Finite Group Databasebeta

← #28348056

Subgroups — #28348056

Order 13696.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812133C212yesmaximal, complemented
2#20293281C211yesmaximal, characteristic, complemented
3#20293262C211yesmaximal, characteristic, complemented
4#20292999C211yesmaximal, characteristic, complemented
5#20292872C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292779C212yesmaximal, complemented
8#20292769C212yesmaximal, complemented
9C4×#13133803C211yesmaximal, characteristic, complemented
10#20292704C211yesmaximal, characteristic, complemented
11#459×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yescomplemented
13C2×D4×D107—21no—
14C2×D4×D107C2212yes—
15#16812133—28no—
16C2×#13133807C2212yes—
17C2×#13133806C2212yescomplemented
18C2×#13133806C2212yescomplemented
19C2×#13133803C2211yescharacteristic
20C2×#13133801C2212yescomplemented
21#16812109C2211yescharacteristic, complemented
22#16812093—22no—
23#16812089C2212yescomplemented
24#16812083—21no—
25#16812080—21no—
26#16812080C2212yes—
27#16812077—22no—
28C4×D428C2211yescharacteristic
29C4×D428C2211yescharacteristic
30C2×C4⋊Dic107—21no—
31#16812042C2211yescharacteristic, complemented
32#16812034C2212yescomplemented
33#16812030C2211yescharacteristic, complemented
34#16812029C2211yescharacteristic
35(C2×D107)⋊C8C2212yescomplemented
36C4×C107⋊C8C2211yescharacteristic, complemented
37C2×D8×C107C2212yescomplemented
38#127×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic, complemented
40C4⋊C8×C107C2211yescharacteristic, complemented
41#102×C107C2212yescomplemented
42C23×D107—22no—
43#13133817—24no—
44#13133817—42no—
45C2×Dic214C2312yes—
46C2×D428D411yescharacteristic
47C2×D428C2311yescharacteristic
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51D4×D107—24no—
52D4×D107—42no—
53D4×D107—44no—
54C22×Dic107—22no—
55C2×C4×D107—21no—
56C2×C4×D107C2312yes—
57#13133807—24no—
58#13133806—24no—
59#13133806—42no—
60#13133803D414yescomplemented
61#13133801—24no—
62C2×C107⋊C8C2311yescharacteristic
63C2×C107⋊C8C2311yescharacteristic
64C4⋊Dic107—21no—
65C4⋊Dic107—21no—
66C4⋊Dic107—22no—
67C4⋊Dic107D411yescharacteristic
68C4⋊Dic107C2311yescharacteristic
69C22⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4D412yes—
73C2×D4×C107—21no—
74C2×D4×C107—21no—
75C2×D4×C107C2312yes—
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—21no—
78D8×C107—24no—
79C2×C8×C107C2312yes—
80C4⋊C4×C107C2311yescharacteristic
81C22⋊C4×C107—21no—
82C42×C107C2311yescharacteristic
83C23×C107—21no—
84C23×C107—22no—
85C22×D107—21no—
86C22×D107—44no—
87C22×D107—44no—
88C22×D107—82no—
89C22×D107C2×D412yes—
90D4×C107—22no—
91D4×C107—21no—
92D4×C107—22no—
93D4×C107—42no—
94D4×C107—81no—
95D4×C107C2×D412yescomplemented
96C2×C4×C107—21no—
97C2×C4×C107—42no—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—44no—
102C22⋊D107—44no—
103C22⋊D107—44no—
104C2×Dic107—21no—
105C2×Dic107—22no—
106C2×Dic107—44no—
107C2×Dic107C2×D412yes—
108D428—22no—
109D428—21no—
110D428C2×D412yes—
111C4×D107—22no—
112C4×D107—22no—
113C4×D107—42no—
114Dic214—22no—
115Dic214—22no—
116C8×C107—22no—
117C107⋊C8—21no—
118C107⋊C8C2×D412yescomplemented
119C22×C107—22no—
120C22×C107—41no—
121C22×C107—41no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107C22×D411yescharacteristic
126C2×D107—22no—
127C2×D107—22no—
128C2×D107—22no—
129C2×D107—22no—
130C2×D107—41no—
131C2×D107—41no—
132C2×D107—88no—
133C4×C107—21no—
134C4×C107—41no—
135C4×C107AGL(1,ℤ/8)12yescomplemented
136C4×C107C22×D411yescharacteristic
137C4×C107H(5,𝔽2)11yescharacteristic
138Dic107—22no—
139Dic107—42no—
140Dic107—41no—
141C2×C107—42no—
142C2×C107—82no—
143C2×C107D4211yescharacteristic
144C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C107#57711yescharacteristic
146D107—42no—
147D107—42no—
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#459D10711yescharacteristic, complemented
157#450—1072no—
158#449—1072no—
159C4×D8—1071no—
160#436—1071no—
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×D8C2×D10712yescomplemented
170C2×M16—1072no—
171C4⋊Q8—1071no—
172#127C2×D10711yescharacteristic
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, complemented
182C2×C4⋊C4—2141no—
183#107—1071no—
184C4⋊C8C2×D10711yescharacteristic, complemented
185#103—1072no—
186#102—1071no—
187#102—1071no—
188#102C2×D10712yescomplemented
189C22⋊C8—1072no—
190C4×C8—1071no—
191C24—2142no—
192#41—4282no—
193#41—2144no—
194C2×Q8—1072no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—1071no—
198C2×D4—1071no—
199C2×D4—4284no—
200C2×D4—4282no—
201C2×D4—2142no—
202C2×D4—2144no—
203C2×D4—2142no—
204C2×D4—2142no—
205C2×D4C22×D10712yes—
206C2×D4C22×D10711yescharacteristic
207C22×C4—21no—
208C22×C4—1072no—
209C22×C4—2142no—
210C22×C4—2141no—
211SD16—4282no—
212SD16—2144no—
213SD16—2144no—
214D8—24no—
215D8—1074no—
216M16—2144no—
217C2×C8—1071no—
218C2×C8—1071no—
219C2×C8C22×D10712yes—
220C4⋊C4—1071no—
221C4⋊C4—1071no—
222C4⋊C4—2142no—
223C4⋊C4—2141no—
224C4⋊C4—2141no—
225C4⋊C4C22×D10711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1072no—
228C22⋊C4—2142no—
229C22⋊C4—2142no—
230C22⋊C4—2141no—
231C42C22×D10711yescharacteristic
232C23—21no—
233C23—22no—
234C23—8562no—
235C23—1072no—
236C23—4284no—
237C23—4284no—
238C23—2141no—
239Q8—2142no—
240Q8—2142no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—1072no—
247D4—4284no—
248D4—4284no—
249D4—4284no—
250D4—2142no—
251D4—2141no—
252D4D4×D10712yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—1072no—
256C2×C4—4284no—
257C2×C4—4282no—
258C2×C4—2141no—
259C2×C4—2142no—
260C2×C4—2142no—
261C2×C4—2142no—
262C2×C4D4×D10711yescharacteristic
263C2×C4D4×D10711yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—22no—
266C8—1072no—
267C8—2141no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—8568no—
275C22—4281no—
276C22—4281no—
277C22—2142no—
278C22—2142no—
279C22—2142no—
280C22—2142no—
281C22C2×D4×D10711yescharacteristic
282C4—21no—
283C4—41no—
284C4—4281no—
285C4—4282no—
286C4—2142no—
287C4#1681213312yescomplemented
288C4C2×D4×D10711yescharacteristic
289C4#1681215511yescharacteristic
290C2—42no—
291C2—82no—
292C2—4282no—
293C2—4282no—
294C2#2029326111yescharacteristic
295C2C2×#1681213311yescharacteristic
296C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3272
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293281C2semidirect
3. #20293262C2semidirect
4. #20292999C2semidirect
5. #20292872C2semidirect
6. #20292833C2semidirect
7. #20292779C2semidirect
8. #20292769C2semidirect
9. C4×#13133803C2semidirect
10. #20292704C2semidirect
11. #459×C107C2semidirect
12. C2×#13133817C22semidirect
17. C2×#13133806C22semidirect
18. C2×#13133806C22semidirect
20. C2×#13133801C22semidirect
21. #16812109C22semidirect
23. #16812089C22semidirect
31. #16812042C22semidirect
32. #16812034C22semidirect
33. #16812030C22semidirect
35. (C2×D107)⋊C8C22semidirect
36. C4×C107⋊C8C22semidirect
37. C2×D8×C107C22semidirect
39. C4×D4×C107C22semidirect
40. C4⋊C8×C107C22semidirect
41. #102×C107C22semidirect
60. #13133803D4semidirect
95. D4×C107C2×D4semidirect
118. C107⋊C8C2×D4semidirect
135. C4×C107AGL(1,ℤ/8)semidirect
149. C107#3272semidirect
156. #459D107semidirect
169. C2×D8C2×D107semidirect
181. C4×D4C2×D107semidirect
184. C4⋊C8C2×D107semidirect
188. #102C2×D107semidirect
252. D4D4×D107semidirect
287. C4#16812133semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932812
#202932622
#202929992
#202928722
#202928332
#202927792
#202927692
C4×#131338032
#202927042
#459×C1072
#3272107