Finite Group Databasebeta

← #18006197

Subgroups — #18006197

Order 4736.

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×#11924560C212yesmaximal, complemented
2#14558187C211yesmaximal, characteristic, complemented
3D4×D148C211yesmaximal, characteristic, complemented
4#14557906C211yesmaximal, characteristic, complemented
5#14557780C212yesmaximal, complemented
6#14557738C211yesmaximal, characteristic, complemented
7#14557728C211yesmaximal, characteristic, complemented
8#14557685C212yesmaximal, complemented
9#14557679C212yesmaximal, complemented
10#14557612C211yesmaximal, characteristic, complemented
11C4×D8×C37C211yesmaximal, characteristic, complemented
12C2×#10596964C2212yescomplemented
13C22×D148—21no—
14C2×D4×D37C2212yes—
15#11924560—28no—
16C2×#10596930C2212yescomplemented
17C2×#10596930C2212yescomplemented
18C2×#10596940C2212yes—
19C2×#10596939C2212yescomplemented
20C2×#10596929C2212yescomplemented
21#11924536C2211yescharacteristic, complemented
22#11924533C2211yescharacteristic
23#11924520—22no—
24#11924516C2212yescomplemented
25#11924511C2212yes—
26#11924510—21no—
27#11924510—21no—
28#11924506—22no—
29C4×D148C2211yescharacteristic
30C2×C4⋊Dic37—21no—
31C4⋊#64436C2211yescharacteristic, complemented
32#11924459C2212yescomplemented
33#11924456C2212yescomplemented
34#11924448C2212yescomplemented
35C2×D8×C37C2211yescharacteristic
36C4×D4×C37C2211yescharacteristic
37C4×D4×C37C2211yescharacteristic, complemented
38#107×C37C2211yescharacteristic, complemented
39#102×C37C2211yescharacteristic
40#102×C37C2211yescharacteristic, complemented
41C4×C8×C37C2211yescharacteristic, complemented
42C23×D37—22no—
43C2×D4×C37C2311yescharacteristic
44C2×D4×C37C2311yescharacteristic
45C22×C4×C37—21no—
46C22×C4×C37—21no—
47C2×C22⋊D37—22no—
48C2×C22⋊D37—22no—
49C22×Dic37—22no—
50#10596964—24no—
51#10596964—42no—
52D4×D37—24no—
53D4×D37—42no—
54C2×D148—22no—
55C2×D148—21no—
56C2×D148—21no—
57C2×D148—44no—
58C2×D148D411yescharacteristic
59C2×D148C2312yes—
60C2×C4×D37C2312yes—
61C2×Dic74C2312yes—
62D8×C37D414yescomplemented
63C2×C8×C37C2311yescharacteristic
64C2×C8×C37C2311yescharacteristic
65C4⋊C4×C37C2311yescharacteristic
66C4⋊C4×C37C2311yescharacteristic
67C22⋊C4×C37—21no—
68C22⋊C4×C37—21no—
69C42×C37C2311yescharacteristic
70#10596940—24no—
71#10596939—24no—
72#10596938—22no—
73#10596938—22no—
74#10596938D412yes—
75C4⋊Dic37—21no—
76C4⋊Dic37—21no—
77C4⋊Dic37—22no—
78C4⋊Dic37D411yescharacteristic
79C2×#64436C2312yes—
80#10596930—24no—
81#10596930—42no—
82#10596929—24no—
83C23×C37—21no—
84C23×C37—21no—
85C22×D37—22no—
86C22×D37—44no—
87C22×D37—44no—
88C22×D37—82no—
89C22×D37C2×D412yes—
90D4×C37—21no—
91D4×C37—21no—
92D4×C37C2×D412yes—
93D4×C37C2×D412yescomplemented
94C2×C4×C37—21no—
95C2×C4×C37—21no—
96C2×C4×C37—42no—
97C2×C4×C37—42no—
98C2×C4×C37C2×D411yescharacteristic
99C2×C4×C37C2×D411yescharacteristic
100C2×C4×C37C2411yescharacteristic
101C22⋊D37—44no—
102C22⋊D37—44no—
103C2×Dic37—22no—
104C2×Dic37—44no—
105C2×Dic37C2×D412yes—
106D148—22no—
107D148—22no—
108D148—22no—
109D148—44no—
110D148—42no—
111D148—81no—
112C4×D37—22no—
113C4×D37—22no—
114Dic74—22no—
115Dic74—22no—
116C8×C37—21no—
117C8×C37C2×D412yescomplemented
118#64436—22no—
119C22×C37—22no—
120C22×C37—22no—
121C22×C37—41no—
122C22×C37—41no—
123C22×C37—41no—
124C22×C37—41no—
125C22×C37C22×D411yescharacteristic
126C2×D37—22no—
127C2×D37—22no—
128C2×D37—22no—
129C2×D37—42no—
130C2×D37—42no—
131C2×D37—42no—
132C2×D37—88no—
133C4×C37—21no—
134C4×C37—41no—
135C4×C37—41no—
136C4×C37AGL(1,ℤ/8)12yescomplemented
137C4×C37C22×D411yescharacteristic
138C4×C37H(5,𝔽2)11yescharacteristic
139Dic37—22no—
140Dic37—42no—
141#3272—371nomaximal, Sylow
142C2×C37—42no—
143C2×C37—42no—
144C2×C37D4211yescharacteristic
145C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C37#57711yescharacteristic
147D37—42no—
148D37—82no—
149C2×AGL(1,ℤ/8)—372no—
150#547—371no—
151D42—371no—
152#492—371no—
153#468—372no—
154#461—371no—
155#459—371no—
156#450—372no—
157#449—372no—
158C4×D8D3711yescharacteristic, complemented
159#436—371no—
160C37#327211yescharacteristic, complemented, Sylow
161C2×#41—372no—
162C22×D4—372no—
163C22×D4—741no—
164AGL(1,ℤ/8)—748no—
165C2×SD16—372no—
166C2×SD16—372no—
167C2×SD16—372no—
168C2×D8—372no—
169C2×D8C2×D3711yescharacteristic
170C2×M16—372no—
171C4⋊Q8—371no—
172#127—371no—
173#123—742no—
174C22⋊Q8—372no—
175#121—372no—
176#121—741no—
177#121—741no—
178#120—742no—
179C4×D4—371no—
180C4×D4C2×D3711yescharacteristic
181C4×D4C2×D3711yescharacteristic, complemented
182C2×C4⋊C4—741no—
183#107C2×D3711yescharacteristic, complemented
184C4⋊C8—371no—
185#103—372no—
186#102—372no—
187#102C2×D3711yescharacteristic
188#102C2×D3711yescharacteristic, complemented
189C22⋊C8—372no—
190C4×C8C2×D3711yescharacteristic, complemented
191C24—742no—
192#41—744no—
193#41—1482no—
194C2×Q8—372no—
195C2×D4—371no—
196C2×D4—372no—
197C2×D4—742no—
198C2×D4—742no—
199C2×D4—744no—
200C2×D4—742no—
201C2×D4—741no—
202C2×D4—741no—
203C2×D4—1484no—
204C2×D4—1482no—
205C2×D4C22×D3711yescharacteristic
206C2×D4C22×D3711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—372no—
210C22×C4—742no—
211SD16—744no—
212SD16—744no—
213SD16—1482no—
214D8—744no—
215D8D14814yescomplemented
216M16—744no—
217C2×C8—372no—
218C2×C8C22×D3711yescharacteristic
219C2×C8C22×D3711yescharacteristic
220C4⋊C4—371no—
221C4⋊C4—742no—
222C4⋊C4—741no—
223C4⋊C4—741no—
224C4⋊C4C22×D3711yescharacteristic
225C4⋊C4C22×D3711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—372no—
229C22⋊C4—742no—
230C22⋊C4—742no—
231C42C22×D3711yescharacteristic
232C23—21no—
233C23—21no—
234C23—372no—
235C23—2962no—
236C23—742no—
237C23—1484no—
238C23—1484no—
239Q8—742no—
240Q8—742no—
241D4—21no—
242D4—21no—
243D4—2961no—
244D4—742no—
245D4—742no—
246D4—742no—
247D4—1484no—
248D4—1484no—
249D4—1484no—
250D4—1482no—
251D4C2×D14812yes—
252D4C2×D14812yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—372no—
258C2×C4—742no—
259C2×C4—742no—
260C2×C4—742no—
261C2×C4—1484no—
262C2×C4D4×D3711yescharacteristic
263C2×C4D4×D3711yescharacteristic
264C2×C4C23×D3711yescharacteristic
265C8—21no—
266C8—742no—
267C8C2×D14812yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—2968no—
275C22—742no—
276C22—742no—
277C22—742no—
278C22—1482no—
279C22—1482no—
280C22—1482no—
281C22C2×D4×D3711yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—742no—
286C4—1482no—
287C4#1192456012yescomplemented
288C4C22×D14811yescharacteristic
289C4#1192458211yescharacteristic
290C2—42no—
291C2—42no—
292C2—2962no—
293C2—1482no—
294C2D4×D14811yescharacteristic
295C2C2×#1192456011yescharacteristic
296C2#1455834011yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC4×D8×C37C2
socleC22×C37C22×D4
hypercenterC4×D8D37
hypocenterC37#3272
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#11924560C2semidirect
2. #14558187C2semidirect
3. D4×D148C2semidirect
4. #14557906C2semidirect
5. #14557780C2semidirect
6. #14557738C2semidirect
7. #14557728C2semidirect
8. #14557685C2semidirect
9. #14557679C2semidirect
10. #14557612C2semidirect
11. C4×D8×C37C2semidirect
12. C2×#10596964C22semidirect
16. C2×#10596930C22semidirect
17. C2×#10596930C22semidirect
19. C2×#10596939C22semidirect
20. C2×#10596929C22semidirect
21. #11924536C22semidirect
24. #11924516C22semidirect
31. C4⋊#64436C22semidirect
32. #11924459C22semidirect
33. #11924456C22semidirect
34. #11924448C22semidirect
37. C4×D4×C37C22semidirect
38. #107×C37C22semidirect
40. #102×C37C22semidirect
41. C4×C8×C37C22semidirect
62. D8×C37D4semidirect
93. D4×C37C2×D4semidirect
117. C8×C37C2×D4semidirect
136. C4×C37AGL(1,ℤ/8)semidirect
158. C4×D8D37semidirect
160. C37#3272semidirect
181. C4×D4C2×D37semidirect
183. #107C2×D37semidirect
188. #102C2×D37semidirect
190. C4×C8C2×D37semidirect
215. D8D148semidirect
252. D4C2×D148semidirect
267. C8C2×D148semidirect
287. C4#11924560semidirect

Maximal subgroups

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

subgroupindex
C2×#119245602
#145581872
D4×D1482
#145579062
#145577802
#145577382
#145577282
#145576852
#145576792
#145576122
C4×D8×C372
#327237