
#237574542
#237574432
#237574392
#237573892
#237573882
#471×C1572
#3291157