Finite Group Databasebeta

← #28348061

Subgroups — #28348061

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×#16812134C212yesmaximal, complemented
2D4×D428C211yesmaximal, characteristic, complemented
3#20293001C211yesmaximal, characteristic, complemented
4#20292875C212yesmaximal, complemented
5#20292829C211yesmaximal, characteristic, complemented
6#20292822C211yesmaximal, characteristic, complemented
7#20292777C212yesmaximal, complemented
8#20292772C212yesmaximal, complemented
9C4×#13133803C211yesmaximal, characteristic, complemented
10#20292705C211yesmaximal, characteristic, complemented
11#547×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C2×D4×D107—21no—
14#16812134—28no—
15C2×#13133805C2212yes—
16C2×#13133805C2212yescomplemented
17C2×#13133805C2212yescomplemented
18C2×#13133803C2211yescharacteristic
19C2×#13133803C2212yescomplemented
20C2×#13133802C2212yescomplemented
21#16812106C2211yescharacteristic
22#16812084—21no—
23#16812083C2212yes—
24#16812079—22no—
25C4×D428C2211yescharacteristic
26#16812042C2211yescharacteristic, complemented
27C4⋊(C107⋊C8)C2211yescharacteristic, complemented
28#16812033C2212yescomplemented
29#16812030C2211yescharacteristic, complemented
30#16812029C2211yescharacteristic
31#16812029C2212yescomplemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34C2×#41×C107C2212yescomplemented
35C4⋊Q8×C107C2211yescharacteristic, complemented
36#123×C107—22no—
37C22⋊Q8×C107C2212yescomplemented
38#121×C107—21no—
39C4×D4×C107C2211yescharacteristic
40C4×D4×C107C2211yescharacteristic, complemented
41C2×C4⋊C4×C107—21no—
42C23×D107—22no—
43C2×D428—24no—
44C2×D428—22no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428—42no—
48C2×D428C2312yes—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—22no—
51D4×D107—44no—
52C2×C4×D107—21no—
53#13133805—24no—
54#13133805—24no—
55#13133805—42no—
56#13133803—24no—
57#13133803D414yescomplemented
58#13133802—24no—
59C2×C107⋊C8C2312yes—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—22no—
65#41×C107—24no—
66#41×C107—42no—
67C2×Q8×C107C2312yes—
68C2×D4×C107—22no—
69C2×D4×C107D411yescharacteristic
70C2×D4×C107C2311yescharacteristic
71C22×C4×C107—21no—
72C22×C4×C107—22no—
73C22×C4×C107C2312yes—
74C4⋊C4×C107—21no—
75C4⋊C4×C107—22no—
76C4⋊C4×C107—21no—
77C4⋊C4×C107D411yescharacteristic
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—22no—
80C22⋊C4×C107—21no—
81C22⋊C4×C107D412yes—
82C42×C107C2311yescharacteristic
83C23×C107—21no—
84C23×C107C2×D412yes—
85C22×D107—21no—
86C22×D107—22no—
87C22×D107—44no—
88C22×D107—44no—
89C22×D107—82no—
90Q8×C107—22no—
91Q8×C107—22no—
92D4×C107—22no—
93D4×C107—21no—
94D4×C107—44no—
95D4×C107C2×D412yescomplemented
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—21no—
99C2×C4×C107—22no—
100C2×C4×C107—44no—
101C2×C4×C107—42no—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—44no—
107C2×Dic107—21no—
108D428—22no—
109D428—21no—
110D428—22no—
111D428—44no—
112D428—42no—
113D428—81no—
114D428C2×D412yes—
115C4×D107—42no—
116C107⋊C8—22no—
117C107⋊C8—21no—
118C107⋊C8C2×D412yescomplemented
119C22×C107—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107C22×D411yescharacteristic
126C2×D107—22no—
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×C107—42no—
136C4×C107—41no—
137C4×C107AGL(1,ℤ/8)12yescomplemented
138C4×C107C22×D411yescharacteristic
139C4×C107H(5,𝔽2)11yescharacteristic
140Dic107—41no—
141C2×C107—42no—
142C2×C107—42no—
143C2×C107D4211yescharacteristic
144C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C107#57711yescharacteristic
146D107—42no—
147D107—82no—
148#3272—1071nomaximal, Sylow
149C107#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1072no—
151#547D10711yescharacteristic, complemented
152D42—1071no—
153#492—1071no—
154#468—1072no—
155#461—1071no—
156#459—1071no—
157#450—1072no—
158#449—1072no—
159C4×D8—1071no—
160#436—1071no—
161C2×#41C2×D10712yescomplemented
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×M16—1072no—
171C4⋊Q8C2×D10711yescharacteristic, complemented
172#127—1071no—
173#123—22no—
174C22⋊Q8C2×D10712yescomplemented
175#121—21no—
176#121—1072no—
177#121—2141no—
178#120—2142no—
179C4×D4—1071no—
180C4×D4C2×D10711yescharacteristic
181C4×D4C2×D10711yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1071no—
184C4⋊C8—1071no—
185#103—1072no—
186#102—1071no—
187#102—1071no—
188#102—1072no—
189C22⋊C8—1072no—
190C4×C8—1071no—
191C24—2142no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D10712yes—
195C2×D4—22no—
196C2×D4—1071no—
197C2×D4—1072no—
198C2×D4—4282no—
199C2×D4—4284no—
200C2×D4—2144no—
201C2×D4—2142no—
202C2×D4—2141no—
203C2×D4—2142no—
204C2×D4—2141no—
205C2×D4C22⋊D10711yescharacteristic
206C2×D4C22×D10711yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—2141no—
210C22×C4C22×D10712yes—
211SD16—4282no—
212SD16—2144no—
213SD16—2144no—
214D8—1074no—
215D8—2144no—
216M16—2144no—
217C2×C8—1071no—
218C2×C8—1071no—
219C2×C8—1072no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1071no—
224C4⋊C4C22⋊D10711yescharacteristic
225C4⋊C4C22×D10711yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—2141no—
229C22⋊C4—2142no—
230C22⋊C4C22⋊D10712yes—
231C42C22×D10711yescharacteristic
232C23—21no—
233C23—8562no—
234C23—4284no—
235C23—4284no—
236C23—2142no—
237C23—2141no—
238C23C2×C22⋊D10712yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—8561no—
245D4—1072no—
246D4—4284no—
247D4—4284no—
248D4—4282no—
249D4—2142no—
250D4—2142no—
251D4—2141no—
252D4D4×D10712yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—4282no—
260C2×C4—2141no—
261C2×C4C2×C22⋊D10711yescharacteristic
262C2×C4C2×C22⋊D10712yes—
263C2×C4C2×C22⋊D10711yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—1072no—
266C8—2142no—
267C8—2141no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—8568no—
275C22—4282no—
276C22—4282no—
277C22—4282no—
278C22—4281no—
279C22—4281no—
280C22—2142no—
281C22C22×C22⋊D10711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—4281no—
287C4#1681213412yescomplemented
288C4C2×D4×D10711yescharacteristic
289C4#1681215411yescharacteristic
290C2—42no—
291C2—42no—
292C2—8562no—
293C2—4282no—
294C2D4×C22⋊D10711yescharacteristic
295C2C2×#1681213411yescharacteristic
296C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3272
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. D4×D428C2semidirect
3. #20293001C2semidirect
4. #20292875C2semidirect
5. #20292829C2semidirect
6. #20292822C2semidirect
7. #20292777C2semidirect
8. #20292772C2semidirect
9. C4×#13133803C2semidirect
10. #20292705C2semidirect
11. #547×C107C2semidirect
16. C2×#13133805C22semidirect
17. C2×#13133805C22semidirect
19. C2×#13133803C22semidirect
20. C2×#13133802C22semidirect
26. #16812042C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. #16812033C22semidirect
29. #16812030C22semidirect
31. #16812029C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
33. C4×C107⋊C8C22semidirect
34. C2×#41×C107C22semidirect
35. C4⋊Q8×C107C22semidirect
37. C22⋊Q8×C107C22semidirect
40. C4×D4×C107C22semidirect
57. #13133803D4semidirect
95. D4×C107C2×D4semidirect
118. C107⋊C8C2×D4semidirect
137. C4×C107AGL(1,ℤ/8)semidirect
149. C107#3272semidirect
151. #547D107semidirect
161. C2×#41C2×D107semidirect
171. C4⋊Q8C2×D107semidirect
174. C22⋊Q8C2×D107semidirect
181. C4×D4C2×D107semidirect
252. D4D4×D107semidirect
287. C4#16812134semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
D4×D4282
#202930012
#202928752
#202928292
#202928222
#202927772
#202927722
C4×#131338032
#202927052
#547×C1072
#3272107