Finite Group Databasebeta

← #19146211

Subgroups — #19146211

Order 5504.

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

normal subgroups only

This group has 556 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×#13063351C212yesmaximal, complemented
2#14583570C211yesmaximal, characteristic, complemented
3#14583275C211yesmaximal, characteristic, complemented
4#14583150C212yesmaximal, complemented
5#14583096C211yesmaximal, characteristic, complemented
6#14583094C211yesmaximal, characteristic, complemented
7#14583042C212yesmaximal, complemented
8#14583033C212yesmaximal, complemented
9C4×#10621883C211yesmaximal, characteristic, complemented
10#14582977C211yesmaximal, characteristic, complemented
11D42×C43C211yesmaximal, characteristic, complemented
12C2×#10621898C2212yescomplemented
13#13063351—28no—
14C2×#10621883C2211yescharacteristic
15C2×#10621883C2212yescomplemented
16C2×#10621882C2212yes—
17C2×#10621882C2212yescomplemented
18C2×#10621882C2212yescomplemented
19C2×#10621877C2212yescomplemented
20#13063322C2211yescharacteristic, complemented
21#13063321—21no—
22#13063299C2212yescomplemented
23#13063298—22no—
24C4×Dic86C2211yescharacteristic, complemented
25#13063287—21no—
26#13063256C2211yescharacteristic, complemented
27C4⋊(C43⋊C8)C2211yescharacteristic, complemented
28#13063248C2211yescharacteristic, complemented
29#13063246C2211yescharacteristic
30#13063246C2212yescomplemented
31#13063245C2212yescomplemented
32C22⋊(C43⋊C8)C2212yescomplemented
33C4×C43⋊C8C2211yescharacteristic, complemented
34C22×D4×C43—21no—
35C22×D4×C43C2212yes—
36#127×C43C2211yescharacteristic
37#121×C43—21no—
38#121×C43C2212yes—
39#120×C43—22no—
40C4×D4×C43C2211yescharacteristic
41C4×D4×C43C2211yescharacteristic
42C24×C43—22no—
43C2×D4×C43—22no—
44C2×D4×C43—21no—
45C2×D4×C43—21no—
46C2×D4×C43—22no—
47C2×D4×C43—24no—
48C2×D4×C43—42no—
49C2×D4×C43—44no—
50C2×D4×C43D411yescharacteristic
51C2×D4×C43C2312yes—
52C2×D4×C43C2311yescharacteristic
53C22×C4×C43—21no—
54C22×C4×C43C2312yes—
55C2×C22⋊D43—22no—
56C2×C22⋊D43—22no—
57#10621898—24no—
58#10621898—42no—
59C2×D172—21no—
60C2×D172—21no—
61C2×D172C2312yes—
62C2×C4×D43—22no—
63C2×Dic86C2311yescharacteristic
64C4⋊C4×C43D411yescharacteristic
65C4⋊C4×C43C2311yescharacteristic
66C22⋊C4×C43—21no—
67C22⋊C4×C43—22no—
68C22⋊C4×C43D412yes—
69C42×C43C2311yescharacteristic
70#10621883—24no—
71#10621883D414yescomplemented
72#10621882—24no—
73#10621882—24no—
74#10621882—42no—
75(C2×D43)⋊C4—22no—
76C4⋊Dic43C2311yescharacteristic
77Dic43⋊C4—21no—
78C4×Dic43—21no—
79#10621877—24no—
80C2×C43⋊C8C2311yescharacteristic
81C2×C43⋊C8C2312yes—
82C2×C43⋊C8C2311yescharacteristic
83C23×C43—21no—
84C23×C43—22no—
85C23×C43—44no—
86C23×C43—44no—
87C23×C43—82no—
88C23×C43C2×D412yes—
89C22×D43—22no—
90D4×C43—22no—
91D4×C43—22no—
92D4×C43—21no—
93D4×C43—22no—
94D4×C43—44no—
95D4×C43—44no—
96D4×C43—42no—
97D4×C43—81no—
98D4×C43C2×D412yes—
99C2×C4×C43—22no—
100C2×C4×C43—22no—
101C2×C4×C43—21no—
102C2×C4×C43—42no—
103C2×C4×C43C2×D411yescharacteristic
104C2×C4×C43C2×D412yes—
105C2×C4×C43C2×D411yescharacteristic
106C2×C4×C43C2411yescharacteristic
107C22⋊D43—44no—
108C22⋊D43—82no—
109C2×Dic43—21no—
110C2×Dic43—21no—
111D172—22no—
112D172—22no—
113D172—42no—
114D172—81no—
115C4×D43—44no—
116Dic86—21no—
117Dic86C2×D412yescomplemented
118C43⋊C8—22no—
119C43⋊C8—21no—
120C43⋊C8C2×D412yescomplemented
121C22×C43—22no—
122C22×C43—22no—
123C22×C43—22no—
124C22×C43—22no—
125C22×C43—41no—
126C22×C43—41no—
127C22×C43—42no—
128C22×C43—42no—
129C22×C43—42no—
130C22×C43—88no—
131C22×C43C22×D411yescharacteristic
132C2×D43—42no—
133C2×D43—42no—
134C2×D43—42no—
135C4×C43—22no—
136C4×C43—21no—
137C4×C43—41no—
138C4×C43AGL(1,ℤ/8)12yescomplemented
139C4×C43C22×D411yescharacteristic
140C4×C43H(5,𝔽2)11yescharacteristic
141Dic43—22no—
142Dic43—41no—
143#3274—431nomaximal, Sylow
144C2×C43—42no—
145C2×C43—42no—
146C2×C43—82no—
147C2×C43D4211yescharacteristic
148C2×C43C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C43#57611yescharacteristic
150D43—82no—
151C2×AGL(1,ℤ/8)—432no—
152#550—431no—
153D42D4311yescharacteristic, complemented
154#493—431no—
155#469—432no—
156#460—431no—
157#459—431no—
158#449—432no—
159#447—432no—
160C4×SD16—431no—
161#436—431no—
162C43#327411yescharacteristic, complemented, Sylow
163C2×#41—432no—
164C22×D4—21no—
165C22×D4C2×D4312yes—
166AGL(1,ℤ/8)—868no—
167C2×SD16—431no—
168C2×SD16—432no—
169C2×D8—432no—
170C2×D8—432no—
171C2×D8—432no—
172C2×M16—432no—
173#127—431no—
174#127—861no—
175#127C2×D4311yescharacteristic
176#121—21no—
177#121—432no—
178#121—862no—
179#121C2×D4312yes—
180#120—22no—
181C4×Q8—431no—
182C4×D4—861no—
183C4×D4C2×D4311yescharacteristic
184C4×D4C2×D4311yescharacteristic
185#106—431no—
186C4⋊C8—431no—
187#103—431no—
188#102—431no—
189#102—432no—
190#102—432no—
191C22⋊C8—432no—
192C4×C8—431no—
193C24—22no—
194#41—864no—
195#41—1722no—
196C2×Q8—431no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—432no—
205C2×D4—862no—
206C2×D4—862no—
207C2×D4—861no—
208C2×D4—861no—
209C2×D4C22⋊D4311yescharacteristic
210C2×D4C22×D4312yes—
211C2×D4C22×D4311yescharacteristic
212C22×C4—21no—
213C22×C4—862no—
214C22×C4C22×D4312yes—
215SD16—434no—
216SD16—864no—
217D8—864no—
218D8—864no—
219D8—1722no—
220M16—864no—
221C2×C8—431no—
222C2×C8—431no—
223C2×C8—432no—
224C4⋊C4—431no—
225C4⋊C4—861no—
226C4⋊C4C22⋊D4311yescharacteristic
227C4⋊C4C22×D4311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—862no—
231C22⋊C4C22⋊D4312yes—
232C42—861no—
233C42C22×D4311yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—862no—
240C23C2×C22⋊D4312yes—
241Q8—432no—
242Q8—861no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—862no—
252D4—862no—
253D4—3441no—
254D4—3442no—
255D4—1724no—
256D4—1722no—
257D4D4×D4312yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—861no—
263C2×C4—861no—
264C2×C4—1724no—
265C2×C4C2×C22⋊D4311yescharacteristic
266C2×C4C2×C22⋊D4312yes—
267C2×C4C2×C22⋊D4311yescharacteristic
268C2×C4C23×D4311yescharacteristic
269C8—432no—
270C8—862no—
271C8—861no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—1722no—
283C22—1722no—
284C22—1722no—
285C22C22×C22⋊D4311yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—862no—
290C4—1721no—
291C4#1306335112yescomplemented
292C4C2×D4×D4311yescharacteristic
293C4#1306337011yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—3442no—
298C2D4×C22⋊D4311yescharacteristic
299C2C2×#1306335111yescharacteristic
300C2#1458370211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingD42×C43C2
socleC22×C43C22×D4
hypercenterD42D43
hypocenterC43#3274
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#13063351C2semidirect
2. #14583570C2semidirect
3. #14583275C2semidirect
4. #14583150C2semidirect
5. #14583096C2semidirect
6. #14583094C2semidirect
7. #14583042C2semidirect
8. #14583033C2semidirect
9. C4×#10621883C2semidirect
10. #14582977C2semidirect
11. D42×C43C2semidirect
12. C2×#10621898C22semidirect
15. C2×#10621883C22semidirect
17. C2×#10621882C22semidirect
18. C2×#10621882C22semidirect
19. C2×#10621877C22semidirect
20. #13063322C22semidirect
22. #13063299C22semidirect
24. C4×Dic86C22semidirect
26. #13063256C22semidirect
27. C4⋊(C43⋊C8)C22semidirect
28. #13063248C22semidirect
30. #13063246C22semidirect
31. #13063245C22semidirect
32. C22⋊(C43⋊C8)C22semidirect
33. C4×C43⋊C8C22semidirect
71. #10621883D4semidirect
117. Dic86C2×D4semidirect
120. C43⋊C8C2×D4semidirect
138. C4×C43AGL(1,ℤ/8)semidirect
153. D42D43semidirect
162. C43#3274semidirect
291. C4#13063351semidirect