Finite Group Databasebeta

← #28348149

Subgroups — #28348149

Order 13696.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16812131C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293274C212yesmaximal, complemented
4#20293010C211yesmaximal, characteristic, complemented
5#20292869C211yesmaximal, characteristic, complemented
6#20292841C212yesmaximal, complemented
7#20292779C212yesmaximal, complemented
8#20292770C212yesmaximal, complemented
9#20292709C211yesmaximal, characteristic, complemented
10C8×C22⋊D107C211yesmaximal, characteristic, complemented
11#466×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yescomplemented
13C2×D4×D107C2212yes—
14#16812131—24no—
15C2×Dic428C2211yescharacteristic, complemented
16C2×#13133807C2212yescomplemented
17C2×#13133806C2212yescomplemented
18C2×D856C2211yescharacteristic
19C2×#13133803C2212yes—
20D8×D107—24no—
21C2×C8×D107C2211yescharacteristic
22#16812096C2212yescomplemented
23#16812091—22no—
24#16812089C2212yes—
25#16812080—22no—
26#16812077—22no—
27#16812071C2212yescomplemented
28C4×C22⋊D107C2211yescharacteristic
29C22⋊C4×D107—22no—
30Dic107⋊C8C2211yescharacteristic, complemented
31#16812032C2212yescomplemented
32#16812029C2212yescomplemented
33C22⋊(C107⋊C8)C2211yescharacteristic, complemented
34(C2×D107)⋊C8C2211yescharacteristic
35C8×Dic107C2211yescharacteristic, complemented
36C2×D8×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic, complemented
38#121×C107C2212yescomplemented
39#107×C107C2211yescharacteristic, complemented
40#102×C107C2212yescomplemented
41C23×D107—22no—
42#13133818—24no—
43#13133818—42no—
44C2×Dic214C2312yes—
45C2×D428C2312yes—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107D411yescharacteristic
49D4×D107—24no—
50D4×D107—42no—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107C2311yescharacteristic
54Dic428—22no—
55#13133807—24no—
56#13133806—42no—
57D856D414yescomplemented
58#13133803—24no—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—22no—
62C4⋊Dic107—22no—
63Dic107⋊C4D411yescharacteristic
64C22⋊Dic107—22no—
65C22⋊Dic107—22no—
66C22⋊Dic107D411yescharacteristic
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4D411yescharacteristic
70C4×Dic107C2311yescharacteristic
71C2×D4×C107—22no—
72C2×D4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74D8×C107—22no—
75C2×C8×C107—22no—
76C2×C8×C107C2311yescharacteristic
77C2×C8×C107C2311yescharacteristic
78C4⋊C4×C107C2312yes—
79C22⋊C4×C107—22no—
80C23×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107—22no—
83C22×D107—44no—
84C22×D107—44no—
85C22×D107—82no—
86C22×D107C2×D411yescharacteristic
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—82no—
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—22no—
96C22⋊D107—44no—
97C22⋊D107—44no—
98C2×Dic107—22no—
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102D428—22no—
103D428C2×D414yes—
104C4×D107—21no—
105C4×D107—21no—
106C4×D107—44no—
107C4×D107—82no—
108Dic214—22no—
109Dic214—22no—
110C8×C107—21no—
111C8×C107C2×D412yes—
112C107⋊C8—21no—
113C107⋊C8—21no—
114C22×C107—21no—
115C22×C107—21no—
116C22×C107—21no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107C22×D411yescharacteristic
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—24no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—88no—
127C2×D107#13512yescomplemented
128C4×C107—21no—
129C4×C107—42no—
130C4×C107C22×D411yescharacteristic
131C4×C107H(5,𝔽2)11yescharacteristic
132Dic107—21no—
133Dic107—21no—
134Dic107—21no—
135Dic107—42no—
136C2×C107—41no—
137C2×C107—82no—
138C2×C107D4211yescharacteristic
139C2×C107C2×#13511yescharacteristic
140C2×C107#57611yescharacteristic
141D107—22no—
142D107—44no—
143#3280—1071nomaximal, Sylow
144C107#328011yescharacteristic, complemented, Sylow
145C2×#135—1071no—
146C22×D8—1071no—
147#546—1072no—
148#495—1071no—
149#467—1071no—
150#466D10711yescharacteristic, complemented
151#463—1072no—
152#450—1072no—
153#449—1072no—
154C4×D8—1071no—
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×C8C2×D10711yescharacteristic, complemented
168#124—1072no—
169#123—2142no—
170C22⋊Q8—1072no—
171#121—2142no—
172#121C2×D10712yescomplemented
173#120—2142no—
174C4×D4—1071no—
175C4×D4—1072no—
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×C8—1071no—
185C24—2142no—
186#41—4282no—
187#41—2144no—
188C2×Q8—1072no—
189C2×D4—22no—
190C2×D4—1071no—
191C2×D4—1072no—
192C2×D4—4282no—
193C2×D4—2142no—
194C2×D4—2144no—
195C2×D4—2142no—
196C2×D4C22×D10712yes—
197C22×C4—1071no—
198C22×C4—2142no—
199C22×C4—2142no—
200C22×C4C22×D10711yescharacteristic
201Q16—2142no—
202SD16—4282no—
203SD16—2144no—
204D8—22no—
205D8—1074no—
206D8—2144no—
207C2×C8—22no—
208C2×C8—1071no—
209C2×C8—1071no—
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—
223C42—1071no—
224C23—22no—
225C23—8562no—
226C23—1071no—
227C23—4284no—
228C23—4284no—
229C23—2142no—
230C23D4×D10711yescharacteristic
231Q8—2142no—
232Q8—2142no—
233D4—22no—
234D4—22no—
235D4—82no—
236D4—1074no—
237D4—4284no—
238D4—4284no—
239D4—2142no—
240D4—2142no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—8562no—
245C2×C4—1071no—
246C2×C4—1071no—
247C2×C4—1071no—
248C2×C4—4284no—
249C2×C4—2142no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4D4×D10711yescharacteristic
253C2×C4C23×D10711yescharacteristic
254C8—21no—
255C8—2141no—
256C8—2141no—
257C8D4×D10712yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—8568no—
265C22—1072no—
266C22—4282no—
267C22—4282no—
268C22—2144no—
269C22—2141no—
270C22—2141no—
271C22C2×D4×D10711yescharacteristic
272C4—21no—
273C4—42no—
274C4—4282no—
275C4—2141no—
276C4—2141no—
277C4—2141no—
278C4C2×D4×D10711yescharacteristic
279C4#1681215511yescharacteristic
280C2—41no—
281C2—82no—
282C2—4284no—
283C2—2142no—
284C2#2029326111yescharacteristic
285C2#135×D10711yescharacteristic
286C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#466×C107C2
socleC22×C107C22×D4
hypercenter#466D107
hypocenterC107#3280
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16812131C2semidirect
2. C2×D8×D107C2semidirect
3. #20293274C2semidirect
4. #20293010C2semidirect
5. #20292869C2semidirect
6. #20292841C2semidirect
7. #20292779C2semidirect
8. #20292770C2semidirect
9. #20292709C2semidirect
10. C8×C22⋊D107C2semidirect
11. #466×C107C2semidirect
12. C2×#13133818C22semidirect
15. C2×Dic428C22semidirect
16. C2×#13133807C22semidirect
17. C2×#13133806C22semidirect
22. #16812096C22semidirect
27. #16812071C22semidirect
30. Dic107⋊C8C22semidirect
31. #16812032C22semidirect
32. #16812029C22semidirect
33. C22⋊(C107⋊C8)C22semidirect
35. C8×Dic107C22semidirect
37. C22×C8×C107C22semidirect
38. #121×C107C22semidirect
39. #107×C107C22semidirect
40. #102×C107C22semidirect
57. D856D4semidirect
127. C2×D107#135semidirect
144. C107#3280semidirect
150. #466D107semidirect
167. C22×C8C2×D107semidirect
172. #121C2×D107semidirect
177. #107C2×D107semidirect
181. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121312
C2×D8×D1072
#202932742
#202930102
#202928692
#202928412
#202927792
#202927702
#202927092
C8×C22⋊D1072
#466×C1072
#3280107