Finite Group Databasebeta

← #28348222

Subgroups — #28348222

Order 13696.

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#20293438C211yesmaximal, characteristic, complemented
2#20293437C211yesmaximal, characteristic, complemented
3#20293431C211yesmaximal, characteristic, complemented
4#20293430C211yesmaximal, characteristic, complemented
5#20293014C211yesmaximal, characteristic, complemented
6#20292896C211yesmaximal, characteristic, complemented
7#20292895C211yesmaximal, characteristic, complemented
8#20292889C211yesmaximal, characteristic, complemented
9#20292888C211yesmaximal, characteristic, complemented
10#20292806C211yesmaximal, characteristic, complemented
11#20292801C211yesmaximal, characteristic, complemented
12#20292794C211yesmaximal, characteristic, complemented
13#20292746C211yesmaximal, characteristic, complemented
14#20292746C211yesmaximal, characteristic, complemented
15#453×C107C211yesmaximal, characteristic, complemented
16#16812156C2211yescharacteristic, complemented
17#16812155C2211yescharacteristic
18#16812154C2211yescharacteristic, complemented
19#16812142C2211yescharacteristic, complemented
20#16812139—21no—
21#16812139C2211yescharacteristic, complemented
22#16812138—21no—
23#16812137C2211yescharacteristic, complemented
24#16812136—21no—
25#16812135—21no—
26#16812135—21no—
27#16812135C2211yescharacteristic
28#16812135C2211yescharacteristic, complemented
29#16812134—21no—
30#16812134C2211yescharacteristic, complemented
31#16812133—21no—
32#16812133—21no—
33#16812130—21no—
34#16812129—21no—
35#16812129—21no—
36#16812129C2211yescharacteristic
37#16812129C2211yescharacteristic
38#16812128—21no—
39C2×#13133806—21no—
40C2×#13133806C2211yescharacteristic, complemented
41C2×#13133806C2211yescharacteristic, complemented
42SD16×D107—21no—
43C2×#13133803—21no—
44C2×#13133803C2211yescharacteristic, complemented
45C2×#13133803C2211yescharacteristic, complemented
46D8×D107—21no—
47#16812117C2211yescharacteristic, complemented
48#16812117C2211yescharacteristic, complemented
49#16812116C2211yescharacteristic, complemented
50#16812116C2211yescharacteristic, complemented
51#16812097C2211yescharacteristic, complemented
52#16812044C2211yescharacteristic, complemented
53#16812044C2211yescharacteristic, complemented
54#16812037C2211yescharacteristic, complemented
55#16812037C2211yescharacteristic, complemented
56#16812027C2211yescharacteristic, complemented
57#16812026C2211yescharacteristic, complemented
58#16812025C2211yescharacteristic, complemented
59#16812020C2211yescharacteristic, complemented
60H(5,𝔽2)×C107C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C107C2211yescharacteristic
62AGL(1,ℤ/8)×C107C2211yescharacteristic, complemented
63#127×C107C2211yescharacteristic, complemented
64#104×C107C2211yescharacteristic, complemented
65#104×C107C2211yescharacteristic, complemented
66#100×C107C2211yescharacteristic, complemented
67#13133819—21no—
68#13133818—21no—
69#13133818—21no—
70#13133818C2311yescharacteristic
71#13133818C2311yescharacteristic
72#13133817—21no—
73#13133817—21no—
74#13133817—21no—
75#13133817—41no—
76C2×Dic214—21no—
77C2×Dic214C2311yescharacteristic
78Q8×D107—21no—
79C2×D428—21no—
80C2×D428C2311yescharacteristic
81C2×C22⋊D107—41no—
82D4×D107—21no—
83D4×D107—21no—
84D4×D107—21no—
85D4×D107—41no—
86Dic428—21no—
87#13133808—21no—
88#13133808D411yescharacteristic, complemented
89#13133808D411yescharacteristic, complemented
90#13133807—21no—
91#13133807—21no—
92#13133806—21no—
93#13133806—21no—
94#13133806—21no—
95#13133806—21no—
96#13133806—21no—
97#13133806—21no—
98#13133806—21no—
99#13133806D411yescharacteristic, complemented
100#13133806D411yescharacteristic, complemented
101#13133805—21no—
102#13133805D411yescharacteristic, complemented
103#13133805D411yescharacteristic, complemented
104D856—21no—
105#13133803—21no—
106#13133803—21no—
107#13133803—21no—
108#13133803—21no—
109#13133803—21no—
110#13133803—21no—
111#13133803—21no—
112#13133803D411yescharacteristic, complemented
113#13133803D411yescharacteristic, complemented
114#13133802—21no—
115#13133802—21no—
116#13133802C2311yescharacteristic
117#13133802C2311yescharacteristic
118#13133801—21no—
119#13133801—21no—
120C2×C107⋊C8—21no—
121C2×C107⋊C8—21no—
122C2×C107⋊C8C2311yescharacteristic
123C2×C107⋊C8C2311yescharacteristic
124C8×D107—21no—
125C8×D107—21no—
126(C2×D107)⋊C4—41no—
127#41×C107—21no—
128#41×C107—21no—
129#41×C107C2311yescharacteristic
130#41×C107C2311yescharacteristic
131C2×D4×C107—21no—
132C2×D4×C107—21no—
133C2×D4×C107—41no—
134C2×D4×C107—41no—
135C2×D4×C107C2311yescharacteristic
136C2×D4×C107C2311yescharacteristic
137SD16×C107—21no—
138SD16×C107—21no—
139D8×C107—21no—
140D8×C107—21no—
141M16×C107C2311yescharacteristic
142M16×C107C2311yescharacteristic
143C42×C107C2311yescharacteristic
144C23×C107—21no—
145C23×C107—21no—
146C23×C107—41no—
147C22×D107—21no—
148C22×D107—41no—
149C22×D107—41no—
150Q8×C107C2×D411yescharacteristic
151Q8×C107C2×D411yescharacteristic
152D4×C107—21no—
153D4×C107—21no—
154D4×C107—21no—
155D4×C107—21no—
156D4×C107—21no—
157D4×C107—21no—
158D4×C107—41no—
159D4×C107—41no—
160D4×C107—82no—
161D4×C107C2×D411yescharacteristic
162D4×C107C2×D411yescharacteristic
163C2×C4×C107—21no—
164C2×C4×C107—21no—
165C2×C4×C107—21no—
166C2×C4×C107—41no—
167C2×C4×C107C2411yescharacteristic
168C22⋊D107—21no—
169C22⋊D107—21no—
170C22⋊D107—41no—
171C22⋊D107—41no—
172C22⋊D107—41no—
173C22⋊D107—41no—
174C2×Dic107—21no—
175C2×Dic107—41no—
176C2×Dic107—41no—
177D428—21no—
178D428—21no—
179D428—41no—
180D428C2×D411yescharacteristic
181D428C2×D411yescharacteristic
182C4×D107—21no—
183C4×D107—21no—
184C4×D107—41no—
185C4×D107—41no—
186Dic214—21no—
187Dic214—21no—
188Dic214—41no—
189Dic214C2×D411yescharacteristic
190Dic214C2×D411yescharacteristic
191C8×C107—21no—
192C8×C107—21no—
193C107⋊C8—21no—
194C107⋊C8—21no—
195C107⋊C8C2×D411yescharacteristic
196C107⋊C8C2×D411yescharacteristic
197C107⋊C8C2×D411yescharacteristic
198C107⋊C8C2×D411yescharacteristic
199C22×C107—21no—
200C22×C107—21no—
201C22×C107—41no—
202C22×C107—41no—
203C22×C107—81no—
204C22×C107—81no—
205C22×C107—161no—
206C22×C107H(5,𝔽2)11yescharacteristic
207C2×D107—21no—
208C2×D107—21no—
209C2×D107—41no—
210C2×D107—81no—
211C2×D107—161no—
212C2×D107—161no—
213C4×C107—21no—
214C4×C107—21no—
215C4×C107—41no—
216C4×C107C22×D411yescharacteristic
217C4×C107C22×D411yescharacteristic
218Dic107—21no—
219Dic107—21no—
220Dic107—41no—
221C2×C107—21no—
222C2×C107—41no—
223C2×C107—41no—
224C2×C107—81no—
225C2×C107—81no—
226C2×C107D4211yescharacteristic
227D107—41no—
228D107—41no—
229D107—81no—
230#3288—1071nomaximal, Sylow
231C107#328811yescharacteristic, complemented, Sylow
232#577—1071no—
233#577—1071no—
234#576—1071no—
235#576—1071no—
236#496—1071no—
237#472—1071no—
238#472—1071no—
239#471—1071no—
240#471—1071no—
241#455—1071no—
242#454—1071no—
243#453—1071no—
244#453D10711yescharacteristic, complemented
245#444—1071no—
246#444—1071no—
2472−1+4—1071no—
248H(5,𝔽2)—1071no—
249H(5,𝔽2)—1071no—
250H(5,𝔽2)C2×D10711yescharacteristic, complemented
251#137—1071no—
252#137—1071no—
253#137—2141no—
254#137—2141no—
255AGL(1,ℤ/8)—1071no—
256AGL(1,ℤ/8)—1071no—
257AGL(1,ℤ/8)—1071no—
258AGL(1,ℤ/8)—1071no—
259AGL(1,ℤ/8)—2141no—
260AGL(1,ℤ/8)—2141no—
261AGL(1,ℤ/8)—2141no—
262AGL(1,ℤ/8)—2141no—
263AGL(1,ℤ/8)—2141no—
264AGL(1,ℤ/8)—2141no—
265AGL(1,ℤ/8)C2×D10711yescharacteristic
266AGL(1,ℤ/8)C2×D10711yescharacteristic, complemented
267#135—1071no—
268#135—1071no—
269#135—2141no—
270#135—2141no—
271#135—2141no—
272#135—2141no—
273C2×SD16—1071no—
274C2×SD16—1071no—
275C2×SD16—2141no—
276C2×SD16—2141no—
277C2×D8—1071no—
278C2×D8—1071no—
279C2×D8—2141no—
280C2×D8—2141no—
281#131—1071no—
282#131—1071no—
283#131—1071no—
284#131—1071no—
285#127C2×D10711yescharacteristic, complemented
286#124—1071no—
287#108—1071no—
288#108—1071no—
289#104—1071no—
290#104—1071no—
291#104C2×D10711yescharacteristic, complemented
292#104C2×D10711yescharacteristic, complemented
293#101—1071no—
294#100—1071no—
295#100—1071no—
296#100C2×D10711yescharacteristic, complemented
297#97—1071no—
298#41—21no—
299#41—21no—
300#41—1071no—
301#41—1071no—
302#41—4281no—
303#41—2141no—
304#41—2141no—
305#41—2141no—
306#41—2141no—
307#41—2141no—
308#41—2141no—
309#41C22×D10711yescharacteristic
310#41C22×D10711yescharacteristic
311C2×Q8—1071no—
312C2×Q8—2141no—
313C2×Q8—2141no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—1071no—
319C2×D4—4281no—
320C2×D4—4281no—
321C2×D4—2141no—
322C2×D4—2141no—
323C2×D4—2141no—
324C2×D4—2141no—
325C2×D4C22×D10711yescharacteristic
326C2×D4C22×D10711yescharacteristic
327Q16—1071no—
328Q16—1071no—
329Q16—2141no—
330Q16—2141no—
331SD16—21no—
332SD16—21no—
333SD16—1071no—
334SD16—1071no—
335SD16—1071no—
336SD16—1071no—
337SD16—2141no—
338SD16—2141no—
339SD16—2141no—
340SD16—2141no—
341SD16—2141no—
342SD16—2141no—
343SD16—2141no—
344SD16—2141no—
345SD16—2141no—
346SD16—2141no—
347D8—21no—
348D8—21no—
349D8—1071no—
350D8—1071no—
351D8—2141no—
352D8—2141no—
353D8—2141no—
354D8—2141no—
355D8—2141no—
356D8—2141no—
357D8—2141no—
358D8—2141no—
359M16—1071no—
360M16—1071no—
361M16—2141no—
362M16—2141no—
363M16—2141no—
364M16—2141no—
365M16C22×D10711yescharacteristic
366M16C22×D10711yescharacteristic
367C2×C8—1071no—
368C2×C8—1071no—
369C2×C8—2141no—
370C2×C8—2141no—
371C2×C8—2141no—
372C2×C8—2141no—
373C22⋊C4—4281no—
374C42C22×D10711yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—4281no—
379C23—4281no—
380C23—2141no—
381Q8—1071no—
382Q8—1071no—
383Q8—4281no—
384Q8—2141no—
385Q8—2141no—
386Q8D4×D10711yescharacteristic
387Q8D4×D10711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—1071no—
398D4—1071no—
399D4—4281no—
400D4—4281no—
401D4—4281no—
402D4—4281no—
403D4—4281no—
404D4—2141no—
405D4—2141no—
406D4—2141no—
407D4—2141no—
408D4D4×D10711yescharacteristic
409D4D4×D10711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—4281no—
415C2×C4—4281no—
416C2×C4—4281no—
417C2×C4—4281no—
418C2×C4—2141no—
419C2×C4—2141no—
420C2×C4—2141no—
421C2×C4C23×D10711yescharacteristic
422C8—21no—
423C8—21no—
424C8—1071no—
425C8—1071no—
426C8—1071no—
427C8—1071no—
428C8—2141no—
429C8—2141no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—8561no—
438C22—4281no—
439C22—17121no—
440C22—17121no—
441C22—2141no—
442C22—2141no—
443C22#1681215411yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—4281no—
448C4—2141no—
449C4—2141no—
450C4C2×D4×D10711yescharacteristic
451C4C2×D4×D10711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—8561no—
458C2—4281no—
459C2—4281no—
460C2#2029325911yescharacteristic
subgroupG/H
centerC2#20293259
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#453×C107C2
socleC2×C107D42
hypercenter#453D107
hypocenterC107#3288
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20293438C2semidirect
2. #20293437C2semidirect
3. #20293431C2semidirect
4. #20293430C2semidirect
5. #20293014C2semidirect
6. #20292896C2semidirect
7. #20292895C2semidirect
8. #20292889C2semidirect
9. #20292888C2semidirect
10. #20292806C2semidirect
11. #20292801C2semidirect
12. #20292794C2semidirect
13. #20292746C2semidirect
14. #20292746C2semidirect
15. #453×C107C2semidirect
16. #16812156C22semidirect
18. #16812154C22semidirect
19. #16812142C22semidirect
21. #16812139C22semidirect
23. #16812137C22semidirect
28. #16812135C22semidirect
30. #16812134C22semidirect
40. C2×#13133806C22semidirect
41. C2×#13133806C22semidirect
44. C2×#13133803C22semidirect
45. C2×#13133803C22semidirect
47. #16812117C22semidirect
48. #16812117C22semidirect
49. #16812116C22semidirect
50. #16812116C22semidirect
51. #16812097C22semidirect
52. #16812044C22semidirect
53. #16812044C22semidirect
54. #16812037C22semidirect
55. #16812037C22semidirect
56. #16812027C22semidirect
57. #16812026C22semidirect
58. #16812025C22semidirect
59. #16812020C22semidirect
60. H(5,𝔽2)×C107C22semidirect
62. AGL(1,ℤ/8)×C107C22semidirect
63. #127×C107C22semidirect
64. #104×C107C22semidirect
65. #104×C107C22semidirect
66. #100×C107C22semidirect
88. #13133808D4semidirect
89. #13133808D4semidirect
99. #13133806D4semidirect
100. #13133806D4semidirect
102. #13133805D4semidirect
103. #13133805D4semidirect
112. #13133803D4semidirect
113. #13133803D4semidirect
231. C107#3288semidirect
244. #453D107semidirect
250. H(5,𝔽2)C2×D107semidirect
266. AGL(1,ℤ/8)C2×D107semidirect
285. #127C2×D107semidirect
291. #104C2×D107semidirect
292. #104C2×D107semidirect
296. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934382
#202934372
#202934312
#202934302
#202930142
#202928962
#202928952
#202928892
#202928882
#202928062
#202928012
#202927942
#202927462
#453×C1072
#3288107