Finite Group Databasebeta

← #28348224

Subgroups — #28348224

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#20293439C211yesmaximal, characteristic, complemented
2#20293437C211yesmaximal, characteristic, complemented
3#20293431C211yesmaximal, characteristic, complemented
4#20293428C211yesmaximal, characteristic, complemented
5#20293015C211yesmaximal, characteristic, complemented
6#20292896C211yesmaximal, characteristic, complemented
7#20292891C211yesmaximal, characteristic, complemented
8#20292889C211yesmaximal, characteristic, complemented
9#20292807C211yesmaximal, characteristic, complemented
10#20292798C211yesmaximal, characteristic, complemented
11#20292797C211yesmaximal, characteristic, complemented
12#20292793C211yesmaximal, characteristic, complemented
13#20292748C211yesmaximal, characteristic, complemented
14#20292747C211yesmaximal, characteristic, complemented
15#472×C107C211yesmaximal, characteristic, complemented
16#16812156C2211yescharacteristic, complemented
17#16812155C2211yescharacteristic, complemented
18#16812154C2211yescharacteristic
19#16812154C2211yescharacteristic, complemented
20#16812142—21no—
21#16812142C2211yescharacteristic, complemented
22#16812142C2211yescharacteristic, complemented
23#16812138—21no—
24#16812137—21no—
25#16812137C2211yescharacteristic
26#16812137C2211yescharacteristic, complemented
27#16812136—21no—
28#16812135C2211yescharacteristic
29#16812135C2211yescharacteristic, complemented
30#16812133—21no—
31#16812133—21no—
32#16812133—21no—
33#16812133—21no—
34#16812131—21no—
35#16812131C2211yescharacteristic
36#16812130—21no—
37#16812129C2211yescharacteristic
38#16812128—21no—
39#16812128—21no—
40C2×#13133807—21no—
41C2×#13133807C2211yescharacteristic, complemented
42C2×#13133806C2211yescharacteristic, complemented
43SD16×D107—21no—
44C2×#13133803C2211yescharacteristic, complemented
45D8×D107—21no—
46D8×D107—21no—
47#16812117C2211yescharacteristic, complemented
48#16812117C2211yescharacteristic, complemented
49#16812115C2211yescharacteristic, complemented
50#16812105C2211yescharacteristic, complemented
51#16812096C2211yescharacteristic, complemented
52#16812044C2211yescharacteristic, complemented
53#16812036C2211yescharacteristic, complemented
54#16812035C2211yescharacteristic, complemented
55#16812035C2211yescharacteristic, complemented
56#16812035C2211yescharacteristic, complemented
57#16812027C2211yescharacteristic, complemented
58#16812025C2211yescharacteristic, complemented
59#16812019C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C107C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C107C2211yescharacteristic, complemented
62C2×D8×C107C2211yescharacteristic, complemented
63#131×C107C2211yescharacteristic, complemented
64#108×C107C2211yescharacteristic, complemented
65#100×C107C2211yescharacteristic, complemented
66#100×C107C2211yescharacteristic, complemented
67#13133819—21no—
68#13133818—21no—
69#13133818—21no—
70#13133818C2311yescharacteristic
71#13133818C2311yescharacteristic
72#13133818C2311yescharacteristic
73#13133817—21no—
74#13133817—21no—
75#13133817—21no—
76#13133817—21no—
77#13133817—21no—
78#13133817—41no—
79C2×Dic214—21no—
80C2×Dic214C2311yescharacteristic
81Q8×D107—21no—
82C2×D428—21no—
83C2×D428C2311yescharacteristic
84C2×C22⋊D107—41no—
85C2×C22⋊D107—41no—
86C2×C22⋊D107—41no—
87D4×D107—21no—
88D4×D107—21no—
89D4×D107—21no—
90D4×D107—21no—
91D4×D107—21no—
92D4×D107—41no—
93Dic428—21no—
94Dic428—21no—
95Dic428D411yescharacteristic, complemented
96#13133808D411yescharacteristic, complemented
97#13133807—21no—
98#13133807—21no—
99#13133807—21no—
100#13133807—21no—
101#13133807—21no—
102#13133807—21no—
103#13133807D411yescharacteristic, complemented
104#13133807D411yescharacteristic, complemented
105#13133806—21no—
106#13133806—21no—
107#13133806—21no—
108#13133806—21no—
109#13133806D411yescharacteristic, complemented
110#13133805D411yescharacteristic, complemented
111D856—21no—
112D856—21no—
113D856D411yescharacteristic, complemented
114#13133803—21no—
115#13133803—21no—
116#13133803—21no—
117#13133803—21no—
118#13133803D411yescharacteristic, complemented
119#13133802C2311yescharacteristic
120#13133801—21no—
121#13133801—21no—
122#13133801—21no—
123C2×C107⋊C8C2311yescharacteristic
124C8×D107—21no—
125C8×D107—21no—
126C8×D107—21no—
127C22⋊Dic107—41no—
128C4×Dic107C2311yescharacteristic
129#41×C107C2311yescharacteristic
130C2×D4×C107C2311yescharacteristic
131C2×D4×C107C2311yescharacteristic
132SD16×C107—21no—
133SD16×C107—21no—
134D8×C107—21no—
135D8×C107—21no—
136D8×C107—21no—
137D8×C107—21no—
138M16×C107—21no—
139M16×C107C2311yescharacteristic
140M16×C107C2311yescharacteristic
141M16×C107C2311yescharacteristic
142C2×C8×C107—21no—
143C2×C8×C107C2311yescharacteristic
144C23×C107—21no—
145C23×C107—21no—
146C22×D107—21no—
147C22×D107—41no—
148C22×D107—41no—
149C22×D107—41no—
150Q8×C107C2×D411yescharacteristic
151D4×C107—21no—
152D4×C107—21no—
153D4×C107—21no—
154D4×C107—21no—
155D4×C107—21no—
156D4×C107C2×D411yescharacteristic
157C2×C4×C107—21no—
158C2×C4×C107C2411yescharacteristic
159C22⋊D107—21no—
160C22⋊D107—21no—
161C22⋊D107—21no—
162C22⋊D107—41no—
163C22⋊D107—41no—
164C22⋊D107—41no—
165C22⋊D107—41no—
166C22⋊D107—41no—
167C22⋊D107—41no—
168C22⋊D107—82no—
169C2×Dic107—21no—
170C2×Dic107—21no—
171C2×Dic107—41no—
172C2×Dic107—41no—
173C2×Dic107—41no—
174D428—21no—
175D428—21no—
176D428—41no—
177D428C2×D411yescharacteristic
178D428C2×D411yescharacteristic
179D428C2×D411yescharacteristic
180C4×D107—21no—
181C4×D107—21no—
182C4×D107—21no—
183C4×D107—41no—
184C4×D107—41no—
185Dic214—21no—
186Dic214—21no—
187Dic214—41no—
188Dic214C2×D411yescharacteristic
189Dic214C2×D411yescharacteristic
190Dic214C2×D411yescharacteristic
191C8×C107—21no—
192C8×C107—21no—
193C8×C107—21no—
194C8×C107C2×D411yescharacteristic
195C8×C107C2×D411yescharacteristic
196C107⋊C8—21no—
197C107⋊C8C2×D411yescharacteristic
198C107⋊C8C2×D411yescharacteristic
199C22×C107—21no—
200C22×C107—41no—
201C22×C107—41no—
202C22×C107—81no—
203C22×C107—81no—
204C22×C107H(5,𝔽2)11yescharacteristic
205C2×D107—21no—
206C2×D107—21no—
207C2×D107—21no—
208C2×D107—41no—
209C2×D107—81no—
210C2×D107—161no—
211C2×D107—161no—
212C2×D107—161no—
213C4×C107—21no—
214C4×C107C22×D411yescharacteristic
215C4×C107C22×D411yescharacteristic
216Dic107—21no—
217Dic107—21no—
218Dic107—21no—
219Dic107—41no—
220Dic107—41no—
221C2×C107—21no—
222C2×C107—41no—
223C2×C107—81no—
224C2×C107—81no—
225C2×C107D4211yescharacteristic
226D107—41no—
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#472D10711yescharacteristic, complemented
239#471—1071no—
240#471—1071no—
241#455—1071no—
242#454—1071no—
243#453—1071no—
244#453—1071no—
245#444—1071no—
246#444—1071no—
2472−1+4—1071no—
248H(5,𝔽2)—1071no—
249H(5,𝔽2)—1071no—
250H(5,𝔽2)—1071no—
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, complemented
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—2141no—
279C2×D8—2141no—
280C2×D8C2×D10711yescharacteristic, complemented
281#131—1071no—
282#131—1071no—
283#131—1071no—
284#131C2×D10711yescharacteristic, complemented
285#127—1071no—
286#124—1071no—
287#108—1071no—
288#108C2×D10711yescharacteristic, complemented
289#104—1071no—
290#104—1071no—
291#104—1071no—
292#104—1071no—
293#101—1071no—
294#100—1071no—
295#100C2×D10711yescharacteristic, complemented
296#100C2×D10711yescharacteristic, complemented
297#97—1071no—
298#41—1071no—
299#41—1071no—
300#41—1071no—
301#41—4281no—
302#41—2141no—
303#41—2141no—
304#41—2141no—
305#41—2141no—
306#41—2141no—
307#41—2141no—
308#41—2141no—
309#41—2141no—
310#41C22×D10711yescharacteristic
311C2×Q8—1071no—
312C2×Q8—2141no—
313C2×Q8—2141no—
314C2×D4—1071no—
315C2×D4—4281no—
316C2×D4—4281no—
317C2×D4—4281no—
318C2×D4—4281no—
319C2×D4—2141no—
320C2×D4—2141no—
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—21no—
350D8—21no—
351D8—1071no—
352D8—1071no—
353D8—2141no—
354D8—2141no—
355D8—2141no—
356D8—2141no—
357D8—2141no—
358D8—2141no—
359M16—21no—
360M16—1071no—
361M16—2141no—
362M16—2141no—
363M16—2141no—
364M16C22×D10711yescharacteristic
365M16C22×D10711yescharacteristic
366M16C22×D10711yescharacteristic
367C2×C8—21no—
368C2×C8—1071no—
369C2×C8—2141no—
370C2×C8—2141no—
371C2×C8—2141no—
372C2×C8C22×D10711yescharacteristic
373C22⋊C4—4281no—
374C42—1071no—
375C23—21no—
376C23—21no—
377C23—4281no—
378C23—4281no—
379C23—4281no—
380C23—2141no—
381Q8—1071no—
382Q8—1071no—
383Q8—1071no—
384Q8—4281no—
385Q8—2141no—
386Q8—2141no—
387Q8D4×D10711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—8562no—
394D4—1071no—
395D4—1071no—
396D4—1071no—
397D4—4281no—
398D4—4281no—
399D4—4281no—
400D4—4281no—
401D4—4281no—
402D4—4281no—
403D4—4281no—
404D4—2141no—
405D4—2141no—
406D4—2141no—
407D4—2141no—
408D4—2141no—
409D4D4×D10711yescharacteristic
410C2×C4—21no—
411C2×C4—4281no—
412C2×C4—4281no—
413C2×C4—4281no—
414C2×C4—4281no—
415C2×C4—4281no—
416C2×C4—2141no—
417C2×C4—2141no—
418C2×C4—2141no—
419C2×C4—2141no—
420C2×C4—2141no—
421C2×C4C23×D10711yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—1071no—
426C8—1071no—
427C8—2141no—
428C8D4×D10711yescharacteristic
429C8D4×D10711yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—8561no—
436C22—4281no—
437C22—17121no—
438C22—17121no—
439C22—17121no—
440C22—2141no—
441C22—2141no—
442C22—2141no—
443C22#1681215511yescharacteristic
444C4—21no—
445C4—4281no—
446C4—4281no—
447C4—2141no—
448C4—2141no—
449C4—2141no—
450C4C2×D4×D10711yescharacteristic
451C4C2×D4×D10711yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—8561no—
457C2—4281no—
458C2—4281no—
459C2—4281no—
460C2#2029326111yescharacteristic
subgroupG/H
centerC2#20293261
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#472×C107C2
socleC2×C107D42
hypercenter#472D107
hypocenterC107#3288
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20293439C2semidirect
2. #20293437C2semidirect
3. #20293431C2semidirect
4. #20293428C2semidirect
5. #20293015C2semidirect
6. #20292896C2semidirect
7. #20292891C2semidirect
8. #20292889C2semidirect
9. #20292807C2semidirect
10. #20292798C2semidirect
11. #20292797C2semidirect
12. #20292793C2semidirect
13. #20292748C2semidirect
14. #20292747C2semidirect
15. #472×C107C2semidirect
16. #16812156C22semidirect
17. #16812155C22semidirect
19. #16812154C22semidirect
21. #16812142C22semidirect
22. #16812142C22semidirect
26. #16812137C22semidirect
29. #16812135C22semidirect
41. C2×#13133807C22semidirect
42. C2×#13133806C22semidirect
44. C2×#13133803C22semidirect
47. #16812117C22semidirect
48. #16812117C22semidirect
49. #16812115C22semidirect
50. #16812105C22semidirect
51. #16812096C22semidirect
52. #16812044C22semidirect
53. #16812036C22semidirect
54. #16812035C22semidirect
55. #16812035C22semidirect
56. #16812035C22semidirect
57. #16812027C22semidirect
58. #16812025C22semidirect
59. #16812019C22semidirect
60. AGL(1,ℤ/8)×C107C22semidirect
61. AGL(1,ℤ/8)×C107C22semidirect
62. C2×D8×C107C22semidirect
63. #131×C107C22semidirect
64. #108×C107C22semidirect
65. #100×C107C22semidirect
66. #100×C107C22semidirect
95. Dic428D4semidirect
96. #13133808D4semidirect
103. #13133807D4semidirect
104. #13133807D4semidirect
109. #13133806D4semidirect
110. #13133805D4semidirect
113. D856D4semidirect
118. #13133803D4semidirect
231. C107#3288semidirect
238. #472D107semidirect
265. AGL(1,ℤ/8)C2×D107semidirect
266. AGL(1,ℤ/8)C2×D107semidirect
280. C2×D8C2×D107semidirect
284. #131C2×D107semidirect
288. #108C2×D107semidirect
295. #100C2×D107semidirect
296. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934392
#202934372
#202934312
#202934282
#202930152
#202928962
#202928912
#202928892
#202928072
#202927982
#202927972
#202927932
#202927482
#202927472
#472×C1072
#3288107