Finite Group Databasebeta

← #471×D157 #34109831

Subgroups — #471×D157 #34109831

Order 20096.

468 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#137×D157C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D157C211yesmaximal, characteristic
3C2×SD16×D157C211yesmaximal, characteristic, complemented
4#131×D157C211yesmaximal, characteristic, complemented
5#23757539C211yesmaximal, characteristic, complemented
6#23757538C211yesmaximal, characteristic, complemented
7#23757537C211yesmaximal, characteristic, complemented
8#23757536C211yesmaximal, characteristic, complemented
9#23757535C211yesmaximal, characteristic, complemented
10#23757534C211yesmaximal, characteristic, complemented
11#23757533C211yesmaximal, characteristic, complemented
12#108×D157C211yesmaximal, characteristic, complemented
13#101×D157C211yesmaximal, characteristic, complemented
14#100×D157C211yesmaximal, characteristic, complemented
15#471×C157C211yesmaximal, characteristic, complemented
16#41×D157C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097804C2211yescharacteristic, complemented
20#19097803—21no—
21#19097802C2211yescharacteristic, complemented
22#19097801C2211yescharacteristic, complemented
23#19097800—21no—
24#19097799C2211yescharacteristic
25#19097798—21no—
26#19097797C2211yescharacteristic
27#19097796C2211yescharacteristic
28#19097795—21no—
29Q16×D157—21no—
30C2×#14578943C2211yescharacteristic, complemented
31C2×#14578942C2211yescharacteristic, complemented
32C2×#14578941C2211yescharacteristic, complemented
33SD16×D157—21no—
34SD16×D157—21no—
35SD16×D157—21no—
36SD16×D157—21no—
37SD16×D157—21no—
38SD16×D157—21no—
39D8×D157—21no—
40#19097779—21no—
41#19097779C2211yescharacteristic, complemented
42#19097778C2211yescharacteristic, complemented
43#19097777—21no—
44#19097777C2211yescharacteristic, complemented
45M16×D157—21no—
46M16×D157C2211yescharacteristic
47M16×D157C2211yescharacteristic
48M16×D157C2211yescharacteristic
49C2×C8×D157—21no—
50C2×C8×D157C2211yescharacteristic
51#19097707C2211yescharacteristic, complemented
52#19097706C2211yescharacteristic, complemented
53#19097706C2211yescharacteristic, complemented
54#19097690C2211yescharacteristic, complemented
55#19097689C2211yescharacteristic, complemented
56#19097689C2211yescharacteristic, complemented
57#19097688C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59#19097687C2211yescharacteristic, 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
67C23×D157—21no—
68#14578955D411yescharacteristic
69#14578954—21no—
70#14578954C2311yescharacteristic
71#14578953—21no—
72#14578953D411yescharacteristic
73C2×Dic314—21no—
74C2×Dic314C2311yescharacteristic
75Q8×D157—21no—
76Q8×D157—21no—
77Q8×D157—21no—
78Q8×D157—21no—
79Q8×D157D411yescharacteristic
80C2×D628C2311yescharacteristic
81C2×C22⋊D157—21no—
82D4×D157—21no—
83D4×D157—21no—
84D4×D157—21no—
85D4×D157—21no—
86D4×D157—21no—
87D4×D157D411yescharacteristic
88C2×C4×D157—21no—
89C2×C4×D157—21no—
90C2×C4×D157C2311yescharacteristic
91Dic628—21no—
92#14578944—21no—
93#14578944—21no—
94#14578943—21no—
95#14578943—21no—
96#14578943—21no—
97#14578943—21no—
98#14578942—21no—
99#14578942—21no—
100#14578942—21no—
101#14578942—21no—
102#14578941—21no—
103#14578941—21no—
104#14578941—21no—
105#14578941—21no—
106D1256—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—
118C2×#11929434—21no—
119C2×#11929434C2311yescharacteristic
120C8×D157—21no—
121C8×D157—21no—
122C8×D157—21no—
123C8×D157—21no—
124C8×D157—21no—
125C8×D157D411yescharacteristic
126C8×D157D411yescharacteristic
127C8×D157D411yescharacteristic
128C8×D157D411yescharacteristic
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—82no—
149C22×D157#4111yescharacteristic
150Dic314—21no—
151Dic314—21no—
152Dic314—21no—
153Dic314—41no—
154Dic314—41no—
155Dic314C2×D411yescharacteristic
156D628—21no—
157D628—21no—
158D628—21no—
159D628C2×D411yescharacteristic
160C22⋊D157—21no—
161C22⋊D157—21no—
162C22⋊D157—21no—
163C22⋊D157—41no—
164C22⋊D157—41no—
165C2×Dic157—21no—
166C2×Dic157—21no—
167C2×Dic157#4111yescharacteristic
168C4×D157—21no—
169C4×D157—21no—
170C4×D157—21no—
171C4×D157—41no—
172C4×D157—41no—
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—
183Q8×C157—21no—
184Q8×C157—21no—
185Q8×C157C2×D411yescharacteristic
186D4×C157—21no—
187D4×C157—21no—
188D4×C157—21no—
189D4×C157C2×D411yescharacteristic
190C2×C4×C157—21no—
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—81no—
201C22×C157C2×#4111yescharacteristic
202C2×D157—22no—
203C2×D157—21no—
204C2×D157—42no—
205C2×D157—41no—
206C2×D157—82no—
207C2×D157—82no—
208C2×D157—81no—
209C2×D157#12111yescharacteristic
210C2×D157#12111yescharacteristic
211C4×C157—21no—
212C4×C157—41no—
213C4×C157C22×D411yescharacteristic
214C4×C157C22×D411yescharacteristic
215Dic157—21no—
216Dic157—41no—
217Dic157#12111yescharacteristic
218Dic157#12111yescharacteristic
219C2×C157—21no—
220C2×C157—41no—
221C2×C157—81no—
222C2×C157C2×#12111yescharacteristic
223D157—21no—
224D157—41no—
225D157—81no—
226D157#47112yesdirect product factor
227C157C2×#47111yescharacteristic, complemented, Sylow
228C2×#471—1571nomaximal, Sylow
229C2×#137—1571no—
230C2×AGL(1,ℤ/8)—1571no—
231C22×SD16—1571no—
232C2×#131—1571no—
233#471—1571no—
234#471—1571no—
235#471—1571no—
236#471—1571no—
237#471—1571no—
238#471—1571no—
239#471—1571no—
240#471D15711yescharacteristic, direct product factor
241C2×#108—1571no—
242C2×#101—1571no—
243C2×#100—1571no—
244C2×#41—1571no—
245C22×Q8—1571no—
246C22×D4—1571no—
247#137—3141no—
248#137—3141no—
249#137—1571no—
250#137—1571no—
251#137—1571no—
252#137C2×D15711yescharacteristic, complemented
253AGL(1,ℤ/8)—3141no—
254AGL(1,ℤ/8)—3141no—
255AGL(1,ℤ/8)—1571no—
256AGL(1,ℤ/8)—1571no—
257AGL(1,ℤ/8)—1571no—
258AGL(1,ℤ/8)C2×D15711yescharacteristic
259C2×Q16—3141no—
260C2×SD16—3141no—
261C2×SD16—3141no—
262C2×SD16—3141no—
263C2×SD16—3141no—
264C2×SD16—3141no—
265C2×SD16—3141no—
266C2×SD16—1571no—
267C2×SD16—1571no—
268C2×SD16—1571no—
269C2×SD16C2×D15711yescharacteristic, complemented
270C2×D8—3141no—
271#131—3141no—
272#131—3141no—
273#131—1571no—
274#131—1571no—
275#131—1571no—
276#131C2×D15711yescharacteristic, complemented
277C2×M16—3141no—
278C2×M16—1571no—
279C2×M16—1571no—
280C2×M16—1571no—
281C22×C8—3141no—
282C22×C8—1571no—
283#108—1571no—
284#108—1571no—
285#108—1571no—
286#108C2×D15711yescharacteristic, complemented
287#101—1571no—
288#101—1571no—
289#101—1571no—
290#101C2×D15711yescharacteristic, complemented
291#100—1571no—
292#100—1571no—
293#100—1571no—
294#100C2×D15711yescharacteristic, complemented
295C24—3141no—
296#41—3141no—
297#41—3141no—
298#41—1571no—
299#41—1571no—
300#41—1571no—
301#41C22×D15711yescharacteristic
302C2×Q8—3141no—
303C2×Q8—3141no—
304C2×Q8—3141no—
305C2×Q8—3141no—
306C2×Q8—3141no—
307C2×Q8—1571no—
308C2×Q8—1571no—
309C2×Q8C22×D15711yescharacteristic
310C2×D4—3141no—
311C2×D4—3141no—
312C2×D4—3141no—
313C2×D4—3141no—
314C2×D4—3141no—
315C2×D4—3141no—
316C2×D4—1571no—
317C2×D4—1571no—
318C2×D4C22×D15711yescharacteristic
319C22×C4—3141no—
320C22×C4—3141no—
321C22×C4—1571no—
322Q16—21no—
323Q16—3141no—
324Q16—3141no—
325Q16—3141no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—3141no—
331SD16—3141no—
332SD16—3141no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342D8—21no—
343D8—3141no—
344D8—3141no—
345D8—3141no—
346M16—21no—
347M16—3141no—
348M16—3141no—
349M16—3141no—
350M16—3141no—
351M16—3141no—
352M16—3141no—
353M16—1571no—
354M16—1571no—
355M16—1571no—
356M16C22×D15711yescharacteristic
357M16C22×D15711yescharacteristic
358M16C22×D15711yescharacteristic
359C2×C8—21no—
360C2×C8—3141no—
361C2×C8—3141no—
362C2×C8—3141no—
363C2×C8—3141no—
364C2×C8—3141no—
365C2×C8—3141no—
366C2×C8—1571no—
367C2×C8—1571no—
368C2×C8—1571no—
369C2×C8—1571no—
370C2×C8—1571no—
371C2×C8C22×D15711yescharacteristic
372C23—21no—
373C23—3141no—
374C23—3141no—
375C23—6281no—
376C23—6281no—
377C23—1571no—
378C23—12562no—
379Q8—21no—
380Q8—21no—
381Q8—3141no—
382Q8—3141no—
383Q8—3141no—
384Q8—6281no—
385Q8—6281no—
386Q8—1571no—
387Q8D4×D15711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—3141no—
392D4—3141no—
393D4—3141no—
394D4—3141no—
395D4—3141no—
396D4—3141no—
397D4—6281no—
398D4—6281no—
399D4—1571no—
400D4D4×D15711yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—3141no—
404C2×C4—3141no—
405C2×C4—3141no—
406C2×C4—3141no—
407C2×C4—3141no—
408C2×C4—6281no—
409C2×C4—6281no—
410C2×C4—1571no—
411C2×C4—1571no—
412C2×C4—1571no—
413C2×C4—1571no—
414C2×C4—1571no—
415C2×C4C23×D15711yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—3141no—
420C8—3141no—
421C8—3141no—
422C8—1571no—
423C8—1571no—
424C8D4×D15711yescharacteristic
425C8D4×D15711yescharacteristic
426C22—21no—
427C22—41no—
428C22—81no—
429C22—3142no—
430C22—3141no—
431C22—6282no—
432C22—6281no—
433C22—1571no—
434C22—1571no—
435C22—12561no—
436C22—12562no—
437C22—12562no—
438C22#41×D15711yescharacteristic
439C4—21no—
440C4—41no—
441C4—3141no—
442C4—6281no—
443C4—1571no—
444C4—1571no—
445C4C2×D4×D15711yescharacteristic
446C4C2×D4×D15711yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—3141no—
451C2—6281no—
452C2—1572no—
453C2—12561no—
454C2#121×D15711yescharacteristic
subgroupG/H
centerC2#121×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#471×C157C2
socleC2×C157C2×#121
prime radicalAGL(1,ℤ/8)×D157C2
hypercenter#471D157
hypocenterC157C2×#471
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. #137×D157C2semidirect
3. C2×SD16×D157C2semidirect
4. #131×D157C2semidirect
5. #23757539C2semidirect
6. #23757538C2semidirect
7. #23757537C2semidirect
8. #23757536C2semidirect
9. #23757535C2semidirect
10. #23757534C2semidirect
11. #23757533C2semidirect
12. #108×D157C2semidirect
13. #101×D157C2semidirect
14. #100×D157C2semidirect
15. #471×C157C2semidirect
19. #19097804C22semidirect
21. #19097802C22semidirect
22. #19097801C22semidirect
30. C2×#14578943C22semidirect
31. C2×#14578942C22semidirect
32. C2×#14578941C22semidirect
41. #19097779C22semidirect
42. #19097778C22semidirect
44. #19097777C22semidirect
51. #19097707C22semidirect
52. #19097706C22semidirect
53. #19097706C22semidirect
54. #19097690C22semidirect
55. #19097689C22semidirect
56. #19097689C22semidirect
57. #19097688C22semidirect
58. #19097687C22semidirect
59. #19097687C22semidirect
60. #137×C157C22semidirect
62. C2×SD16×C157C22semidirect
63. #131×C157C22semidirect
64. #108×C157C22semidirect
65. #101×C157C22semidirect
66. #100×C157C22semidirect
226. D157#471direct (N × H)
227. C157C2×#471semidirect
240. #471D157direct (N × H)
252. #137C2×D157semidirect
269. C2×SD16C2×D157semidirect
276. #131C2×D157semidirect
286. #108C2×D157semidirect
290. #101C2×D157semidirect
294. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#137×D1572
AGL(1,ℤ/8)×D1572
C2×SD16×D1572
#131×D1572
#237575392
#237575382
#237575372
#237575362
#237575352
#237575342
#237575332
#108×D1572
#101×D1572
#100×D1572
#471×C1572
C2×#471157