Finite Group Databasebeta

← #34109862

Subgroups — #34109862

Order 20096.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758069C211yesmaximal, characteristic, complemented
2C2×#19097795C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D157C211yesmaximal, characteristic, complemented
4#23758038C211yesmaximal, characteristic, complemented
5#23757542C211yesmaximal, characteristic, complemented
6#23757541C211yesmaximal, characteristic, complemented
7#23757540C211yesmaximal, characteristic, complemented
8#23757537C211yesmaximal, characteristic, complemented
9#23757536C211yesmaximal, characteristic, complemented
10#23757535C211yesmaximal, characteristic, complemented
11#23757534C211yesmaximal, characteristic, complemented
12#23757320C211yesmaximal, characteristic, complemented
13#23757254C211yesmaximal, characteristic, complemented
14#100×D157C211yesmaximal, characteristic, complemented
15#472×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17#41×D157C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097804C2211yescharacteristic, complemented
20#19097801C2211yescharacteristic, complemented
21#19097800—21no—
22#19097799C2211yescharacteristic
23#19097798—21no—
24#19097797C2211yescharacteristic
25#19097797C2211yescharacteristic, complemented
26#19097796C2211yescharacteristic
27#19097795—21no—
28#19097795—21no—
29#19097795—21no—
30#19097795—21no—
31#19097795—21no—
32#19097795—21no—
33#19097792—21no—
34#19097790—21no—
35C2×#14578943C2211yescharacteristic, complemented
36C2×#14578942C2211yescharacteristic, complemented
37SD16×D157—21no—
38C2×#14578939C2211yescharacteristic, complemented
39D8×D157—21no—
40#19097779—21no—
41#19097779C2211yescharacteristic, complemented
42#19097778C2211yescharacteristic, complemented
43#19097777—21no—
44#19097777C2211yescharacteristic, complemented
45C2×#14578937—21no—
46C2×#14578937C2211yescharacteristic
47M16×D157—21no—
48M16×D157C2211yescharacteristic
49M16×D157C2211yescharacteristic
50M16×D157C2211yescharacteristic
51#19097707C2211yescharacteristic, complemented
52#19097706C2211yescharacteristic, complemented
53#19097706C2211yescharacteristic, complemented
54#19097689C2211yescharacteristic, complemented
55#19097689C2211yescharacteristic, complemented
56#19097688C2211yescharacteristic, complemented
57#19097688C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59#19097687C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C157C2211yescharacteristic
61AGL(1,ℤ/8)×C157C2211yescharacteristic, complemented
62C2×D8×C157C2211yescharacteristic, complemented
63#131×C157C2211yescharacteristic, complemented
64#108×C157C2211yescharacteristic, complemented
65#100×C157C2211yescharacteristic, complemented
66#100×C157C2211yescharacteristic, complemented
67C23×D157—21no—
68#14578955D411yescharacteristic
69#14578954—21no—
70#14578954C2311yescharacteristic
71#14578953—21no—
72#14578953—21no—
73#14578953—21no—
74#14578953—21no—
75#14578953—21no—
76#14578953D411yescharacteristic
77C2×Dic314C2311yescharacteristic
78Q8×D157D411yescharacteristic
79C2×D628C2311yescharacteristic
80C2×C22⋊D157—21no—
81C2×C22⋊D157—21no—
82D4×D157—21no—
83D4×D157—21no—
84D4×D157—21no—
85D4×D157—21no—
86D4×D157—21no—
87D4×D157D411yescharacteristic
88C22×Dic157—21no—
89C2×C4×D157—21no—
90C2×C4×D157C2311yescharacteristic
91Dic628—21no—
92#14578944—21no—
93#14578943—21no—
94#14578943—21no—
95#14578943—21no—
96#14578943—21no—
97#14578942—21no—
98#14578942—21no—
99#14578942—21no—
100#14578942—21no—
101#14578942—21no—
102#14578941—21no—
103D1256—21no—
104#14578939—21no—
105#14578939—21no—
106#14578939—21no—
107#14578939—21no—
108#14578939—21no—
109#14578938—21no—
110#14578938C2311yescharacteristic
111#14578938C2311yescharacteristic
112#14578938C2311yescharacteristic
113#14578937—21no—
114#14578937—21no—
115#14578937—21no—
116#14578937—21no—
117#14578937—21no—
118#14578937D411yescharacteristic
119#14578937D411yescharacteristic
120#14578937D411yescharacteristic
121#14578937D411yescharacteristic
122C2×#11929434—21no—
123C2×#11929434C2311yescharacteristic
124C8×D157—21no—
125C8×D157—21no—
126C8×D157—21no—
127C8×D157—21no—
128C8×D157—21no—
129#41×C157C2311yescharacteristic
130C2×D4×C157C2311yescharacteristic
131C2×D4×C157C2311yescharacteristic
132SD16×C157—21no—
133SD16×C157—21no—
134D8×C157—21no—
135D8×C157—21no—
136D8×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—82no—
149C22×D157#4111yescharacteristic
150Dic314—21no—
151Dic314—21no—
152Dic314—21no—
153Dic314C2×D411yescharacteristic
154D628—21no—
155D628—21no—
156D628—21no—
157D628C2×D411yescharacteristic
158C22⋊D157—21no—
159C22⋊D157—21no—
160C22⋊D157—21no—
161C22⋊D157—41no—
162C22⋊D157—41no—
163C22⋊D157—41no—
164C22⋊D157—41no—
165C2×Dic157—21no—
166C2×Dic157—21no—
167C2×Dic157—41no—
168C2×Dic157—41no—
169C2×Dic157#4111yescharacteristic
170C4×D157—21no—
171C4×D157—21no—
172C4×D157—21no—
173C4×D157C2×D411yescharacteristic
174C4×D157C2×D411yescharacteristic
175C4×D157C2×D411yescharacteristic
176C4×D157C2×D411yescharacteristic
177#11929434—21no—
178#11929434—21no—
179#11929434—21no—
180#11929434C2×D411yescharacteristic
181#11929434C2×D411yescharacteristic
182C23×C157—21no—
183C23×C157—21no—
184Q8×C157C2×D411yescharacteristic
185D4×C157—21no—
186D4×C157—21no—
187D4×C157—21no—
188D4×C157—21no—
189D4×C157—21no—
190D4×C157C2×D411yescharacteristic
191C2×C4×C157—21no—
192C2×C4×C157C2411yescharacteristic
193C8×C157—21no—
194C8×C157—21no—
195C8×C157—21no—
196C8×C157C2×D411yescharacteristic
197C8×C157C2×D411yescharacteristic
198C22×C157—21no—
199C22×C157—41no—
200C22×C157—41no—
201C22×C157—81no—
202C22×C157—81no—
203C22×C157C2×#4111yescharacteristic
204C2×D157—21no—
205C2×D157—41no—
206C2×D157—41no—
207C2×D157—81no—
208C2×D157—82no—
209C2×D157—82no—
210C2×D157—81no—
211C2×D157#12111yescharacteristic
212C2×D157#12111yescharacteristic
213C4×C157—21no—
214C4×C157C22×D411yescharacteristic
215C4×C157C22×D411yescharacteristic
216Dic157—21no—
217Dic157—41no—
218Dic157#12111yescharacteristic
219Dic157#12111yescharacteristic
220C2×C157—21no—
221C2×C157—41no—
222C2×C157—81no—
223C2×C157—81no—
224C2×C157C2×#12111yescharacteristic
225D157—21no—
226D157—21no—
227D157—41no—
228D157—81no—
229C157#306811yescharacteristic, complemented, Sylow
230#3068—1571nomaximal, Sylow
231#575—1571no—
232C2×AGL(1,ℤ/8)—1571no—
233C2×AGL(1,ℤ/8)—1571no—
234#568—1571no—
235#472—1571no—
236#472—1571no—
237#472—1571no—
238#472D15711yescharacteristic, complemented
239#471—1571no—
240#471—1571no—
241#471—1571no—
242#471—1571no—
243#430—1571no—
244#413—1571no—
245C2×#100—1571no—
246C2×#41—1571no—
247C2×#41—1571no—
248C22×D4—1571no—
249#137—3141no—
250#137—1571no—
251#137—1571no—
252AGL(1,ℤ/8)—3141no—
253AGL(1,ℤ/8)—3141no—
254AGL(1,ℤ/8)—3141no—
255AGL(1,ℤ/8)—3141no—
256AGL(1,ℤ/8)—3141no—
257AGL(1,ℤ/8)—3141no—
258AGL(1,ℤ/8)—3141no—
259AGL(1,ℤ/8)—1571no—
260AGL(1,ℤ/8)—1571no—
261AGL(1,ℤ/8)—1571no—
262AGL(1,ℤ/8)—1571no—
263AGL(1,ℤ/8)C2×D15711yescharacteristic
264AGL(1,ℤ/8)C2×D15711yescharacteristic, complemented
265#135—3141no—
266#135—3141no—
267C2×SD16—3141no—
268C2×SD16—1571no—
269C2×SD16—1571no—
270C2×D8—3141no—
271C2×D8—1571no—
272C2×D8C2×D15711yescharacteristic, complemented
273#131—3141no—
274#131—3141no—
275#131—1571no—
276#131—1571no—
277#131—1571no—
278#131C2×D15711yescharacteristic, complemented
279C2×M16—3141no—
280C2×M16—3141no—
281C2×M16—1571no—
282C2×M16—1571no—
283C2×M16—1571no—
284C2×M16—1571no—
285#108—1571no—
286#108—1571no—
287#108—1571no—
288#108C2×D15711yescharacteristic, complemented
289#101—1571no—
290#101—1571no—
291#100—1571no—
292#100—1571no—
293#100—1571no—
294#100—1571no—
295#100C2×D15711yescharacteristic, complemented
296#100C2×D15711yescharacteristic, complemented
297C24—3141no—
298#41—3141no—
299#41—3141no—
300#41—3141no—
301#41—3141no—
302#41—3141no—
303#41—3141no—
304#41—1571no—
305#41—1571no—
306#41—1571no—
307#41C22×D15711yescharacteristic
308C2×Q8—1571no—
309C2×Q8—1571no—
310C2×D4—3141no—
311C2×D4—3141no—
312C2×D4—3141no—
313C2×D4—3141no—
314C2×D4—3141no—
315C2×D4—3141no—
316C2×D4—3141no—
317C2×D4—1571no—
318C2×D4—1571no—
319C2×D4C22×D15711yescharacteristic
320C2×D4C22×D15711yescharacteristic
321C22×C4—3141no—
322C22×C4—3141no—
323C22×C4—1571no—
324Q16—3141no—
325Q16—3141no—
326SD16—21no—
327SD16—21no—
328SD16—3141no—
329SD16—3141no—
330SD16—3141no—
331SD16—3141no—
332SD16—3141no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—3141no—
343D8—3141no—
344D8—3141no—
345D8—3141no—
346D8—3141no—
347D8—3141no—
348M16—21no—
349M16—3141no—
350M16—3141no—
351M16—3141no—
352M16—3141no—
353M16—3141no—
354M16—3141no—
355M16—1571no—
356M16—1571no—
357M16—1571no—
358M16—1571no—
359M16—1571no—
360M16—1571no—
361M16—1571no—
362M16C22×D15711yescharacteristic
363M16C22×D15711yescharacteristic
364M16C22×D15711yescharacteristic
365C2×C8—21no—
366C2×C8—3141no—
367C2×C8—3141no—
368C2×C8—3141no—
369C2×C8—3141no—
370C2×C8—3141no—
371C2×C8—3141no—
372C2×C8—1571no—
373C2×C8C22×D15711yescharacteristic
374C23—21no—
375C23—21no—
376C23—3141no—
377C23—3141no—
378C23—6281no—
379C23—6281no—
380C23—1571no—
381C23—12562no—
382Q8—3141no—
383Q8—3141no—
384Q8—3141no—
385Q8—1571no—
386Q8D4×D15711yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—3141no—
393D4—3141no—
394D4—3141no—
395D4—3141no—
396D4—3141no—
397D4—3141no—
398D4—6281no—
399D4—6281no—
400D4—6281no—
401D4—6281no—
402D4—1571no—
403D4D4×D15711yescharacteristic
404C2×C4—21no—
405C2×C4—3141no—
406C2×C4—3141no—
407C2×C4—3141no—
408C2×C4—3141no—
409C2×C4—3141no—
410C2×C4—6281no—
411C2×C4—6281no—
412C2×C4—1571no—
413C2×C4—1571no—
414C2×C4—1571no—
415C2×C4—1571no—
416C2×C4—1571no—
417C2×C4C23×D15711yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—3141no—
422C8—3141no—
423C8—3141no—
424C8—1571no—
425C8—1571no—
426C8D4×D15711yescharacteristic
427C8D4×D15711yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—3141no—
434C22—6281no—
435C22—6281no—
436C22—1571no—
437C22—1571no—
438C22—12561no—
439C22—12561no—
440C22—12562no—
441C22—12562no—
442C22#41×D15711yescharacteristic
443C4—21no—
444C4—3141no—
445C4—6281no—
446C4—1571no—
447C4—1571no—
448C4C2×D4×D15711yescharacteristic
449C4C2×D4×D15711yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—3141no—
455C2—3141no—
456C2—6281no—
457C2—12561no—
458C2#121×D15711yescharacteristic
subgroupG/H
centerC2#121×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#472×C157C2
socleC2×C157C2×#121
hypercenter#472D157
hypocenterC157#3068
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #23758069C2semidirect
2. C2×#19097795C2semidirect
3. AGL(1,ℤ/8)×D157C2semidirect
4. #23758038C2semidirect
5. #23757542C2semidirect
6. #23757541C2semidirect
7. #23757540C2semidirect
8. #23757537C2semidirect
9. #23757536C2semidirect
10. #23757535C2semidirect
11. #23757534C2semidirect
12. #23757320C2semidirect
13. #23757254C2semidirect
14. #100×D157C2semidirect
15. #472×C157C2semidirect
19. #19097804C22semidirect
20. #19097801C22semidirect
25. #19097797C22semidirect
35. C2×#14578943C22semidirect
36. C2×#14578942C22semidirect
38. C2×#14578939C22semidirect
41. #19097779C22semidirect
42. #19097778C22semidirect
44. #19097777C22semidirect
51. #19097707C22semidirect
52. #19097706C22semidirect
53. #19097706C22semidirect
54. #19097689C22semidirect
55. #19097689C22semidirect
56. #19097688C22semidirect
57. #19097688C22semidirect
58. #19097687C22semidirect
59. #19097687C22semidirect
61. AGL(1,ℤ/8)×C157C22semidirect
62. C2×D8×C157C22semidirect
63. #131×C157C22semidirect
64. #108×C157C22semidirect
65. #100×C157C22semidirect
66. #100×C157C22semidirect
229. C157#3068semidirect
238. #472D157semidirect
264. AGL(1,ℤ/8)C2×D157semidirect
272. C2×D8C2×D157semidirect
278. #131C2×D157semidirect
288. #108C2×D157semidirect
295. #100C2×D157semidirect
296. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580692
C2×#190977952
AGL(1,ℤ/8)×D1572
#237580382
#237575422
#237575412
#237575402
#237575372
#237575362
#237575352
#237575342
#237573202
#237572542
#100×D1572
#472×C1572
#3068157