Finite Group Databasebeta

← #28348063

Subgroups — #28348063

Order 13696.

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

normal subgroups only

This group has 546 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×#16812137C212yesmaximal, complemented
2#20293282C211yesmaximal, characteristic, complemented
3D4×D428C211yesmaximal, characteristic, complemented
4#20293000C211yesmaximal, characteristic, complemented
5#20292873C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292823C211yesmaximal, characteristic, complemented
8#20292780C212yesmaximal, complemented
9#20292774C212yesmaximal, complemented
10C4×D856C211yesmaximal, characteristic, complemented
11#436×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yescomplemented
13C22×D428C2212yes—
14C2×D4×D107—21no—
15#16812137—28no—
16C2×#13133807C2212yes—
17C2×#13133807C2212yescomplemented
18C2×#13133807C2212yescomplemented
19C2×D856C2211yescharacteristic
20C2×D856C2212yescomplemented
21#16812109C2211yescharacteristic, complemented
22#16812106C2211yescharacteristic
23#16812091—22no—
24#16812088C2212yescomplemented
25#16812084—21no—
26#16812083C2212yes—
27#16812080—21no—
28#16812079—22no—
29C4×D428C2211yescharacteristic
30C4×D428C2211yescharacteristic, complemented
31C4⋊C4×D107—21no—
32#16812043C2211yescharacteristic, complemented
33#16812034C2212yescomplemented
34#16812031C2211yescharacteristic
35#16812031C2212yescomplemented
36#16812031C2211yescharacteristic, complemented
37C2×M16×C107C2212yescomplemented
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic, complemented
40C22⋊C8×C107C2212yescomplemented
41C4×C8×C107C2211yescharacteristic, complemented
42C23×D107—22no—
43#13133818—24no—
44#13133818—42no—
45C2×Dic214C2312yes—
46C2×D428—24no—
47C2×D428—22no—
48C2×D428—21no—
49C2×D428—21no—
50C2×D428—42no—
51C2×D428C2312yes—
52C2×D428C2311yescharacteristic
53C2×D428C2311yescharacteristic
54C2×C22⋊D107—22no—
55C2×C22⋊D107—22no—
56D4×D107—44no—
57C2×C4×D107—21no—
58C2×C4×D107—22no—
59C2×C4×D107—21no—
60#13133807—24no—
61#13133807—24no—
62#13133807—42no—
63D856—24no—
64D856D414yescomplemented
65C4⋊Dic107—21no—
66C4⋊Dic107—21no—
67C4⋊Dic107C2311yescharacteristic
68C4⋊Dic107C2311yescharacteristic
69Dic107⋊C4—22no—
70C22⋊Dic107—22no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4—21no—
73(C2×D107)⋊C4—22no—
74C2×D4×C107D411yescharacteristic
75C22×C4×C107C2312yes—
76M16×C107—24no—
77C2×C8×C107C2312yes—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C4⋊C4×C107D411yescharacteristic
81C22⋊C4×C107D412yes—
82C42×C107C2311yescharacteristic
83C23×C107C2×D412yes—
84C22×D107—21no—
85C22×D107—21no—
86C22×D107—22no—
87C22×D107—44no—
88C22×D107—44no—
89C22×D107—82no—
90D4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—44no—
98C22⋊D107—44no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102D428—22no—
103D428—21no—
104D428—21no—
105D428—22no—
106D428—44no—
107D428—42no—
108D428—81no—
109D428C2×D412yes—
110D428C2×D412yescomplemented
111C4×D107—44no—
112C4×D107—42no—
113C4×D107—42no—
114Dic214—22no—
115Dic214—22no—
116C8×C107—22no—
117C8×C107—21no—
118C8×C107C2×D412yescomplemented
119C22×C107—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107C22×D411yescharacteristic
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—41no—
128C2×D107—41no—
129C2×D107—42no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—88no—
133C4×C107—22no—
134C4×C107—21no—
135C4×C107AGL(1,ℤ/8)12yescomplemented
136C4×C107C22×D411yescharacteristic
137C4×C107H(5,𝔽2)11yescharacteristic
138Dic107—42no—
139Dic107—41no—
140Dic107—41no—
141C2×C107—42no—
142C2×C107D4211yescharacteristic
143C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C107#57711yescharacteristic
145D107—42no—
146D107—42no—
147D107—82no—
148#3272—1071nomaximal, Sylow
149C107#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1072no—
151#547—1071no—
152D42—1071no—
153#492—1071no—
154#468—1072no—
155#461—1071no—
156#459—1071no—
157#450—1072no—
158#449—1072no—
159C4×D8—1071no—
160#436D10711yescharacteristic, complemented
161C2×#41—1072no—
162C22×D4—1072no—
163C22×D4—2141no—
164AGL(1,ℤ/8)—2148no—
165C2×SD16—1072no—
166C2×SD16—1072no—
167C2×SD16—1072no—
168C2×D8—1071no—
169C2×D8—1072no—
170C2×M16C2×D10712yescomplemented
171C4⋊Q8—1071no—
172#127—1071no—
173#123—2142no—
174C22⋊Q8—1072no—
175#121—1072no—
176#121—2141no—
177#121—2141no—
178#120—2142no—
179C4×D4—1071no—
180C4×D4—1071no—
181C4×D4C2×D10711yescharacteristic
182C2×C4⋊C4—2141no—
183#107—1071no—
184C4⋊C8C2×D10711yescharacteristic, complemented
185#103—1072no—
186#102—1071no—
187#102—1071no—
188#102—1072no—
189C22⋊C8C2×D10712yescomplemented
190C4×C8C2×D10711yescharacteristic, complemented
191C24—2142no—
192#41—4282no—
193#41—2144no—
194C2×Q8—1072no—
195C2×D4—1071no—
196C2×D4—1071no—
197C2×D4—1072no—
198C2×D4—4282no—
199C2×D4—4284no—
200C2×D4—2144no—
201C2×D4—2142no—
202C2×D4—2142no—
203C2×D4—2141no—
204C2×D4—2142no—
205C2×D4—2141no—
206C2×D4D42811yescharacteristic
207C22×C4—2141no—
208C22×C4—2142no—
209C22×C4—2141no—
210C22×C4C22×D10712yes—
211SD16—4282no—
212SD16—2144no—
213SD16—2144no—
214D8—1074no—
215D8—2144no—
216M16—24no—
217C2×C8C22×D10711yescharacteristic
218C2×C8C22×D10712yes—
219C2×C8C22×D10711yescharacteristic
220C4⋊C4—1071no—
221C4⋊C4—1071no—
222C4⋊C4—2141no—
223C4⋊C4—2142no—
224C4⋊C4—2141no—
225C4⋊C4D42811yescharacteristic
226C22⋊C4—2142no—
227C22⋊C4—2141no—
228C22⋊C4—2141no—
229C22⋊C4—2142no—
230C22⋊C4D42812yes—
231C42C22×D10711yescharacteristic
232C23—8562no—
233C23—4284no—
234C23—4284no—
235C23—2142no—
236C23—2141no—
237C23—2141no—
238C23C2×D42812yes—
239Q8—2142no—
240Q8—2142no—
241D4—22no—
242D4—8561no—
243D4—1072no—
244D4—1072no—
245D4—4284no—
246D4—4284no—
247D4—4284no—
248D4—4282no—
249D4—2142no—
250D4—2142no—
251D4—2141no—
252D4—2141no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—4284no—
256C2×C4—4282no—
257C2×C4—4282no—
258C2×C4—2141no—
259C2×C4—2141no—
260C2×C4—2142no—
261C2×C4C2×D42811yescharacteristic
262C2×C4C2×D42812yes—
263C2×C4C2×D42811yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D10712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—8568no—
272C22—4282no—
273C22—4282no—
274C22—4282no—
275C22—4281no—
276C22—4281no—
277C22—4281no—
278C22—4281no—
279C22—2142no—
280C22—2142no—
281C22C22×D42811yescharacteristic
282C4—21no—
283C4—22no—
284C4—4281no—
285C4—4281no—
286C4—4282no—
287C4#1681213712yescomplemented
288C4C2×D4×D10711yescharacteristic
289C4#1681215511yescharacteristic
290C2—42no—
291C2—8562no—
292C2—4282no—
293C2—4282no—
294C2D4×D42811yescharacteristic
295C2C2×#1681213711yescharacteristic
296C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#436×C107C2
socleC22×C107C22×D4
hypercenter#436D107
hypocenterC107#3272
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. #20293282C2semidirect
3. D4×D428C2semidirect
4. #20293000C2semidirect
5. #20292873C2semidirect
6. #20292833C2semidirect
7. #20292823C2semidirect
8. #20292780C2semidirect
9. #20292774C2semidirect
10. C4×D856C2semidirect
11. #436×C107C2semidirect
12. C2×#13133818C22semidirect
17. C2×#13133807C22semidirect
18. C2×#13133807C22semidirect
20. C2×D856C22semidirect
21. #16812109C22semidirect
24. #16812088C22semidirect
30. C4×D428C22semidirect
32. #16812043C22semidirect
33. #16812034C22semidirect
35. #16812031C22semidirect
36. #16812031C22semidirect
37. C2×M16×C107C22semidirect
39. C4⋊C8×C107C22semidirect
40. C22⋊C8×C107C22semidirect
41. C4×C8×C107C22semidirect
64. D856D4semidirect
110. D428C2×D4semidirect
118. C8×C107C2×D4semidirect
135. C4×C107AGL(1,ℤ/8)semidirect
149. C107#3272semidirect
160. #436D107semidirect
170. C2×M16C2×D107semidirect
184. C4⋊C8C2×D107semidirect
189. C22⋊C8C2×D107semidirect
190. C4×C8C2×D107semidirect
267. C8D4×D107semidirect
287. C4#16812137semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
#202932822
D4×D4282
#202930002
#202928732
#202928332
#202928232
#202927802
#202927742
C4×D8562
#436×C1072
#3272107