Finite Group Databasebeta

← #28348084

Subgroups — #28348084

Order 13696.

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

normal subgroups only

This group has 556 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#20293297C211yesmaximal, characteristic, complemented
3D4×D428C211yesmaximal, characteristic, complemented
4#20293003C211yesmaximal, characteristic, complemented
5#20292879C212yesmaximal, complemented
6#20292828C211yesmaximal, characteristic, complemented
7#20292773C212yesmaximal, complemented
8#20292759C212yesmaximal, complemented
9C4×#13133806C211yesmaximal, characteristic, complemented
10#20292704C211yesmaximal, characteristic, complemented
11#459×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yescomplemented
13C22×D428—21no—
14C2×D4×D107C2212yes—
15#16812133—28no—
16C2×#13133807C2212yescomplemented
17C2×#13133806C2211yescharacteristic
18C2×#13133803C2212yescomplemented
19C2×#13133803C2212yescomplemented
20C2×#13133801C2212yescomplemented
21#16812106C2211yescharacteristic
22#16812105—21no—
23#16812084C2212yes—
24#16812083—21no—
25#16812081—22no—
26#16812080C2212yescomplemented
27#16812079—22no—
28C4×Dic214C2211yescharacteristic, complemented
29C4×D428C2211yescharacteristic
30D4×Dic107—21no—
31#16812040C2211yescharacteristic, complemented
32#16812032C2211yescharacteristic, complemented
33#16812031C2212yescomplemented
34#16812030C2211yescharacteristic
35(C2×D107)⋊C8C2212yescomplemented
36C4×C107⋊C8C2211yescharacteristic, complemented
37C2×D8×C107C2212yes—
38#127×C107C2211yescharacteristic, complemented
39C4×D4×C107C2211yescharacteristic
40C4⋊C8×C107C2211yescharacteristic, complemented
41#102×C107C2212yescomplemented
42C23×D107—22no—
43#13133817—24no—
44#13133817—42no—
45C2×Dic214C2311yescharacteristic
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—21no—
49C2×D428—44no—
50C2×D428D411yescharacteristic
51C2×D428C2312yes—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55D4×D107—24no—
56D4×D107—42no—
57C22×Dic107—22no—
58C2×C4×D107C2312yes—
59#13133807—24no—
60#13133806D414yescomplemented
61#13133803—24no—
62#13133803—42no—
63#13133801—24no—
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C4⋊Dic107D411yescharacteristic
67C4⋊Dic107C2311yescharacteristic
68Dic107⋊C4—21no—
69C22⋊Dic107—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4D412yes—
72C4×Dic107—21no—
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—22no—
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×D412yes—
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—82no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107—44no—
107C2×Dic107C2×D412yes—
108D428—22no—
109D428—22no—
110D428—22no—
111D428—44no—
112D428—42no—
113D428—81no—
114C4×D107—22no—
115C4×D107—22no—
116Dic214—21no—
117Dic214C2×D412yescomplemented
118C8×C107—22no—
119C107⋊C8—21no—
120C107⋊C8C2×D412yescomplemented
121C22×C107—22no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107C22×D411yescharacteristic
128C2×D107—22no—
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—88no—
135C4×C107—21no—
136C4×C107—41no—
137C4×C107AGL(1,ℤ/8)12yescomplemented
138C4×C107C22×D411yescharacteristic
139C4×C107H(5,𝔽2)11yescharacteristic
140Dic107—22no—
141Dic107—22no—
142Dic107—41no—
143C2×C107—42no—
144C2×C107—82no—
145C2×C107D4211yescharacteristic
146C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C107#57611yescharacteristic
148D107—42no—
149D107—82no—
150#3274—1071nomaximal, Sylow
151C107#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1072no—
153#550—1071no—
154D42—1071no—
155#493—1071no—
156#469—1072no—
157#460—1071no—
158#459D10711yescharacteristic, complemented
159#449—1072no—
160#447—1072no—
161C4×SD16—1071no—
162#436—1071no—
163C2×#41—1072no—
164C22×D4—1072no—
165C22×D4—2141no—
166AGL(1,ℤ/8)—2148no—
167C2×SD16—1071no—
168C2×SD16—1072no—
169C2×D8—1072no—
170C2×D8—1072no—
171C2×D8C2×D10712yes—
172C2×M16—1072no—
173#127—1071no—
174#127—2141no—
175#127C2×D10711yescharacteristic, complemented
176#121—1072no—
177#121—1072no—
178#121—2142no—
179#121—2141no—
180#120—2142no—
181C4×Q8—1071no—
182C4×D4—1071no—
183C4×D4—2141no—
184C4×D4C2×D10711yescharacteristic
185#106—1071no—
186C4⋊C8C2×D10711yescharacteristic, complemented
187#103—1071no—
188#102—1071no—
189#102—1072no—
190#102C2×D10712yescomplemented
191C22⋊C8—1072no—
192C4×C8—1071no—
193C24—2142no—
194#41—4282no—
195#41—2144no—
196C2×Q8—1071no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—1071no—
200C2×D4—1072no—
201C2×D4—4284no—
202C2×D4—4282no—
203C2×D4—2142no—
204C2×D4—2142no—
205C2×D4—2142no—
206C2×D4—2142no—
207C2×D4—2144no—
208C2×D4—2141no—
209C2×D4—2141no—
210C2×D4C22×D10712yes—
211C2×D4C22×D10711yescharacteristic
212C22×C4—21no—
213C22×C4—1072no—
214C22×C4—2142no—
215SD16—1074no—
216SD16—2144no—
217D8—24no—
218D8—4282no—
219D8—2144no—
220M16—2144no—
221C2×C8—1071no—
222C2×C8—1071no—
223C2×C8C22×D10712yes—
224C4⋊C4—1071no—
225C4⋊C4—1071no—
226C4⋊C4—2141no—
227C4⋊C4C22×D10711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1072no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C42—2141no—
233C42C22×D10711yescharacteristic
234C23—21no—
235C23—22no—
236C23—8562no—
237C23—1072no—
238C23—4284no—
239C23—4284no—
240C23—2142no—
241Q8—1072no—
242Q8—2141no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—8561no—
249D4—8562no—
250D4—4284no—
251D4—4284no—
252D4—4284no—
253D4—4282no—
254D4—2142no—
255D4—2142no—
256D4—2142no—
257D4D4×D10712yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—1072no—
261C2×C4—4284no—
262C2×C4—2141no—
263C2×C4—2141no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4D4×D10711yescharacteristic
267C2×C4D4×D10711yescharacteristic
268C2×C4C23×D10711yescharacteristic
269C8—22no—
270C8—1072no—
271C8—2141no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—8568no—
279C22—4282no—
280C22—4282no—
281C22—4282no—
282C22—2142no—
283C22—2142no—
284C22—2142no—
285C22C2×D4×D10711yescharacteristic
286C4—21no—
287C4—41no—
288C4—4281no—
289C4—2142no—
290C4—2142no—
291C4#1681213312yescomplemented
292C4C2×D4×D10711yescharacteristic
293C4#1681215511yescharacteristic
294C2—42no—
295C2—82no—
296C2—8562no—
297C2—4282no—
298C2#2029326111yescharacteristic
299C2C2×#1681213311yescharacteristic
300C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3274
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. #20293297C2semidirect
3. D4×D428C2semidirect
4. #20293003C2semidirect
5. #20292879C2semidirect
6. #20292828C2semidirect
7. #20292773C2semidirect
8. #20292759C2semidirect
9. C4×#13133806C2semidirect
10. #20292704C2semidirect
11. #459×C107C2semidirect
12. C2×#13133817C22semidirect
16. C2×#13133807C22semidirect
18. C2×#13133803C22semidirect
19. C2×#13133803C22semidirect
20. C2×#13133801C22semidirect
26. #16812080C22semidirect
28. C4×Dic214C22semidirect
31. #16812040C22semidirect
32. #16812032C22semidirect
33. #16812031C22semidirect
35. (C2×D107)⋊C8C22semidirect
36. C4×C107⋊C8C22semidirect
38. #127×C107C22semidirect
40. C4⋊C8×C107C22semidirect
41. #102×C107C22semidirect
60. #13133806D4semidirect
117. Dic214C2×D4semidirect
120. C107⋊C8C2×D4semidirect
137. C4×C107AGL(1,ℤ/8)semidirect
151. C107#3274semidirect
158. #459D107semidirect
175. #127C2×D107semidirect
186. C4⋊C8C2×D107semidirect
190. #102C2×D107semidirect
291. C4#16812133semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932972
D4×D4282
#202930032
#202928792
#202928282
#202927732
#202927592
C4×#131338062
#202927042
#459×C1072
#3274107