Finite Group Databasebeta

← #16831114

Subgroups — #16831114

Order 3712.

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×#11745270C212yesmaximal, complemented
2#14271834C211yesmaximal, characteristic, complemented
3D4×D116C211yesmaximal, characteristic, complemented
4#14271540C211yesmaximal, characteristic, complemented
5#14271416C212yesmaximal, complemented
6#14271364C211yesmaximal, characteristic, complemented
7#14271359C211yesmaximal, characteristic, complemented
8#14271310C212yesmaximal, complemented
9#14271299C212yesmaximal, complemented
10C4×#90773C211yesmaximal, characteristic, complemented
11C29×#436C211yesmaximal, characteristic, complemented
12C2×D4×D29—21no—
13C2×#90805C2212yescomplemented
14C22×D116C2212yes—
15C2×M16×C29C2212yescomplemented
16C4×D4×C29C2211yescharacteristic
17#11745302—21no—
18#11745292C2212yes—
19#11745292C2212yescomplemented
20#11745270—28no—
21C2×D232C2212yes—
22C2×D232C2212yescomplemented
23C2×D232C2212yescomplemented
24C2×#90773C2211yescharacteristic
25C2×#90773C2212yescomplemented
26#11745257—21no—
27#11745255—21no—
28#11745246—22no—
29#11745244—22no—
30#11745236C2211yescharacteristic
31#11745236C2211yescharacteristic, complemented
32C4×D116C2211yescharacteristic
33C4×Dic58C2211yescharacteristic, complemented
34C29×C4⋊C8C2211yescharacteristic, complemented
35C29×C22⋊C8C2212yescomplemented
36C4×C8×C29C2211yescharacteristic, complemented
37#11745169C2211yescharacteristic
38#11745169C2212yescomplemented
39#11745169C2212yescomplemented
40#11745165C2211yescharacteristic, complemented
41#11745164C2211yescharacteristic, complemented
42C23×D29—22no—
43C2×D4×C29D411yescharacteristic
44C22×C4×C29C2312yes—
45C2×C22⋊D29—22no—
46C2×C22⋊D29—22no—
47C2×C22⋊D29—22no—
48D4×D29—44no—
49#90805—24no—
50#90805—42no—
51C2×D116—24no—
52C2×D116—21no—
53C2×D116—22no—
54C2×D116—21no—
55C2×D116—21no—
56C2×D116—21no—
57C2×D116—42no—
58C2×D116C2312yes—
59C2×D116C2312yes—
60C2×D116C2311yescharacteristic
61C2×C4×D29—22no—
62C2×C4×D29—21no—
63C2×Dic58C2311yescharacteristic
64M16×C29—24no—
65C2×C8×C29C2312yes—
66C2×C8×C29C2311yescharacteristic
67C2×C8×C29C2311yescharacteristic
68C4⋊C4×C29D411yescharacteristic
69C22⋊C4×C29D412yes—
70C42×C29C2311yescharacteristic
71#90781—22no—
72#90781—22no—
73#90781—21no—
74C4⋊Dic29C2311yescharacteristic
75C4⋊Dic29C2311yescharacteristic
76#90779—21no—
77C4×Dic29—21no—
78D232—24no—
79D232—24no—
80D232—42no—
81#90773—24no—
82#90773D414yescomplemented
83C23×C29C2×D412yes—
84C22×D29—21no—
85C22×D29—22no—
86C22×D29—22no—
87C22×D29—44no—
88C22×D29—44no—
89C22×D29—82no—
90D4×C29—22no—
91C2×C4×C29—22no—
92C2×C4×C29—22no—
93C2×C4×C29C2×D411yescharacteristic
94C2×C4×C29C2×D412yes—
95C2×C4×C29C2×D411yescharacteristic
96C2×C4×C29C2411yescharacteristic
97C22⋊D29—44no—
98C22⋊D29—44no—
99C22⋊D29—82no—
100C2×Dic29—21no—
101C2×Dic29—21no—
102C2×Dic29—21no—
103D116—22no—
104D116—22no—
105D116—21no—
106D116—22no—
107D116—22no—
108D116—44no—
109D116—42no—
110D116—42no—
111D116—81no—
112D116—81no—
113D116C2×D412yes—
114C4×D29—44no—
115C4×D29—42no—
116Dic58—21no—
117Dic58C2×D412yescomplemented
118C8×C29—22no—
119C8×C29—21no—
120C8×C29C2×D412yescomplemented
121#3274—291nomaximal, Sylow
122C22×C29—22no—
123C22×C29—22no—
124C22×C29—22no—
125C22×C29C22×D411yescharacteristic
126C2×D29—22no—
127C2×D29—41no—
128C2×D29—41no—
129C2×D29—42no—
130C2×D29—42no—
131C2×D29—42no—
132C2×D29—42no—
133C2×D29—42no—
134C2×D29—42no—
135C2×D29—88no—
136C4×C29—22no—
137C4×C29—21no—
138C4×C29AGL(1,ℤ/8)12yescomplemented
139C4×C29C22×D411yescharacteristic
140C4×C29H(5,𝔽2)11yescharacteristic
141Dic29—22no—
142Dic29—41no—
143Dic29—41no—
144C2×AGL(1,ℤ/8)—292no—
145#550—291no—
146D42—291no—
147#493—291no—
148#469—292no—
149#460—291no—
150#459—291no—
151#449—292no—
152#447—292no—
153C4×SD16—291no—
154#436D2911yescharacteristic, complemented
155C2×C29—42no—
156C2×C29D4211yescharacteristic
157C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C29#57611yescharacteristic
159D29—42no—
160D29—82no—
161D29—82no—
162C2×#41—292no—
163C22×D4—292no—
164C22×D4—581no—
165AGL(1,ℤ/8)—588no—
166C2×SD16—291no—
167C2×SD16—292no—
168C2×D8—292no—
169C2×D8—292no—
170C2×D8—292no—
171C2×M16C2×D2912yescomplemented
172#127—291no—
173#127—291no—
174#127—581no—
175#121—292no—
176#121—292no—
177#121—581no—
178#121—582no—
179#120—582no—
180C4×Q8—291no—
181C4×D4—291no—
182C4×D4—581no—
183C4×D4C2×D2911yescharacteristic
184#106—291no—
185C4⋊C8C2×D2911yescharacteristic, complemented
186#103—291no—
187#102—291no—
188#102—292no—
189#102—292no—
190C22⋊C8C2×D2912yescomplemented
191C4×C8C2×D2911yescharacteristic, complemented
192C29#327411yescharacteristic, complemented, Sylow
193C24—582no—
194#41—584no—
195#41—1162no—
196C2×Q8—291no—
197C2×D4—291no—
198C2×D4—292no—
199C2×D4—292no—
200C2×D4—584no—
201C2×D4—582no—
202C2×D4—582no—
203C2×D4—582no—
204C2×D4—581no—
205C2×D4—582no—
206C2×D4—581no—
207C2×D4—581no—
208C2×D4—581no—
209C2×D4—1162no—
210C2×D4—1164no—
211C2×D4D11611yescharacteristic
212C22×C4—581no—
213C22×C4—582no—
214C22×C4C22×D2912yes—
215SD16—294no—
216SD16—584no—
217D8—584no—
218D8—584no—
219D8—1162no—
220M16—24no—
221C2×C8C22×D2911yescharacteristic
222C2×C8C22×D2912yes—
223C2×C8C22×D2911yescharacteristic
224C4⋊C4—291no—
225C4⋊C4—291no—
226C4⋊C4—581no—
227C4⋊C4D11611yescharacteristic
228C22⋊C4—581no—
229C22⋊C4—582no—
230C22⋊C4—582no—
231C22⋊C4D11612yes—
232C42—581no—
233C42C22×D2911yescharacteristic
234C23—581no—
235C23—582no—
236C23—582no—
237C23—1164no—
238C23—1164no—
239C23—2322no—
240C23C2×D11612yes—
241Q8—292no—
242Q8—581no—
243D4—22no—
244D4—292no—
245D4—582no—
246D4—582no—
247D4—581no—
248D4—582no—
249D4—582no—
250D4—1164no—
251D4—1164no—
252D4—1164no—
253D4—1162no—
254D4—1162no—
255D4—2321no—
256D4—2321no—
257D4—2322no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—581no—
261C2×C4—581no—
262C2×C4—581no—
263C2×C4—1164no—
264C2×C4—1162no—
265C2×C4C2×D11611yescharacteristic
266C2×C4C2×D11612yes—
267C2×C4C2×D11611yescharacteristic
268C2×C4C23×D2911yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D2912yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—582no—
276C22—1161no—
277C22—1161no—
278C22—1162no—
279C22—1162no—
280C22—1162no—
281C22—1162no—
282C22—1162no—
283C22—1162no—
284C22—2328no—
285C22C22×D11611yescharacteristic
286C4—22no—
287C4—21no—
288C4—582no—
289C4—1161no—
290C4—1161no—
291C4#1174527012yescomplemented
292C4C2×D4×D2911yescharacteristic
293C4#1174536411yescharacteristic
294C2—42no—
295C2—1162no—
296C2—2322no—
297C2—2322no—
298C2D4×D11611yescharacteristic
299C2C2×#1174527011yescharacteristic
300C2#1427196811yescharacteristic
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#436C2
socleC22×C29C22×D4
hypercenter#436D29
hypocenterC29#3274
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#11745270C2semidirect
2. #14271834C2semidirect
3. D4×D116C2semidirect
4. #14271540C2semidirect
5. #14271416C2semidirect
6. #14271364C2semidirect
7. #14271359C2semidirect
8. #14271310C2semidirect
9. #14271299C2semidirect
10. C4×#90773C2semidirect
11. C29×#436C2semidirect
13. C2×#90805C22semidirect
15. C2×M16×C29C22semidirect
19. #11745292C22semidirect
22. C2×D232C22semidirect
23. C2×D232C22semidirect
25. C2×#90773C22semidirect
31. #11745236C22semidirect
33. C4×Dic58C22semidirect
34. C29×C4⋊C8C22semidirect
35. C29×C22⋊C8C22semidirect
36. C4×C8×C29C22semidirect
38. #11745169C22semidirect
39. #11745169C22semidirect
40. #11745165C22semidirect
41. #11745164C22semidirect
82. #90773D4semidirect
117. Dic58C2×D4semidirect
120. C8×C29C2×D4semidirect
138. C4×C29AGL(1,ℤ/8)semidirect
154. #436D29semidirect
171. C2×M16C2×D29semidirect
185. C4⋊C8C2×D29semidirect
190. C22⋊C8C2×D29semidirect
191. C4×C8C2×D29semidirect
192. C29#3274semidirect
271. C8D4×D29semidirect
291. C4#11745270semidirect

Maximal subgroups

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

subgroupindex
C2×#117452702
#142718342
D4×D1162
#142715402
#142714162
#142713642
#142713592
#142713102
#142712992
C4×#907732
C29×#4362
#327429