Finite Group Databasebeta

← #28348147

Subgroups — #28348147

Order 13696.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16812128C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293272C212yesmaximal, complemented
4#20293011C211yesmaximal, characteristic, complemented
5#20292869C211yesmaximal, characteristic, complemented
6#20292864C211yesmaximal, characteristic, complemented
7#20292842C212yesmaximal, complemented
8#20292779C212yesmaximal, complemented
9#20292773C212yesmaximal, complemented
10C8×D428C211yesmaximal, characteristic, complemented
11C4×D8×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yescomplemented
13C2×D4×D107C2212yes—
14#16812128—24no—
15C2×Dic428C2211yescharacteristic, complemented
16C2×#13133807C2212yescomplemented
17C2×#13133806C2212yescomplemented
18C2×D856C2211yescharacteristic
19C2×#13133803C2212yes—
20D8×D107—24no—
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic, complemented
23#16812097C2212yescomplemented
24#16812093—22no—
25#16812089C2212yes—
26#16812084C2212yescomplemented
27#16812083—22no—
28#16812079—22no—
29C4×D428C2211yescharacteristic
30C2×(C2×D107)⋊C4—22no—
31C4⋊(C107⋊C8)C2211yescharacteristic, complemented
32#16812032C2212yescomplemented
33#16812029C2212yescomplemented
34(C2×D107)⋊C8C2211yescharacteristic
35(C2×D107)⋊C8C2211yescharacteristic, complemented
36C2×D8×C107C2211yescharacteristic
37C4×D4×C107C2212yescomplemented
38#107×C107C2211yescharacteristic, complemented
39#102×C107C2212yescomplemented
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42#13133817—24no—
43#13133817—42no—
44C2×Dic214C2312yes—
45C2×D428—22no—
46C2×D428D411yescharacteristic
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50D4×D107—24no—
51D4×D107—42no—
52C22×Dic107—22no—
53C2×C4×D107C2311yescharacteristic
54C2×C4×D107C2311yescharacteristic
55Dic428—22no—
56#13133807—42no—
57#13133806—24no—
58D856—22no—
59#13133803—24no—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C8×D107—22no—
63C8×D107—22no—
64C4⋊Dic107—22no—
65C4⋊Dic107D411yescharacteristic
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4D411yescharacteristic
71(C2×D107)⋊C4D411yescharacteristic
72C2×D4×C107C2312yes—
73C22×C4×C107—22no—
74D8×C107D414yescomplemented
75C2×C8×C107C2311yescharacteristic
76C2×C8×C107C2311yescharacteristic
77C4⋊C4×C107C2312yes—
78C22⋊C4×C107—22no—
79C42×C107C2311yescharacteristic
80C23×C107—22no—
81C22×D107—22no—
82C22×D107—44no—
83C22×D107—44no—
84C22×D107—82no—
85C22×D107C2×D411yescharacteristic
86C22×D107C2×D411yescharacteristic
87D4×C107—22no—
88D4×C107C2×D414yes—
89C2×C4×C107—22no—
90C2×C4×C107—82no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—44no—
95C22⋊D107—44no—
96C2×Dic107—22no—
97C2×Dic107—44no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100D428—22no—
101D428—22no—
102D428—22no—
103D428—82no—
104C4×D107—21no—
105C4×D107—21no—
106C4×D107—21no—
107C4×D107—21no—
108Dic214—22no—
109Dic214—22no—
110C8×C107—21no—
111C8×C107C2×D412yes—
112C107⋊C8—21no—
113C107⋊C8—21no—
114C22×C107—24no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—21no—
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—42no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—88no—
127C2×D107#13512yescomplemented
128C4×C107—21no—
129C4×C107—21no—
130C4×C107—42no—
131C4×C107C22×D411yescharacteristic
132C4×C107H(5,𝔽2)11yescharacteristic
133Dic107—21no—
134Dic107—21no—
135Dic107—42no—
136C2×C107—44no—
137C2×C107D4211yescharacteristic
138C2×C107C2×#13511yescharacteristic
139C2×C107#57611yescharacteristic
140D107—22no—
141D107—41no—
142D107—82no—
143#3280—1071nomaximal, Sylow
144C107#328011yescharacteristic, complemented, Sylow
145C2×#135—1071no—
146C22×D8—1071no—
147#546—1072no—
148#495—1071no—
149#467—1071no—
150#466—1071no—
151#463—1072no—
152#450—1072no—
153#449—1072no—
154C4×D8D10711yescharacteristic, complemented
155C8×D4—1071no—
156C2×#41—1072no—
157C22×D4—1072no—
158#135—2144no—
159C2×Q16—1071no—
160C2×SD16—1072no—
161C2×SD16—1072no—
162C2×D8—1072no—
163C2×D8—1071no—
164C2×D8—2144no—
165C2×D8C2×D10711yescharacteristic
166C22×C8—1071no—
167C22×C8—1071no—
168#124—1072no—
169#123—2142no—
170C22⋊Q8—1072no—
171#121—1072no—
172#121—2142no—
173#120—2142no—
174C4×D4—1071no—
175C4×D4C2×D10712yescomplemented
176C2×C22⋊C4—2142no—
177#107C2×D10711yescharacteristic, complemented
178C4⋊C8—1071no—
179#103—1072no—
180#102—1072no—
181#102C2×D10712yescomplemented
182C22⋊C8—1071no—
183C22⋊C8—1071no—
184C4×C8C2×D10711yescharacteristic, complemented
185C24—2142no—
186#41—4282no—
187#41—2144no—
188C2×Q8—1072no—
189C2×D4—1071no—
190C2×D4—1072no—
191C2×D4—4282no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4—2144no—
196C2×D4C22×D10712yes—
197C22×C4—22no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—2142no—
201Q16—2142no—
202SD16—4282no—
203SD16—2144no—
204D8—2144no—
205D8—2142no—
206D8D42814yescomplemented
207C2×C8—1071no—
208C2×C8—1071no—
209C2×C8—2142no—
210C2×C8—2142no—
211C2×C8C22×D10711yescharacteristic
212C2×C8C22×D10711yescharacteristic
213C4⋊C4—1071no—
214C4⋊C4—2142no—
215C4⋊C4C22×D10712yes—
216C22⋊C4—22no—
217C22⋊C4—1071no—
218C22⋊C4—1071no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C22⋊C4—2142no—
222C22⋊C4—2142no—
223C42C22×D10711yescharacteristic
224C23—22no—
225C23—8562no—
226C23—1071no—
227C23—1071no—
228C23—4284no—
229C23—4284no—
230C23—2142no—
231Q8—2142no—
232Q8—2142no—
233D4—22no—
234D4—8562no—
235D4—4284no—
236D4—4284no—
237D4—2142no—
238D4—2142no—
239D4—2142no—
240D4C2×D42814yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—1071no—
244C2×C4—1071no—
245C2×C4—4284no—
246C2×C4—2142no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4—2141no—
251C2×C4D4×D10711yescharacteristic
252C2×C4D4×D10711yescharacteristic
253C2×C4C23×D10711yescharacteristic
254C8—21no—
255C8—2141no—
256C8—2141no—
257C8C2×D42812yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—8568no—
262C22—1072no—
263C22—4282no—
264C22—4282no—
265C22—4282no—
266C22—2141no—
267C22—2141no—
268C22—2141no—
269C22—2141no—
270C22—2141no—
271C22C2×D4×D10711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—4282no—
276C4—2141no—
277C4—2141no—
278C4C22×D42811yescharacteristic
279C4#1681215511yescharacteristic
280C2—44no—
281C2—8562no—
282C2—4281no—
283C2—2142no—
284C2D4×D42811yescharacteristic
285C2#135×D10711yescharacteristic
286C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×D8×C107C2
socleC22×C107C22×D4
hypercenterC4×D8D107
hypocenterC107#3280
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16812128C2semidirect
2. C2×D8×D107C2semidirect
3. #20293272C2semidirect
4. #20293011C2semidirect
5. #20292869C2semidirect
6. #20292864C2semidirect
7. #20292842C2semidirect
8. #20292779C2semidirect
9. #20292773C2semidirect
10. C8×D428C2semidirect
11. C4×D8×C107C2semidirect
12. C2×#13133817C22semidirect
15. C2×Dic428C22semidirect
16. C2×#13133807C22semidirect
17. C2×#13133806C22semidirect
22. C2×C8×D107C22semidirect
23. #16812097C22semidirect
26. #16812084C22semidirect
31. C4⋊(C107⋊C8)C22semidirect
32. #16812032C22semidirect
33. #16812029C22semidirect
35. (C2×D107)⋊C8C22semidirect
37. C4×D4×C107C22semidirect
38. #107×C107C22semidirect
39. #102×C107C22semidirect
40. C4×C8×C107C22semidirect
74. D8×C107D4semidirect
127. C2×D107#135semidirect
144. C107#3280semidirect
154. C4×D8D107semidirect
175. C4×D4C2×D107semidirect
177. #107C2×D107semidirect
181. #102C2×D107semidirect
184. C4×C8C2×D107semidirect
206. D8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121282
C2×D8×D1072
#202932722
#202930112
#202928692
#202928642
#202928422
#202927792
#202927732
C8×D4282
C4×D8×C1072
#3280107