Finite Group Databasebeta

← #28348227

Subgroups — #28348227

Order 13696.

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

normal subgroups only

This group has 464 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#20293433C212yesmaximal, complemented
2#20293428C212yesmaximal, complemented
3#20292894C212yesmaximal, complemented
4#20292887C212yesmaximal, complemented
5#20292804C211yesmaximal, characteristic, complemented
6#20292799C211yesmaximal, characteristic, complemented
7#20292794C212yesmaximal, complemented
8#20292750C212yesmaximal, complemented
9#496×C107C211yesmaximal, characteristic, complemented
10#16812157C2211yescharacteristic, complemented
11#16812154C2211yescharacteristic
12#16812154C2212yescomplemented
13#16812140C2212yescomplemented
14#16812138—22no—
15#16812136—22no—
16#16812135C2212yes—
17#16812135C2212yescomplemented
18#16812135C2212yescomplemented
19#16812133—22no—
20#16812133—22no—
21#16812131C2212yes—
22#16812130—22no—
23#16812128—22no—
24SD16×D107—22no—
25D8×D107—22no—
26#16812115C2212yescomplemented
27#16812115C2212yescomplemented
28#16812045C2212yescomplemented
29#16812037C2212yescomplemented
30#16812037C2212yescomplemented
31#16812028C2211yescharacteristic, complemented
32#16812026C2212yescomplemented
33#16812026C2211yescharacteristic, complemented
34C2×SD16×C107C2212yescomplemented
35C2×D8×C107C2212yescomplemented
36#127×C107C2211yescharacteristic, complemented
37#124×C107C2211yescharacteristic, complemented
38#97×C107C2211yescharacteristic, complemented
39#13133819—22no—
40#13133818—41no—
41#13133818C2312yes—
42#13133818C2312yes—
43#13133817—22no—
44#13133817—22no—
45#13133817—22no—
46Q8×D107—22no—
47C2×C22⋊D107—41no—
48C2×C22⋊D107—42no—
49D4×D107—22no—
50D4×D107—22no—
51D4×D107—22no—
52Dic428D412yescomplemented
53#13133808—22no—
54#13133807D412yescomplemented
55#13133807D412yescomplemented
56#13133806—22no—
57#13133806—22no—
58#13133806—22no—
59#13133805—22no—
60D856D412yescomplemented
61#13133803—22no—
62#13133803—22no—
63#13133803—22no—
64#13133802C2312yes—
65#13133802C2312yes—
66#13133801—22no—
67#13133801—22no—
68C8×D107—22no—
69C8×D107—22no—
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107—41no—
72C2×D4×C107C2312yes—
73C2×D4×C107C2311yescharacteristic
74SD16×C107—22no—
75SD16×C107—22no—
76D8×C107—22no—
77D8×C107—22no—
78C2×C8×C107C2312yes—
79C22⋊C4×C107—41no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C23×C107—22no—
83C22×D107—42no—
84C22×D107—41no—
85Q8×C107—22no—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—82no—
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—22no—
94C22⋊D107—22no—
95C22⋊D107—42no—
96C22⋊D107—42no—
97C22⋊D107—42no—
98C2×Dic107—42no—
99C2×Dic107—41no—
100D428—41no—
101D428C2×D412yes—
102D428C2×D412yes—
103C4×D107—22no—
104C4×D107—22no—
105C4×D107—42no—
106Dic214—41no—
107Dic214C2×D412yes—
108Dic214C2×D412yes—
109C8×C107C2×D412yes—
110C8×C107C2×D412yes—
111C107⋊C8—22no—
112C107⋊C8—22no—
113C22×C107—41no—
114C22×C107—42no—
115C22×C107—82no—
116C22×C107—81no—
117C22×C107H(5,𝔽2)11yescharacteristic
118C2×D107—22no—
119C2×D107—22no—
120C2×D107—162no—
121C2×D107—161no—
122C4×C107—41no—
123C4×C107—41no—
124C4×C107C22×D412yes—
125Dic107—22no—
126Dic107—22no—
127C2×C107—21no—
128C2×C107—82no—
129C2×C107—81no—
130C2×C107D4211yescharacteristic
131D107—42no—
132D107—42no—
133#3288—1071nomaximal, Sylow
134C107#328811yescharacteristic, complemented, Sylow
135#577—1072no—
136#576—1072no—
137#496D10711yescharacteristic, complemented
138#472—1072no—
139#471—1072no—
140#455—1071no—
141#454—1071no—
142#453—1072no—
143#444—1072no—
1442−1+4—1071no—
145H(5,𝔽2)—1071no—
146H(5,𝔽2)—1072no—
147#137—1072no—
148#137—2142no—
149AGL(1,ℤ/8)—1072no—
150AGL(1,ℤ/8)—1072no—
151AGL(1,ℤ/8)—1072no—
152AGL(1,ℤ/8)—2142no—
153AGL(1,ℤ/8)—2142no—
154AGL(1,ℤ/8)—2142no—
155#135—1072no—
156#135—2142no—
157#135—2142no—
158C2×SD16—2142no—
159C2×SD16C2×D10712yescomplemented
160C2×D8—2142no—
161C2×D8C2×D10712yescomplemented
162#131—1072no—
163#131—1072no—
164#127C2×D10711yescharacteristic, complemented
165#124C2×D10711yescharacteristic, complemented
166#108—1072no—
167#104—1072no—
168#104—1072no—
169#101—1071no—
170#100—1072no—
171#100—1071no—
172#97C2×D10711yescharacteristic, complemented
173#41—1072no—
174#41—1072no—
175#41—4281no—
176#41—2142no—
177#41—2142no—
178#41—2142no—
179#41—2142no—
180C2×Q8—2142no—
181C2×Q8C22×D10711yescharacteristic
182C2×D4—41no—
183C2×D4—4281no—
184C2×D4—4282no—
185C2×D4—2142no—
186C2×D4—2142no—
187C2×D4—2142no—
188C2×D4C22×D10711yescharacteristic
189C2×D4C22×D10712yes—
190Q16—1072no—
191Q16—2142no—
192SD16—22no—
193SD16—22no—
194SD16—1072no—
195SD16—1072no—
196SD16—2142no—
197SD16—2142no—
198SD16—2142no—
199SD16—2142no—
200D8—22no—
201D8—22no—
202D8—1072no—
203D8—2142no—
204D8—2142no—
205D8—2142no—
206M16—1072no—
207M16—1072no—
208M16—2142no—
209M16—2142no—
210C2×C8—2142no—
211C2×C8—2142no—
212C2×C8C22×D10712yes—
213C22⋊C4—41no—
214C42C22×D10711yescharacteristic
215C23—21no—
216C23—22no—
217C23—4282no—
218C23—4281no—
219Q8—22no—
220Q8—1072no—
221Q8—1072no—
222Q8—4281no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—1072no—
228D4—1072no—
229D4—4281no—
230D4—4282no—
231D4—4282no—
232D4—4282no—
233D4—2142no—
234D4—2142no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—4282no—
238C2×C4—4281no—
239C2×C4—4282no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4C23×D10711yescharacteristic
243C8—2142no—
244C8—2142no—
245C8D4×D10712yes—
246C8D4×D10712yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—17121no—
252C22—17122no—
253C22—2142no—
254C22—2142no—
255C22#1681215411yescharacteristic
256C4—41no—
257C4—41no—
258C4—2142no—
259C4—2142no—
260C4C2×D4×D10712yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—4282no—
265C2—4282no—
266C2#2029326211yescharacteristic
subgroupG/H
centerC2#20293262
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC2×C107D42
hypercenter#496D107
hypocenterC107#3288
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20293433C2semidirect
2. #20293428C2semidirect
3. #20292894C2semidirect
4. #20292887C2semidirect
5. #20292804C2semidirect
6. #20292799C2semidirect
7. #20292794C2semidirect
8. #20292750C2semidirect
9. #496×C107C2semidirect
10. #16812157C22semidirect
12. #16812154C22semidirect
13. #16812140C22semidirect
17. #16812135C22semidirect
18. #16812135C22semidirect
26. #16812115C22semidirect
27. #16812115C22semidirect
28. #16812045C22semidirect
29. #16812037C22semidirect
30. #16812037C22semidirect
31. #16812028C22semidirect
32. #16812026C22semidirect
33. #16812026C22semidirect
34. C2×SD16×C107C22semidirect
35. C2×D8×C107C22semidirect
36. #127×C107C22semidirect
37. #124×C107C22semidirect
38. #97×C107C22semidirect
52. Dic428D4semidirect
54. #13133807D4semidirect
55. #13133807D4semidirect
60. D856D4semidirect
134. C107#3288semidirect
137. #496D107semidirect
159. C2×SD16C2×D107semidirect
161. C2×D8C2×D107semidirect
164. #127C2×D107semidirect
165. #124C2×D107semidirect
172. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934332
#202934282
#202928942
#202928872
#202928042
#202927992
#202927942
#202927502
#496×C1072
#3288107