Finite Group Databasebeta

← #28346941

Subgroups — #28346941

Order 13696.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#16812133C212yesmaximal, complemented
2C2×#16812128C211yesmaximal, characteristic, complemented
3C2×D8×D107C211yesmaximal, characteristic, complemented
4#20293190C212yesmaximal, complemented
5#20293015C212yesmaximal, complemented
6#20293014C212yesmaximal, complemented
7#20293011C211yesmaximal, characteristic, complemented
8#20293010C211yesmaximal, characteristic, complemented
9#20293002C211yesmaximal, characteristic, complemented
10#20292581C211yesmaximal, characteristic, complemented
11#493×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×D4×D107C2212yes—
14#16812133—28no—
15#16812128—24no—
16C2×Dic428C2211yescharacteristic, complemented
17C2×#13133807C2212yescomplemented
18C2×#13133806C2212yescomplemented
19C2×#13133806C2212yescomplemented
20C2×D856C2211yescharacteristic
21C2×#13133803C2212yes—
22C2×#13133803C2212yescomplemented
23D8×D107—24no—
24C2×#13133801C2211yescharacteristic
25C2×C8×D107C2211yescharacteristic
26#16812105C2212yescomplemented
27#16812097C2212yescomplemented
28#16812096C2212yescomplemented
29#16812081—24no—
30#16812077—24no—
31#16812066C2211yescharacteristic
32#16812020C2211yescharacteristic, complemented
33#16812019C2211yescharacteristic, complemented
34C8×Dic107C2211yescharacteristic, complemented
35C2×D8×C107C2211yescharacteristic
36C2×D8×C107C2211yescharacteristic, complemented
37C2×D8×C107C2212yescomplemented
38#127×C107C2212yescomplemented
39C4×C8×C107C2211yescharacteristic, complemented
40C23×D107—22no—
41#13133817—24no—
42#13133817—24no—
43C2×Dic214C2312yes—
44C2×D428C2312yes—
45C2×C22⋊D107—22no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—24no—
48D4×D107—24no—
49D4×D107—24no—
50C22×Dic107—22no—
51C2×C4×D107C2311yescharacteristic
52Dic428—22no—
53#13133807—24no—
54#13133806—24no—
55#13133806—24no—
56D856—22no—
57#13133803—24no—
58#13133803—24no—
59#13133801D414yescomplemented
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C8×D107D414yescomplemented
63Dic107⋊C4D412yes—
64C22⋊Dic107—24no—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4D412yes—
67C4×Dic107C2311yescharacteristic
68C2×D4×C107—24no—
69C2×D4×C107C2312yes—
70C2×D4×C107C2312yes—
71D8×C107—22no—
72D8×C107—22no—
73D8×C107—24no—
74C2×C8×C107C2311yescharacteristic
75C2×C8×C107C2311yescharacteristic
76C42×C107C2311yescharacteristic
77C23×C107—22no—
78C23×C107—22no—
79C22×D107—22no—
80C22×D107—44no—
81C22×D107—84no—
82C22×D107C2×D411yescharacteristic
83D4×C107—22no—
84D4×C107—22no—
85D4×C107—22no—
86D4×C107—22no—
87D4×C107—84no—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—44no—
91C22⋊D107—44no—
92C22⋊D107—84no—
93C2×Dic107—22no—
94C2×Dic107—44no—
95C2×Dic107C2×D412yes—
96C2×Dic107C2×D411yescharacteristic
97D428—22no—
98D428—22no—
99C4×D107C2×D412yes—
100C4×D107C2×D412yes—
101Dic214—22no—
102Dic214—22no—
103C8×C107C2×D412yes—
104C8×C107C2×D412yes—
105C107⋊C8C2×D412yes—
106C107⋊C8C2×D412yes—
107C22×C107—42no—
108C22×C107—42no—
109C22×C107—44no—
110C22×C107—44no—
111C22×C107C22×D411yescharacteristic
112C2×D107—22no—
113C2×D107—42no—
114C2×D107—44no—
115C2×D107—88no—
116C2×D107#12712yes—
117C4×C107—22no—
118C4×C107C22×D411yescharacteristic
119C4×C107C22×D411yescharacteristic
120Dic107—22no—
121Dic107—42no—
122Dic107#12712yes—
123C2×C107—82no—
124C2×C107—82no—
125C2×C107C2×#12711yescharacteristic
126C2×C107#57612yes—
127D107—22no—
128D107—82no—
129#3155—1071nomaximal, Sylow
130C107#315511yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1072no—
132C2×#135—1071no—
133C22×D8—1071no—
134#535—1072no—
135#496—1072no—
136#496—1072no—
137#495—1071no—
138#495—1071no—
139#493—1071no—
140#493D10711yescharacteristic, complemented
141#405—1071no—
142C2×#41—1072no—
143C22×D4—1072no—
144AGL(1,ℤ/8)—2148no—
145#135—2144no—
146C2×Q16—1071no—
147C2×SD16—1072no—
148C2×SD16—1072no—
149C2×SD16—1072no—
150C2×D8—1072no—
151C2×D8—1072no—
152C2×D8—1071no—
153C2×D8—2144no—
154C2×D8C2×D10711yescharacteristic
155C2×D8C2×D10711yescharacteristic, complemented
156C2×D8C2×D10712yescomplemented
157C2×M16—1071no—
158C22×C8—1071no—
159#127—1072no—
160#127C2×D10712yescomplemented
161#124—1072no—
162#124—1072no—
163#121—2144no—
164#120—2144no—
165#117—1071no—
166#97—1071no—
167#97—1071no—
168C4×C8—1071no—
169C4×C8C2×D10711yescharacteristic, complemented
170C24—2142no—
171#41—2144no—
172#41—2144no—
173C2×Q8—1072no—
174C2×D4—24no—
175C2×D4—1072no—
176C2×D4—2144no—
177C2×D4—2144no—
178C2×D4—2144no—
179C2×D4—2142no—
180C2×D4—2142no—
181C2×D4C22×D10712yes—
182C2×D4C22×D10712yes—
183C22×C4—1071no—
184C22×C4—2142no—
185Q16—2142no—
186SD16—2144no—
187SD16—2144no—
188SD16—2144no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—2144no—
193D8—2144no—
194D8—2142no—
195M16—1074no—
196C2×C8—1071no—
197C2×C8—1071no—
198C2×C8—1074no—
199C2×C8C22×D10711yescharacteristic
200C2×C8C22×D10711yescharacteristic
201C4⋊C4—1072no—
202C22⋊C4—1072no—
203C22⋊C4—2144no—
204C22⋊C4—2144no—
205C42—1071no—
206C42C22×D10711yescharacteristic
207C23—22no—
208C23—22no—
209C23—8564no—
210C23—1071no—
211C23—4284no—
212C23—2142no—
213Q8—2142no—
214Q8—2142no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—8564no—
221D4—4284no—
222D4—4284no—
223D4—2142no—
224D4—2142no—
225C2×C4—1071no—
226C2×C4—1072no—
227C2×C4—1072no—
228C2×C4—1072no—
229C2×C4—4284no—
230C2×C4—2142no—
231C2×C4D4×D10712yes—
232C2×C4C23×D10711yescharacteristic
233C8—1072no—
234C8—1072no—
235C8D4×D10712yes—
236C8D4×D10712yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—8568no—
242C22—1072no—
243C22—4282no—
244C22—4284no—
245C22—2142no—
246C22C2×D4×D10711yescharacteristic
247C4—22no—
248C4—1072no—
249C4—4282no—
250C4—2142no—
251C4C2×D4×D10711yescharacteristic
252C4C2×D4×D10711yescharacteristic
253C2—82no—
254C2—82no—
255C2—8562no—
256C2—2142no—
257C2#127×D10711yescharacteristic
258C2#2029342812yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#493×C107C2
socleC22×C107C22×D4
hypercenter#493D107
hypocenterC107#3155
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×#16812128C2semidirect
3. C2×D8×D107C2semidirect
4. #20293190C2semidirect
5. #20293015C2semidirect
6. #20293014C2semidirect
7. #20293011C2semidirect
8. #20293010C2semidirect
9. #20293002C2semidirect
10. #20292581C2semidirect
11. #493×C107C2semidirect
16. C2×Dic428C22semidirect
17. C2×#13133807C22semidirect
18. C2×#13133806C22semidirect
19. C2×#13133806C22semidirect
22. C2×#13133803C22semidirect
26. #16812105C22semidirect
27. #16812097C22semidirect
28. #16812096C22semidirect
32. #16812020C22semidirect
33. #16812019C22semidirect
34. C8×Dic107C22semidirect
36. C2×D8×C107C22semidirect
37. C2×D8×C107C22semidirect
38. #127×C107C22semidirect
39. C4×C8×C107C22semidirect
59. #13133801D4semidirect
62. C8×D107D4semidirect
130. C107#3155semidirect
140. #493D107semidirect
155. C2×D8C2×D107semidirect
156. C2×D8C2×D107semidirect
160. #127C2×D107semidirect
169. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×#168121282
C2×D8×D1072
#202931902
#202930152
#202930142
#202930112
#202930102
#202930022
#202925812
#493×C1072
#3155107