Finite Group Databasebeta

← #496×D107 #28346891

Subgroups — #496×D107 #28346891

Order 13696.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D107C212yesmaximal, complemented
2C2×D8×D107C212yesmaximal, complemented
3#127×D107C211yesmaximal, characteristic, complemented
4#124×D107C211yesmaximal, characteristic, complemented
5#20293017C211yesmaximal, characteristic, complemented
6#20293016C211yesmaximal, characteristic, complemented
7#20293015C212yesmaximal, complemented
8#20293014C211yesmaximal, characteristic, complemented
9#20293013C212yesmaximal, complemented
10#97×D107C211yesmaximal, characteristic, complemented
11#496×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2211yescharacteristic
13C2×D4×D107—22no—
14C2×D4×D107C2211yescharacteristic
15C2×D4×D107C2212yes—
16C2×#13133807C2212yescomplemented
17C2×#13133806C2212yescomplemented
18C2×#13133805C2212yescomplemented
19SD16×D107—28no—
20C2×D856C2212yescomplemented
21C2×#13133803C2212yescomplemented
22C2×#13133803C2212yescomplemented
23D8×D107—28no—
24C2×C8×D107C2212yes—
25#16812106C2211yescharacteristic, complemented
26#16812105—22no—
27#16812105C2212yescomplemented
28#16812098C2211yescharacteristic, complemented
29#16812097C2211yescharacteristic, complemented
30#16812096C2211yescharacteristic, complemented
31#16812095—22no—
32C22⋊C4×D107—22no—
33C42×D107C2211yescharacteristic
34#16812020C2211yescharacteristic, complemented
35#16812019C2212yescomplemented
36C2×SD16×C107C2212yescomplemented
37C2×D8×C107C2212yescomplemented
38#127×C107C2211yescharacteristic, complemented
39#124×C107C2211yescharacteristic, complemented
40#97×C107C2211yescharacteristic, complemented
41C23×D107—21no—
42C23×D107—22no—
43C2×Dic214—21no—
44C2×Dic214C2311yescharacteristic
45Q8×D107—24no—
46C2×D428—22no—
47C2×D428C2311yescharacteristic
48C2×D428C2312yes—
49C2×C22⋊D107—21no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—24no—
52D4×D107—24no—
53D4×D107—24no—
54D4×D107—24no—
55D4×D107—48no—
56C2×C4×D107—21no—
57C2×C4×D107D412yes—
58C2×C4×D107C2311yescharacteristic
59#13133807—24no—
60#13133806—24no—
61#13133805—24no—
62D856—24no—
63#13133803—24no—
64#13133803—24no—
65C2×C107⋊C8C2312yes—
66C8×D107D418yescomplemented
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107D412yes—
71C4×Dic107C2311yescharacteristic
72C2×Q8×C107C2311yescharacteristic
73C2×D4×C107—22no—
74C2×D4×C107C2311yescharacteristic
75C2×D4×C107C2312yes—
76SD16×C107—24no—
77D8×C107—24no—
78C2×C8×C107C2312yes—
79C22⋊C4×C107—22no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C23×C107—22no—
83C22×D107—21no—
84C22×D107—22no—
85C22×D107—42no—
86C22×D107—44no—
87C22×D107—48no—
88C22×D107—44no—
89C22×D107C2×D411yescharacteristic
90Q8×C107—22no—
91D4×C107—22no—
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—44no—
95C2×C4×C107—21no—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—48no—
99C22⋊D107—42no—
100C22⋊D107—44no—
101C2×Dic107—21no—
102C2×Dic107C2×D412yes—
103C2×Dic107C2×D411yescharacteristic
104D428—22no—
105D428—22no—
106D428—22no—
107D428—44no—
108C4×D107—24no—
109C4×D107—42no—
110C4×D107C2×D414yes—
111Dic214—22no—
112Dic214—42no—
113C8×C107C2×D414yescomplemented
114C107⋊C8C2×D414yescomplemented
115C22×C107—41no—
116C22×C107—42no—
117C22×C107—44no—
118C22×C107—42no—
119C22×C107C22×D411yescharacteristic
120C2×D107—41no—
121C2×D107—42no—
122C2×D107—44no—
123C2×D107—42no—
124C2×D107—88no—
125C2×D107—84no—
126C2×D107#12712yes—
127C2×D107AGL(1,ℤ/8)14yes—
128C4×C107—22no—
129C4×C107—41no—
130C4×C107C22×D412yes—
131Dic107—22no—
132Dic107—41no—
133Dic107#12712yes—
134C2×C107—82no—
135C2×C107—81no—
136C2×C107C2×#12711yescharacteristic
137C2×C107C2×AGL(1,ℤ/8)12yes—
138D107—82no—
139D107—81no—
140D107#49614yesdirect product factor
141C2×#496—1071nomaximal, Sylow
142C107C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—1072no—
144C22×D8—1072no—
145C2×#127—1071no—
146C2×#124—1071no—
147#496—1072no—
148#496—1072no—
149#496—1071no—
150#496—1071no—
151#496—1071no—
152#496D10711yescharacteristic, direct product factor
153C2×#97—1071no—
154C22×Q8—1071no—
155C22×D4—1071no—
156C22×D4—1072no—
157C22×D4—2142no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×SD16—1072no—
161C2×SD16—2148no—
162C2×SD16C2×D10712yescomplemented
163C2×D8—1072no—
164C2×D8—1072no—
165C2×D8—1072no—
166C2×D8—2148no—
167C2×D8C2×D10712yescomplemented
168C22×C8—1072no—
169#127—1072no—
170#127—1071no—
171#127—2142no—
172#127C2×D10711yescharacteristic, complemented
173#124—1071no—
174#124—1071no—
175#124—1071no—
176#124—2142no—
177#124C2×D10711yescharacteristic, complemented
178C2×C22⋊C4—2142no—
179C2×C42—1071no—
180#97—1072no—
181#97—1071no—
182#97C2×D10711yescharacteristic, complemented
183C24—2141no—
184C24—2142no—
185C2×Q8—1071no—
186C2×Q8—2141no—
187C2×Q8—2144no—
188C2×Q8C22×D10711yescharacteristic
189C2×D4—22no—
190C2×D4—1071no—
191C2×D4—1072no—
192C2×D4—4288no—
193C2×D4—2144no—
194C2×D4—2144no—
195C2×D4—2144no—
196C2×D4—2141no—
197C2×D4—2144no—
198C2×D4—2142no—
199C2×D4—2142no—
200C2×D4C22×D10711yescharacteristic
201C2×D4C22×D10712yes—
202C22×C4—1071no—
203C22×C4—1072no—
204C22×C4—2141no—
205SD16—24no—
206SD16—2144no—
207SD16—2144no—
208SD16—2144no—
209D8—24no—
210D8—2144no—
211D8—2144no—
212D8—2144no—
213C2×C8—1072no—
214C2×C8—1078no—
215C2×C8C22×D10712yes—
216C22⋊C4—22no—
217C22⋊C4—2142no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C42—1072no—
221C42—1071no—
222C42C22×D10711yescharacteristic
223C23—21no—
224C23—22no—
225C23—1071no—
226C23—4288no—
227C23—4282no—
228C23—4284no—
229C23—4284no—
230C23—2141no—
231C23—2142no—
232Q8—22no—
233Q8—4282no—
234Q8—2142no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—4284no—
240D4—4282no—
241D4—4284no—
242D4—4288no—
243D4—2142no—
244D4—2142no—
245D4—2142no—
246C2×C4—21no—
247C2×C4—1071no—
248C2×C4—1074no—
249C2×C4—1072no—
250C2×C4—4282no—
251C2×C4—2144no—
252C2×C4—2141no—
253C2×C4D4×D10712yes—
254C2×C4C23×D10711yescharacteristic
255C8—1074no—
256C8D4×D10714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—8568no—
262C22—8564no—
263C22—1074no—
264C22—1072no—
265C22—4284no—
266C22—4282no—
267C22—4281no—
268C22—4282no—
269C22C2×D4×D10711yescharacteristic
270C4—22no—
271C4—41no—
272C4—1072no—
273C4—4281no—
274C4—2142no—
275C4C2×D4×D10712yes—
276C2—82no—
277C2—81no—
278C2—8562no—
279C2—8561no—
280C2—1074no—
281C2#127×D10711yescharacteristic
282C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107C2×#496
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D107C2semidirect
2. C2×D8×D107C2semidirect
3. #127×D107C2semidirect
4. #124×D107C2semidirect
5. #20293017C2semidirect
6. #20293016C2semidirect
7. #20293015C2semidirect
8. #20293014C2semidirect
9. #20293013C2semidirect
10. #97×D107C2semidirect
11. #496×C107C2semidirect
16. C2×#13133807C22semidirect
17. C2×#13133806C22semidirect
18. C2×#13133805C22semidirect
20. C2×D856C22semidirect
21. C2×#13133803C22semidirect
22. C2×#13133803C22semidirect
25. #16812106C22semidirect
27. #16812105C22semidirect
28. #16812098C22semidirect
29. #16812097C22semidirect
30. #16812096C22semidirect
34. #16812020C22semidirect
35. #16812019C22semidirect
36. C2×SD16×C107C22semidirect
37. C2×D8×C107C22semidirect
38. #127×C107C22semidirect
39. #124×C107C22semidirect
40. #97×C107C22semidirect
66. C8×D107D4semidirect
113. C8×C107C2×D4semidirect
114. C107⋊C8C2×D4semidirect
140. D107#496direct (N × H)
142. C107C2×#496semidirect
152. #496D107direct (N × H)
162. C2×SD16C2×D107semidirect
167. C2×D8C2×D107semidirect
172. #127C2×D107semidirect
177. #124C2×D107semidirect
182. #97C2×D107semidirect
256. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1072
C2×D8×D1072
#127×D1072
#124×D1072
#202930172
#202930162
#202930152
#202930142
#202930132
#97×D1072
#496×C1072
C2×#496107