Finite Group Databasebeta

← #28348145

Subgroups — #28348145

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×#16812132C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293270C212yesmaximal, complemented
4#20292865C211yesmaximal, characteristic, complemented
5#20292860C211yesmaximal, characteristic, complemented
6#20292839C212yesmaximal, complemented
7#20292776C212yesmaximal, complemented
8#20292771C212yesmaximal, complemented
9C4×D856C211yesmaximal, characteristic, complemented
10C8×D428C211yesmaximal, characteristic, complemented
11#495×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yescomplemented
13C2×D4×D107C2212yes—
14#16812132—24no—
15C2×#13133805C2212yescomplemented
16C2×D856C2211yescharacteristic
17C2×#13133803C2212yes—
18D8×D107—24no—
19C2×C8×D107C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic, complemented
21#16812091—22no—
22#16812085C2212yes—
23#16812082—22no—
24#16812080C2212yescomplemented
25#16812079—22no—
26C4×D428C2211yescharacteristic
27C4×D428C2212yescomplemented
28C22⋊C4×D107—22no—
29#16812043C2211yescharacteristic, complemented
30C4⋊(C107⋊C8)C2211yescharacteristic, complemented
31#16812033C2212yescomplemented
32#16812031C2212yescomplemented
33#16812030C2212yescomplemented
34(C2×D107)⋊C8C2211yescharacteristic
35(C2×D107)⋊C8C2211yescharacteristic, complemented
36C2×Q16×C107C2211yescharacteristic, complemented
37C2×SD16×C107C2212yescomplemented
38C2×D8×C107C2211yescharacteristic
39#124×C107C2212yescomplemented
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42#13133819—24no—
43#13133819—42no—
44C2×D428—22no—
45C2×D428D411yescharacteristic
46C2×D428C2312yes—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49D4×D107—24no—
50D4×D107—42no—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107C2311yescharacteristic
54C2×C4×D107C2311yescharacteristic
55#13133805—24no—
56D856D414yescomplemented
57#13133803—24no—
58C2×C107⋊C8C2311yescharacteristic
59C2×C107⋊C8C2311yescharacteristic
60C8×D107—22no—
61C8×D107—22no—
62C4⋊Dic107D411yescharacteristic
63C4⋊Dic107C2312yes—
64Dic107⋊C4—22no—
65C22⋊Dic107—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4D411yescharacteristic
69(C2×D107)⋊C4D411yescharacteristic
70C2×Q8×C107C2312yes—
71C2×D4×C107C2312yes—
72Q16×C107—22no—
73SD16×C107—42no—
74D8×C107—22no—
75C2×C8×C107C2311yescharacteristic
76C2×C8×C107C2311yescharacteristic
77C22⋊C4×C107—22no—
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
87Q8×C107—22no—
88Q8×C107—22no—
89D4×C107—22no—
90D4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—44no—
96C22⋊D107—82no—
97C2×Dic107—22no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100D428—22no—
101D428—22no—
102D428—44no—
103D428C2×D414yes—
104C4×D107—21no—
105C4×D107—21no—
106C4×D107—21no—
107C4×D107—21no—
108C4×D107—44no—
109C4×D107—82no—
110C8×C107—21no—
111C8×C107C2×D412yes—
112C107⋊C8—21no—
113C107⋊C8—21no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—21no—
121C2×D107—24no—
122C2×D107—21no—
123C2×D107—21no—
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—82no—
137C2×C107D4211yescharacteristic
138C2×C107C2×#13511yescharacteristic
139C2×C107#57611yescharacteristic
140D107—22no—
141D107—41no—
142D107—44no—
143#3280—1071nomaximal, Sylow
144C107#328011yescharacteristic, complemented, Sylow
145C2×#135—1071no—
146C22×D8—1071no—
147#546—1072no—
148#495D10711yescharacteristic, complemented
149#467—1071no—
150#466—1071no—
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×Q16C2×D10711yescharacteristic, complemented
160C2×SD16—1072no—
161C2×SD16C2×D10712yescomplemented
162C2×D8—1072no—
163C2×D8—1071no—
164C2×D8—2144no—
165C2×D8C2×D10711yescharacteristic
166C22×C8—1071no—
167C22×C8—1071no—
168#124C2×D10712yescomplemented
169#123—2142no—
170C22⋊Q8—1072no—
171#121—1072no—
172#121—2142no—
173#120—2142no—
174C4×D4—1071no—
175C4×D4—1072no—
176C2×C22⋊C4—2142no—
177#107—1071no—
178C4⋊C8—1071no—
179#103—1072no—
180#102—1072no—
181#102—1072no—
182C22⋊C8—1071no—
183C22⋊C8—1071no—
184C4×C8C2×D10711yescharacteristic, complemented
185C24—2142no—
186#41—4282no—
187#41—2144no—
188C2×Q8C22×D10712yes—
189C2×D4—1071no—
190C2×D4—1072no—
191C2×D4—4282no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2144no—
195C2×D4—2142no—
196C2×D4C22×D10712yes—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—2142no—
200C22×C4—2142no—
201Q16—22no—
202SD16—42no—
203SD16—2144no—
204D8—22no—
205D8—1074no—
206D8—2144no—
207C2×C8—1071no—
208C2×C8—1071no—
209C2×C8—2142no—
210C2×C8—2142no—
211C2×C8C22×D10711yescharacteristic
212C2×C8C22×D10711yescharacteristic
213C4⋊C4—1071no—
214C4⋊C4—1072no—
215C4⋊C4—2142no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1071no—
219C22⋊C4—1071no—
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—22no—
232Q8—22no—
233D4—22no—
234D4—22no—
235D4—8562no—
236D4—1074no—
237D4—4284no—
238D4—4284no—
239D4—2142no—
240D4—2142no—
241C2×C4—22no—
242C2×C4—8562no—
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—
257C8D4×D10712yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—8568no—
262C22—1072no—
263C22—4282no—
264C22—4282no—
265C22—2141no—
266C22—2141no—
267C22—2144no—
268C22—2141no—
269C22—2141no—
270C22—2141no—
271C22C2×D4×D10711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—4282no—
276C4—2141no—
277C4—2141no—
278C4C2×D4×D10711yescharacteristic
279C4#1681215411yescharacteristic
280C2—82no—
281C2—4284no—
282C2—4281no—
283C2—2142no—
284C2#2029326211yescharacteristic
285C2#135×D10711yescharacteristic
286C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3280
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. C2×D8×D107C2semidirect
3. #20293270C2semidirect
4. #20292865C2semidirect
5. #20292860C2semidirect
6. #20292839C2semidirect
7. #20292776C2semidirect
8. #20292771C2semidirect
9. C4×D856C2semidirect
10. C8×D428C2semidirect
11. #495×C107C2semidirect
12. C2×#13133819C22semidirect
15. C2×#13133805C22semidirect
20. C2×C8×D107C22semidirect
24. #16812080C22semidirect
27. C4×D428C22semidirect
29. #16812043C22semidirect
30. C4⋊(C107⋊C8)C22semidirect
31. #16812033C22semidirect
32. #16812031C22semidirect
33. #16812030C22semidirect
35. (C2×D107)⋊C8C22semidirect
36. C2×Q16×C107C22semidirect
37. C2×SD16×C107C22semidirect
39. #124×C107C22semidirect
40. C4×C8×C107C22semidirect
56. D856D4semidirect
127. C2×D107#135semidirect
144. C107#3280semidirect
148. #495D107semidirect
159. C2×Q16C2×D107semidirect
161. C2×SD16C2×D107semidirect
168. #124C2×D107semidirect
184. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
C2×D8×D1072
#202932702
#202928652
#202928602
#202928392
#202927762
#202927712
C4×D8562
C8×D4282
#495×C1072
#3280107