Finite Group Databasebeta

← #28348146

Subgroups — #28348146

Order 13696.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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
2#20293271C212yesmaximal, complemented
3#20293010C211yesmaximal, characteristic, complemented
4#20292868C211yesmaximal, characteristic, complemented
5#20292860C211yesmaximal, characteristic, complemented
6#20292840C212yesmaximal, complemented
7#20292778C212yesmaximal, complemented
8#20292769C212yesmaximal, complemented
9D8×Dic107C211yesmaximal, characteristic, complemented
10C8×C22⋊D107C211yesmaximal, characteristic, complemented
11C22×D8×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yescomplemented
13#16812128—24no—
14C2×Dic428C2211yescharacteristic, complemented
15C2×#13133806C2212yescomplemented
16C2×#13133806C2212yescomplemented
17C2×C8×D107C2211yescharacteristic, complemented
18#16812096C2212yescomplemented
19#16812090—22no—
20#16812088C2212yes—
21#16812081—22no—
22#16812080C2212yescomplemented
23#16812078—22no—
24D4×Dic107C2212yescomplemented
25C4×C22⋊D107C2211yescharacteristic
26C2×C22⋊Dic107—22no—
27#16812043C2211yescharacteristic, complemented
28Dic107⋊C8C2211yescharacteristic, complemented
29#16812034C2212yescomplemented
30#16812030C2212yescomplemented
31#16812030C2212yescomplemented
32C22⋊(C107⋊C8)C2211yescharacteristic
33(C2×D107)⋊C8C2211yescharacteristic, complemented
34C8×Dic107C2211yescharacteristic, complemented
35C22×D4×C107C2212yes—
36C2×D8×C107—24no—
37C2×D8×C107C2211yescharacteristic
38C2×D8×C107C2212yes—
39C2×D8×C107C2211yescharacteristic
40C22×C8×C107C2211yescharacteristic
41#13133817—24no—
42#13133817—42no—
43C2×Dic214C2312yes—
44C2×C22⋊D107—22no—
45C2×C22⋊D107—22no—
46C2×C22⋊D107D411yescharacteristic
47C22×Dic107—22no—
48C22×Dic107—22no—
49C2×C4×D107C2311yescharacteristic
50Dic428—22no—
51#13133806—24no—
52#13133806—42no—
53C2×C107⋊C8C2311yescharacteristic
54C2×C107⋊C8C2311yescharacteristic
55C8×D107—22no—
56C4⋊Dic107C2312yes—
57Dic107⋊C4—22no—
58Dic107⋊C4D411yescharacteristic
59C22⋊Dic107—22no—
60C22⋊Dic107—22no—
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63C22⋊Dic107—22no—
64C22⋊Dic107D411yescharacteristic
65(C2×D107)⋊C4D411yescharacteristic
66C4×Dic107C2311yescharacteristic
67C24×C107—22no—
68C2×D4×C107—24no—
69C2×D4×C107—22no—
70C2×D4×C107—42no—
71C2×D4×C107C2312yes—
72C2×D4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74D8×C107—24no—
75D8×C107—22no—
76D8×C107D414yescomplemented
77C2×C8×C107—22no—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C23×C107—22no—
81C23×C107—22no—
82C23×C107—44no—
83C23×C107—44no—
84C23×C107—82no—
85C23×C107C2×D411yescharacteristic
86C22×D107C2×D411yescharacteristic
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—22no—
90D4×C107—44no—
91D4×C107C2×D414yes—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—22no—
97C22⋊D107—44no—
98C22⋊D107—82no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—44no—
102C2×Dic107—82no—
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105C2×Dic107C2×D411yescharacteristic
106C4×D107—21no—
107C4×D107—21no—
108Dic214—22no—
109Dic214—22no—
110C8×C107—21no—
111C8×C107C2×D412yes—
112C107⋊C8—21no—
113C107⋊C8—21no—
114C22×C107—21no—
115C22×C107—21no—
116C22×C107—24no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—88no—
123C22×C107#13512yescomplemented
124C22×C107C22×D411yescharacteristic
125C2×D107—21no—
126C2×D107—21no—
127C2×D107—21no—
128C4×C107—21no—
129C4×C107C22×D411yescharacteristic
130C4×C107H(5,𝔽2)11yescharacteristic
131Dic107—21no—
132Dic107—21no—
133Dic107—21no—
134Dic107—42no—
135Dic107—42no—
136C2×C107—22no—
137C2×C107—44no—
138C2×C107—82no—
139C2×C107D4211yescharacteristic
140C2×C107C2×#13511yescharacteristic
141C2×C107#57611yescharacteristic
142D107—41no—
143#3280—1071nomaximal, Sylow
144C107#328011yescharacteristic, complemented, Sylow
145C2×#135—1071no—
146C22×D8D10711yescharacteristic, complemented
147#546—1072no—
148#495—1071no—
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×D4C2×D10712yes—
158#135—2144no—
159C2×Q16—1071no—
160C2×SD16—1072no—
161C2×SD16—1072no—
162C2×D8—24no—
163C2×D8C2×D10712yes—
164C2×D8C2×D10711yescharacteristic
165C2×D8C2×D10711yescharacteristic
166C22×C8—1071no—
167C22×C8C2×D10711yescharacteristic
168#124—1072no—
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×C8—1071no—
185C24—22no—
186#41—4282no—
187#41—2144no—
188C2×Q8—1072no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—1071no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4C22×D10712yes—
196C2×D4C22×D10712yes—
197C22×C4—1071no—
198C22×C4—2142no—
199C22×C4—2142no—
200C22×C4C22×D10711yescharacteristic
201Q16—2142no—
202SD16—4282no—
203SD16—2144no—
204D8—24no—
205D8—22no—
206D8C22⋊D10714yescomplemented
207C2×C8—22no—
208C2×C8—1071no—
209C2×C8—1071no—
210C2×C8—2142no—
211C2×C8C22×D10711yescharacteristic
212C2×C8C22×D10711yescharacteristic
213C4⋊C4—1071no—
214C4⋊C4—1072no—
215C4⋊C4—2142no—
216C22⋊C4—1071no—
217C22⋊C4—1071no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C22⋊C4—2142no—
222C22⋊C4—2142no—
223C42—1071no—
224C23—22no—
225C23—22no—
226C23—44no—
227C23—44no—
228C23—82no—
229C23—1071no—
230C23D4×D10711yescharacteristic
231Q8—2142no—
232Q8—2142no—
233D4—22no—
234D4—22no—
235D4—22no—
236D4—44no—
237D4—8562no—
238D4—4284no—
239D4—2142no—
240D4C2×C22⋊D10714yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—8562no—
244C2×C4—1071no—
245C2×C4—1071no—
246C2×C4—1071no—
247C2×C4—4284no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4D4×D10711yescharacteristic
253C2×C4C23×D10711yescharacteristic
254C8—21no—
255C8—2141no—
256C8—2141no—
257C8C2×C22⋊D10712yes—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—88no—
267C22—2141no—
268C22—2141no—
269C22—2141no—
270C22#1681212812yescomplemented
271C22C2×D4×D10711yescharacteristic
272C4—21no—
273C4—4282no—
274C4—4282no—
275C4—2141no—
276C4—2141no—
277C4—2141no—
278C4C22×C22⋊D10711yescharacteristic
279C4#1681215411yescharacteristic
280C2—22no—
281C2—44no—
282C2—82no—
283C2—4281no—
284C2D4×C22⋊D10711yescharacteristic
285C2C2×#1681212811yescharacteristic
286C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×D8×C107C2
socleC22×C107C22×D4
hypercenterC22×D8D107
hypocenterC107#3280
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16812128C2semidirect
2. #20293271C2semidirect
3. #20293010C2semidirect
4. #20292868C2semidirect
5. #20292860C2semidirect
6. #20292840C2semidirect
7. #20292778C2semidirect
8. #20292769C2semidirect
9. D8×Dic107C2semidirect
10. C8×C22⋊D107C2semidirect
11. C22×D8×C107C2semidirect
12. C2×#13133817C22semidirect
14. C2×Dic428C22semidirect
15. C2×#13133806C22semidirect
16. C2×#13133806C22semidirect
17. C2×C8×D107C22semidirect
18. #16812096C22semidirect
22. #16812080C22semidirect
24. D4×Dic107C22semidirect
27. #16812043C22semidirect
28. Dic107⋊C8C22semidirect
29. #16812034C22semidirect
30. #16812030C22semidirect
31. #16812030C22semidirect
33. (C2×D107)⋊C8C22semidirect
34. C8×Dic107C22semidirect
76. D8×C107D4semidirect
123. C22×C107#135semidirect
144. C107#3280semidirect
146. C22×D8D107semidirect
206. D8C22⋊D107semidirect
270. C22#16812128semidirect

Maximal subgroups

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

subgroupindex
C2×#168121282
#202932712
#202930102
#202928682
#202928602
#202928402
#202927782
#202927692
D8×Dic1072
C8×C22⋊D1072
C22×D8×C1072
#3280107