Finite Group Databasebeta

← #36335899

Subgroups — #36335899

Order 21376.

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#24924080C211yesmaximal, characteristic, complemented
2#24924079C211yesmaximal, characteristic, complemented
3#24924073C211yesmaximal, characteristic, complemented
4#24924072C211yesmaximal, characteristic, complemented
5#24923656C211yesmaximal, characteristic, complemented
6#24923538C211yesmaximal, characteristic, complemented
7#24923537C211yesmaximal, characteristic, complemented
8#24923531C211yesmaximal, characteristic, complemented
9#24923530C211yesmaximal, characteristic, complemented
10#24923448C211yesmaximal, characteristic, complemented
11#24923443C211yesmaximal, characteristic, complemented
12#24923436C211yesmaximal, characteristic, complemented
13#24923388C211yesmaximal, characteristic, complemented
14#24923388C211yesmaximal, characteristic, complemented
15#453×C167C211yesmaximal, characteristic, complemented
16#19128879C2211yescharacteristic, complemented
17#19128878C2211yescharacteristic
18#19128877C2211yescharacteristic, complemented
19#19128865C2211yescharacteristic, complemented
20#19128862—21no—
21#19128862C2211yescharacteristic, complemented
22#19128861—21no—
23#19128860C2211yescharacteristic, complemented
24#19128859—21no—
25#19128858—21no—
26#19128858—21no—
27#19128858C2211yescharacteristic
28#19128858C2211yescharacteristic, complemented
29#19128857—21no—
30#19128857C2211yescharacteristic, complemented
31#19128856—21no—
32#19128856—21no—
33#19128853—21no—
34#19128852—21no—
35#19128852—21no—
36#19128852C2211yescharacteristic
37#19128852C2211yescharacteristic
38#19128851—21no—
39C2×#14581938—21no—
40C2×#14581938C2211yescharacteristic, complemented
41C2×#14581938C2211yescharacteristic, complemented
42SD16×D167—21no—
43C2×#14581935—21no—
44C2×#14581935C2211yescharacteristic, complemented
45C2×#14581935C2211yescharacteristic, complemented
46D8×D167—21no—
47#19128840C2211yescharacteristic, complemented
48#19128840C2211yescharacteristic, complemented
49#19128839C2211yescharacteristic, complemented
50#19128839C2211yescharacteristic, complemented
51#19128820C2211yescharacteristic, complemented
52#19128767C2211yescharacteristic, complemented
53#19128767C2211yescharacteristic, complemented
54#19128760C2211yescharacteristic, complemented
55#19128760C2211yescharacteristic, complemented
56#19128750C2211yescharacteristic, complemented
57#19128749C2211yescharacteristic, complemented
58#19128748C2211yescharacteristic, complemented
59#19128743C2211yescharacteristic, complemented
60H(5,𝔽2)×C167C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C167C2211yescharacteristic
62AGL(1,ℤ/8)×C167C2211yescharacteristic, complemented
63#127×C167C2211yescharacteristic, complemented
64#104×C167C2211yescharacteristic, complemented
65#104×C167C2211yescharacteristic, complemented
66#100×C167C2211yescharacteristic, complemented
67#14581951—21no—
68#14581950—21no—
69#14581950—21no—
70#14581950C2311yescharacteristic
71#14581950C2311yescharacteristic
72#14581949—21no—
73#14581949—21no—
74#14581949—21no—
75#14581949—41no—
76C2×Dic334—21no—
77C2×Dic334C2311yescharacteristic
78Q8×D167—21no—
79C2×D668—21no—
80C2×D668C2311yescharacteristic
81C2×C22⋊D167—41no—
82D4×D167—21no—
83D4×D167—21no—
84D4×D167—21no—
85D4×D167—41no—
86Dic668—21no—
87#14581940—21no—
88#14581940D411yescharacteristic, complemented
89#14581940D411yescharacteristic, complemented
90#14581939—21no—
91#14581939—21no—
92#14581938—21no—
93#14581938—21no—
94#14581938—21no—
95#14581938—21no—
96#14581938—21no—
97#14581938—21no—
98#14581938—21no—
99#14581938D411yescharacteristic, complemented
100#14581938D411yescharacteristic, complemented
101#14581937—21no—
102#14581937D411yescharacteristic, complemented
103#14581937D411yescharacteristic, complemented
104D1336—21no—
105#14581935—21no—
106#14581935—21no—
107#14581935—21no—
108#14581935—21no—
109#14581935—21no—
110#14581935—21no—
111#14581935—21no—
112#14581935D411yescharacteristic, complemented
113#14581935D411yescharacteristic, complemented
114#14581934—21no—
115#14581934—21no—
116#14581934C2311yescharacteristic
117#14581934C2311yescharacteristic
118#14581933—21no—
119#14581933—21no—
120C2×C167⋊C8—21no—
121C2×C167⋊C8—21no—
122C2×C167⋊C8C2311yescharacteristic
123C2×C167⋊C8C2311yescharacteristic
124C8×D167—21no—
125C8×D167—21no—
126(C2×D167)⋊C4—41no—
127#41×C167—21no—
128#41×C167—21no—
129#41×C167C2311yescharacteristic
130#41×C167C2311yescharacteristic
131C2×D4×C167—21no—
132C2×D4×C167—21no—
133C2×D4×C167—41no—
134C2×D4×C167—41no—
135C2×D4×C167C2311yescharacteristic
136C2×D4×C167C2311yescharacteristic
137SD16×C167—21no—
138SD16×C167—21no—
139D8×C167—21no—
140D8×C167—21no—
141M16×C167C2311yescharacteristic
142M16×C167C2311yescharacteristic
143C42×C167C2311yescharacteristic
144C22×D167—21no—
145C22×D167—41no—
146C22×D167—41no—
147Dic334—21no—
148Dic334—21no—
149Dic334—41no—
150Dic334C2×D411yescharacteristic
151Dic334C2×D411yescharacteristic
152D668—21no—
153D668—21no—
154D668—41no—
155D668C2×D411yescharacteristic
156D668C2×D411yescharacteristic
157C22⋊D167—21no—
158C22⋊D167—21no—
159C22⋊D167—41no—
160C22⋊D167—41no—
161C22⋊D167—41no—
162C22⋊D167—41no—
163C2×Dic167—21no—
164C2×Dic167—41no—
165C2×Dic167—41no—
166C4×D167—21no—
167C4×D167—21no—
168C4×D167—41no—
169C4×D167—41no—
170C167⋊C8—21no—
171C167⋊C8—21no—
172C167⋊C8C2×D411yescharacteristic
173C167⋊C8C2×D411yescharacteristic
174C167⋊C8C2×D411yescharacteristic
175C167⋊C8C2×D411yescharacteristic
176C23×C167—21no—
177C23×C167—21no—
178C23×C167—41no—
179Q8×C167C2×D411yescharacteristic
180Q8×C167C2×D411yescharacteristic
181D4×C167—21no—
182D4×C167—21no—
183D4×C167—21no—
184D4×C167—21no—
185D4×C167—21no—
186D4×C167—21no—
187D4×C167—41no—
188D4×C167—41no—
189D4×C167—82no—
190D4×C167C2×D411yescharacteristic
191D4×C167C2×D411yescharacteristic
192C2×C4×C167—21no—
193C2×C4×C167—21no—
194C2×C4×C167—21no—
195C2×C4×C167—41no—
196C2×C4×C167C2411yescharacteristic
197C8×C167—21no—
198C8×C167—21no—
199C22×C167—21no—
200C22×C167—21no—
201C22×C167—41no—
202C22×C167—41no—
203C22×C167—81no—
204C22×C167—81no—
205C22×C167—161no—
206C22×C167H(5,𝔽2)11yescharacteristic
207C2×D167—21no—
208C2×D167—21no—
209C2×D167—41no—
210C2×D167—81no—
211C2×D167—161no—
212C2×D167—161no—
213C4×C167—21no—
214C4×C167—21no—
215C4×C167—41no—
216C4×C167C22×D411yescharacteristic
217C4×C167C22×D411yescharacteristic
218Dic167—21no—
219Dic167—21no—
220Dic167—41no—
221C2×C167—21no—
222C2×C167—41no—
223C2×C167—41no—
224C2×C167—81no—
225C2×C167—81no—
226C2×C167D4211yescharacteristic
227D167—41no—
228D167—41no—
229D167—81no—
230C167#328811yescharacteristic, complemented, Sylow
231#3288—1671nomaximal, Sylow
232#577—1671no—
233#577—1671no—
234#576—1671no—
235#576—1671no—
236#496—1671no—
237#472—1671no—
238#472—1671no—
239#471—1671no—
240#471—1671no—
241#455—1671no—
242#454—1671no—
243#453—1671no—
244#453D16711yescharacteristic, complemented
245#444—1671no—
246#444—1671no—
2472−1+4—1671no—
248H(5,𝔽2)—1671no—
249H(5,𝔽2)—1671no—
250H(5,𝔽2)C2×D16711yescharacteristic, complemented
251#137—3341no—
252#137—3341no—
253#137—1671no—
254#137—1671no—
255AGL(1,ℤ/8)—3341no—
256AGL(1,ℤ/8)—3341no—
257AGL(1,ℤ/8)—3341no—
258AGL(1,ℤ/8)—3341no—
259AGL(1,ℤ/8)—3341no—
260AGL(1,ℤ/8)—3341no—
261AGL(1,ℤ/8)—1671no—
262AGL(1,ℤ/8)—1671no—
263AGL(1,ℤ/8)—1671no—
264AGL(1,ℤ/8)—1671no—
265AGL(1,ℤ/8)C2×D16711yescharacteristic
266AGL(1,ℤ/8)C2×D16711yescharacteristic, complemented
267#135—3341no—
268#135—3341no—
269#135—3341no—
270#135—3341no—
271#135—1671no—
272#135—1671no—
273C2×SD16—3341no—
274C2×SD16—3341no—
275C2×SD16—1671no—
276C2×SD16—1671no—
277C2×D8—3341no—
278C2×D8—3341no—
279C2×D8—1671no—
280C2×D8—1671no—
281#131—1671no—
282#131—1671no—
283#131—1671no—
284#131—1671no—
285#127C2×D16711yescharacteristic, complemented
286#124—1671no—
287#108—1671no—
288#108—1671no—
289#104—1671no—
290#104—1671no—
291#104C2×D16711yescharacteristic, complemented
292#104C2×D16711yescharacteristic, complemented
293#101—1671no—
294#100—1671no—
295#100—1671no—
296#100C2×D16711yescharacteristic, complemented
297#97—1671no—
298#41—21no—
299#41—21no—
300#41—3341no—
301#41—3341no—
302#41—3341no—
303#41—3341no—
304#41—3341no—
305#41—3341no—
306#41—6681no—
307#41—1671no—
308#41—1671no—
309#41C22×D16711yescharacteristic
310#41C22×D16711yescharacteristic
311C2×Q8—3341no—
312C2×Q8—3341no—
313C2×Q8—1671no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—3341no—
319C2×D4—3341no—
320C2×D4—3341no—
321C2×D4—3341no—
322C2×D4—6681no—
323C2×D4—6681no—
324C2×D4—1671no—
325C2×D4C22×D16711yescharacteristic
326C2×D4C22×D16711yescharacteristic
327Q16—3341no—
328Q16—3341no—
329Q16—1671no—
330Q16—1671no—
331SD16—21no—
332SD16—21no—
333SD16—3341no—
334SD16—3341no—
335SD16—3341no—
336SD16—3341no—
337SD16—3341no—
338SD16—3341no—
339SD16—3341no—
340SD16—3341no—
341SD16—3341no—
342SD16—3341no—
343SD16—1671no—
344SD16—1671no—
345SD16—1671no—
346SD16—1671no—
347D8—21no—
348D8—21no—
349D8—3341no—
350D8—3341no—
351D8—3341no—
352D8—3341no—
353D8—3341no—
354D8—3341no—
355D8—3341no—
356D8—3341no—
357D8—1671no—
358D8—1671no—
359M16—3341no—
360M16—3341no—
361M16—3341no—
362M16—3341no—
363M16—1671no—
364M16—1671no—
365M16C22×D16711yescharacteristic
366M16C22×D16711yescharacteristic
367C2×C8—3341no—
368C2×C8—3341no—
369C2×C8—3341no—
370C2×C8—3341no—
371C2×C8—1671no—
372C2×C8—1671no—
373C22⋊C4—6681no—
374C42C22×D16711yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—3341no—
379C23—6681no—
380C23—6681no—
381Q8—3341no—
382Q8—3341no—
383Q8—6681no—
384Q8—1671no—
385Q8—1671no—
386Q8D4×D16711yescharacteristic
387Q8D4×D16711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—3341no—
398D4—3341no—
399D4—3341no—
400D4—3341no—
401D4—6681no—
402D4—6681no—
403D4—6681no—
404D4—6681no—
405D4—6681no—
406D4—1671no—
407D4—1671no—
408D4D4×D16711yescharacteristic
409D4D4×D16711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—3341no—
415C2×C4—3341no—
416C2×C4—3341no—
417C2×C4—6681no—
418C2×C4—6681no—
419C2×C4—6681no—
420C2×C4—6681no—
421C2×C4C23×D16711yescharacteristic
422C8—21no—
423C8—21no—
424C8—3341no—
425C8—3341no—
426C8—1671no—
427C8—1671no—
428C8—1671no—
429C8—1671no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—13361no—
438C22—3341no—
439C22—3341no—
440C22—26721no—
441C22—26721no—
442C22—6681no—
443C22#1912887711yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3341no—
448C4—3341no—
449C4—6681no—
450C4C2×D4×D16711yescharacteristic
451C4C2×D4×D16711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—13361no—
458C2—6681no—
459C2—6681no—
460C2#2492390111yescharacteristic
subgroupG/H
centerC2#24923901
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#453×C167C2
socleC2×C167D42
hypercenter#453D167
hypocenterC167#3288
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #24924080C2semidirect
2. #24924079C2semidirect
3. #24924073C2semidirect
4. #24924072C2semidirect
5. #24923656C2semidirect
6. #24923538C2semidirect
7. #24923537C2semidirect
8. #24923531C2semidirect
9. #24923530C2semidirect
10. #24923448C2semidirect
11. #24923443C2semidirect
12. #24923436C2semidirect
13. #24923388C2semidirect
14. #24923388C2semidirect
15. #453×C167C2semidirect
16. #19128879C22semidirect
18. #19128877C22semidirect
19. #19128865C22semidirect
21. #19128862C22semidirect
23. #19128860C22semidirect
28. #19128858C22semidirect
30. #19128857C22semidirect
40. C2×#14581938C22semidirect
41. C2×#14581938C22semidirect
44. C2×#14581935C22semidirect
45. C2×#14581935C22semidirect
47. #19128840C22semidirect
48. #19128840C22semidirect
49. #19128839C22semidirect
50. #19128839C22semidirect
51. #19128820C22semidirect
52. #19128767C22semidirect
53. #19128767C22semidirect
54. #19128760C22semidirect
55. #19128760C22semidirect
56. #19128750C22semidirect
57. #19128749C22semidirect
58. #19128748C22semidirect
59. #19128743C22semidirect
60. H(5,𝔽2)×C167C22semidirect
62. AGL(1,ℤ/8)×C167C22semidirect
63. #127×C167C22semidirect
64. #104×C167C22semidirect
65. #104×C167C22semidirect
66. #100×C167C22semidirect
88. #14581940D4semidirect
89. #14581940D4semidirect
99. #14581938D4semidirect
100. #14581938D4semidirect
102. #14581937D4semidirect
103. #14581937D4semidirect
112. #14581935D4semidirect
113. #14581935D4semidirect
230. C167#3288semidirect
244. #453D167semidirect
250. H(5,𝔽2)C2×D167semidirect
266. AGL(1,ℤ/8)C2×D167semidirect
285. #127C2×D167semidirect
291. #104C2×D167semidirect
292. #104C2×D167semidirect
296. #100C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249240802
#249240792
#249240732
#249240722
#249236562
#249235382
#249235372
#249235312
#249235302
#249234482
#249234432
#249234362
#249233882
#453×C1672
#3288167