Finite Group Databasebeta

← #28348259

Subgroups — #28348259

Order 13696.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293432C211yesmaximal, characteristic, complemented
2#20293431C212yesmaximal, complemented
3#20293428C211yesmaximal, characteristic, complemented
4#20293002C211yesmaximal, characteristic, complemented
5#20292898C211yesmaximal, characteristic, complemented
6#20292896C212yesmaximal, complemented
7#20292797C212yesmaximal, complemented
8#20292793C212yesmaximal, complemented
9#20292743C211yesmaximal, characteristic, complemented
10#20292742C211yesmaximal, characteristic, complemented
11#472×C107C211yesmaximal, characteristic, complemented
12#16812155C2212yescomplemented
13#16812154C2212yescomplemented
14#16812137—22no—
15#16812137C2212yescomplemented
16#16812137C2212yescomplemented
17#16812136—22no—
18#16812135C2212yescomplemented
19#16812133—22no—
20#16812133—22no—
21#16812131—21no—
22#16812131C2211yescharacteristic
23#16812130—22no—
24#16812129C2211yescharacteristic
25#16812128—21no—
26C2×D856—21no—
27C2×D856C2211yescharacteristic, complemented
28C2×#13133803C2212yescomplemented
29D8×D107—21no—
30D8×D107—22no—
31#16812117C2212yescomplemented
32#16812115C2211yescharacteristic, complemented
33#16812105C2212yescomplemented
34#16812044C2211yescharacteristic, complemented
35#16812036C2211yescharacteristic, complemented
36#16812035C2211yescharacteristic, complemented
37#16812035C2212yescomplemented
38#16812025C2212yescomplemented
39C8×Dic107C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C107C2212yescomplemented
41C2×D8×C107C2211yescharacteristic, complemented
42#131×C107C2211yescharacteristic, complemented
43#108×C107C2211yescharacteristic, complemented
44#100×C107C2212yescomplemented
45#13133819—22no—
46#13133818—22no—
47#13133818C2311yescharacteristic
48#13133818C2312yes—
49#13133817—22no—
50#13133817—22no—
51C2×D428—22no—
52C2×D428C2312yes—
53C2×C22⋊D107—42no—
54C2×C22⋊D107—42no—
55D4×D107—22no—
56D4×D107—22no—
57D4×D107—22no—
58D4×D107—42no—
59Dic428D411yescharacteristic, complemented
60#13133808D411yescharacteristic, complemented
61#13133807—22no—
62#13133807—22no—
63#13133807D412yescomplemented
64#13133806—22no—
65#13133806D411yescharacteristic, complemented
66#13133805D411yescharacteristic, complemented
67D856—22no—
68D856—22no—
69D856—21no—
70D856—21no—
71D856D411yescharacteristic, complemented
72#13133803—22no—
73#13133803—22no—
74#13133803—22no—
75#13133803D411yescharacteristic, complemented
76#13133802C2311yescharacteristic
77#13133801—21no—
78#13133801—22no—
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—21no—
81C8×D107—22no—
82C4×Dic107C2311yescharacteristic
83#41×C107C2311yescharacteristic
84C2×D4×C107C2312yes—
85SD16×C107—22no—
86D8×C107—22no—
87D8×C107—21no—
88D8×C107—21no—
89M16×C107—21no—
90M16×C107C2311yescharacteristic
91M16×C107C2312yes—
92C2×C8×C107—21no—
93C2×C8×C107C2311yescharacteristic
94C23×C107—22no—
95C22×D107—22no—
96C22×D107—42no—
97C22×D107—42no—
98Q8×C107C2×D411yescharacteristic
99D4×C107—21no—
100D4×C107—22no—
101D4×C107—22no—
102D4×C107C2×D411yescharacteristic
103C2×C4×C107—21no—
104C2×C4×C107C2411yescharacteristic
105C22⋊D107—22no—
106C22⋊D107—21no—
107C22⋊D107—42no—
108C22⋊D107—42no—
109C22⋊D107—42no—
110C22⋊D107—84no—
111C2×Dic107—21no—
112C2×Dic107—42no—
113D428—22no—
114D428—22no—
115D428—42no—
116D428C2×D412yes—
117D428C2×D411yescharacteristic
118C4×D107—21no—
119C4×D107—22no—
120C4×D107—42no—
121Dic214C2×D412yes—
122Dic214C2×D411yescharacteristic
123C8×C107—21no—
124C8×C107—22no—
125C8×C107C2×D411yescharacteristic
126C8×C107C2×D411yescharacteristic
127C107⋊C8—21no—
128C107⋊C8C2×D411yescharacteristic
129C107⋊C8C2×D411yescharacteristic
130C22×C107—21no—
131C22×C107—42no—
132C22×C107—82no—
133C22×C107H(5,𝔽2)11yescharacteristic
134C2×D107—22no—
135C2×D107—21no—
136C2×D107—42no—
137C2×D107—82no—
138C2×D107—162no—
139C2×D107—162no—
140C4×C107—21no—
141C4×C107C22×D411yescharacteristic
142C4×C107C22×D411yescharacteristic
143Dic107—22no—
144Dic107—22no—
145Dic107—21no—
146C2×C107—21no—
147C2×C107—41no—
148C2×C107—82no—
149C2×C107D4211yescharacteristic
150D107—42no—
151D107—41no—
152D107—82no—
153#3292—1071nomaximal, Sylow
154C107#329211yescharacteristic, complemented, Sylow
155#576—1072no—
156#576—1071no—
157#576—1071no—
158#493—1071no—
159#472—1071no—
160#472—1072no—
161#472D10711yescharacteristic, complemented
162#453—1072no—
163#453—1072no—
164#443—1071no—
165#443—1071no—
166H(5,𝔽2)—1072no—
167H(5,𝔽2)—1072no—
168AGL(1,ℤ/8)—1072no—
169AGL(1,ℤ/8)—1072no—
170AGL(1,ℤ/8)—1072no—
171AGL(1,ℤ/8)—2142no—
172AGL(1,ℤ/8)—2142no—
173AGL(1,ℤ/8)—2142no—
174AGL(1,ℤ/8)—2142no—
175AGL(1,ℤ/8)C2×D10712yescomplemented
176#135—1071no—
177#135—1071no—
178#135—2141no—
179#135—2141no—
180#135—2142no—
181C2×D8—1072no—
182C2×D8—1071no—
183C2×D8—2141no—
184C2×D8—2142no—
185C2×D8—2141no—
186C2×D8C2×D10711yescharacteristic, complemented
187#131—1072no—
188#131—1071no—
189#131C2×D10711yescharacteristic, complemented
190#127—1072no—
191#108—1071no—
192#108C2×D10711yescharacteristic, complemented
193#104—1071no—
194#104—1072no—
195#104—1071no—
196#100—1072no—
197#100C2×D10712yescomplemented
198C4×C8—1071no—
199#41—1072no—
200#41—1071no—
201#41—2142no—
202#41—2142no—
203#41—2142no—
204#41—2142no—
205#41C22×D10711yescharacteristic
206C2×D4—1072no—
207C2×D4—4282no—
208C2×D4—4282no—
209C2×D4—4282no—
210C2×D4—2142no—
211C2×D4—2142no—
212C2×D4—2142no—
213C2×D4—2142no—
214C2×D4C22×D10712yes—
215Q16—1071no—
216Q16—1071no—
217SD16—22no—
218SD16—1071no—
219SD16—1071no—
220SD16—1072no—
221SD16—2142no—
222SD16—2142no—
223SD16—2142no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1071no—
228D8—1071no—
229D8—2142no—
230D8—2142no—
231D8—2142no—
232D8—2142no—
233D8—2142no—
234D8—2141no—
235D8—2141no—
236M16—21no—
237M16—1071no—
238M16—2142no—
239M16—2141no—
240M16C22×D10712yes—
241M16C22×D10711yescharacteristic
242C2×C8—21no—
243C2×C8—1071no—
244C2×C8—2142no—
245C2×C8—2141no—
246C2×C8C22×D10711yescharacteristic
247C42—1071no—
248C23—22no—
249C23—4282no—
250C23—4282no—
251C23—2142no—
252Q8—1071no—
253Q8—1072no—
254Q8D4×D10711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—8564no—
259D4—1071no—
260D4—1072no—
261D4—4282no—
262D4—4282no—
263D4—4282no—
264D4—4282no—
265D4—2142no—
266D4—2141no—
267D4—2142no—
268D4—2142no—
269D4D4×D10711yescharacteristic
270C2×C4—21no—
271C2×C4—4282no—
272C2×C4—4282no—
273C2×C4—2141no—
274C2×C4—2142no—
275C2×C4—2141no—
276C2×C4C23×D10711yescharacteristic
277C8—21no—
278C8—22no—
279C8—1071no—
280C8—1071no—
281C8—2141no—
282C8D4×D10711yescharacteristic
283C8D4×D10711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—8562no—
288C22—4282no—
289C22—17122no—
290C22—17122no—
291C22—2142no—
292C22—2141no—
293C22#1681215511yescharacteristic
294C4—21no—
295C4—2142no—
296C4—2142no—
297C4—2141no—
298C4C2×D4×D10711yescharacteristic
299C4C2×D4×D10711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—8562no—
304C2—4282no—
305C2—4281no—
306C2#2029326111yescharacteristic
subgroupG/H
centerC2#20293261
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#472×C107C2
socleC2×C107D42
hypercenter#472D107
hypocenterC107#3292
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. #20293432C2semidirect
2. #20293431C2semidirect
3. #20293428C2semidirect
4. #20293002C2semidirect
5. #20292898C2semidirect
6. #20292896C2semidirect
7. #20292797C2semidirect
8. #20292793C2semidirect
9. #20292743C2semidirect
10. #20292742C2semidirect
11. #472×C107C2semidirect
12. #16812155C22semidirect
13. #16812154C22semidirect
15. #16812137C22semidirect
16. #16812137C22semidirect
18. #16812135C22semidirect
27. C2×D856C22semidirect
28. C2×#13133803C22semidirect
31. #16812117C22semidirect
32. #16812115C22semidirect
33. #16812105C22semidirect
34. #16812044C22semidirect
35. #16812036C22semidirect
36. #16812035C22semidirect
37. #16812035C22semidirect
38. #16812025C22semidirect
39. C8×Dic107C22semidirect
40. AGL(1,ℤ/8)×C107C22semidirect
41. C2×D8×C107C22semidirect
42. #131×C107C22semidirect
43. #108×C107C22semidirect
44. #100×C107C22semidirect
59. Dic428D4semidirect
60. #13133808D4semidirect
63. #13133807D4semidirect
65. #13133806D4semidirect
66. #13133805D4semidirect
71. D856D4semidirect
75. #13133803D4semidirect
154. C107#3292semidirect
161. #472D107semidirect
175. AGL(1,ℤ/8)C2×D107semidirect
186. C2×D8C2×D107semidirect
189. #131C2×D107semidirect
192. #108C2×D107semidirect
197. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934322
#202934312
#202934282
#202930022
#202928982
#202928962
#202927972
#202927932
#202927432
#202927422
#472×C1072
#3292107