Finite Group Databasebeta

← #18006198

Subgroups — #18006198

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×#11924564C212yesmaximal, complemented
2#14558188C211yesmaximal, characteristic, complemented
3D4×D148C211yesmaximal, characteristic, complemented
4#14557906C211yesmaximal, characteristic, complemented
5#14557779C212yesmaximal, complemented
6#14557739C211yesmaximal, characteristic, complemented
7#14557729C211yesmaximal, characteristic, complemented
8#14557686C212yesmaximal, complemented
9#14557680C212yesmaximal, complemented
10C4×D296C211yesmaximal, characteristic, complemented
11C37×#436C211yesmaximal, characteristic, complemented
12C2×#10596962C2212yescomplemented
13C22×D148C2212yes—
14C2×D4×D37—21no—
15#11924564—28no—
16C2×#10596930C2212yes—
17C2×#10596930C2212yescomplemented
18C2×#10596930C2212yescomplemented
19C2×D296C2211yescharacteristic
20C2×D296C2212yescomplemented
21#11924536C2211yescharacteristic, complemented
22#11924533C2211yescharacteristic
23#11924518—22no—
24#11924515C2212yescomplemented
25#11924511—21no—
26#11924510C2212yes—
27#11924507—21no—
28#11924506—22no—
29C4×D148C2211yescharacteristic
30C4×D148C2211yescharacteristic, complemented
31C4⋊C4×D37—21no—
32#11924470C2211yescharacteristic, complemented
33#11924461C2212yescomplemented
34#11924458C2211yescharacteristic
35#11924458C2212yescomplemented
36#11924458C2211yescharacteristic, complemented
37C2×M16×C37C2212yescomplemented
38C4×D4×C37C2211yescharacteristic
39C4⋊C8×C37C2211yescharacteristic, complemented
40C22⋊C8×C37C2212yescomplemented
41C4×C8×C37C2211yescharacteristic, complemented
42C23×D37—22no—
43C2×D4×C37D411yescharacteristic
44C22×C4×C37C2312yes—
45C2×C22⋊D37—22no—
46C2×C22⋊D37—22no—
47D4×D37—44no—
48#10596962—24no—
49#10596962—42no—
50C2×D148—24no—
51C2×D148—22no—
52C2×D148—21no—
53C2×D148—21no—
54C2×D148—42no—
55C2×D148C2312yes—
56C2×D148C2311yescharacteristic
57C2×D148C2311yescharacteristic
58C2×C4×D37—21no—
59C2×C4×D37—22no—
60C2×C4×D37—21no—
61C2×Dic74C2312yes—
62M16×C37—24no—
63C2×C8×C37C2312yes—
64C2×C8×C37C2311yescharacteristic
65C2×C8×C37C2311yescharacteristic
66C4⋊C4×C37D411yescharacteristic
67C22⋊C4×C37D412yes—
68C42×C37C2311yescharacteristic
69C22⋊Dic37—22no—
70#10596938—21no—
71#10596938—21no—
72#10596938—22no—
73C4⋊Dic37—21no—
74C4⋊Dic37—21no—
75C4⋊Dic37C2311yescharacteristic
76C4⋊Dic37C2311yescharacteristic
77#10596936—22no—
78D296—24no—
79D296D414yescomplemented
80#10596930—24no—
81#10596930—24no—
82#10596930—42no—
83C23×C37C2×D412yes—
84C22×D37—21no—
85C22×D37—21no—
86C22×D37—22no—
87C22×D37—44no—
88C22×D37—44no—
89C22×D37—82no—
90D4×C37—22no—
91C2×C4×C37—22no—
92C2×C4×C37—22no—
93C2×C4×C37C2×D412yes—
94C2×C4×C37C2×D411yescharacteristic
95C2×C4×C37C2×D411yescharacteristic
96C2×C4×C37C2411yescharacteristic
97C22⋊D37—44no—
98C22⋊D37—44no—
99C2×Dic37—21no—
100C2×Dic37—22no—
101C2×Dic37—21no—
102D148—22no—
103D148—21no—
104D148—21no—
105D148—22no—
106D148—44no—
107D148—42no—
108D148—81no—
109D148C2×D412yes—
110D148C2×D412yescomplemented
111C4×D37—44no—
112C4×D37—42no—
113C4×D37—42no—
114Dic74—22no—
115Dic74—22no—
116C8×C37—22no—
117C8×C37—21no—
118C8×C37C2×D412yescomplemented
119C22×C37—22no—
120C22×C37—22no—
121C22×C37—22no—
122C22×C37C22×D411yescharacteristic
123C2×D37—22no—
124C2×D37—22no—
125C2×D37—41no—
126C2×D37—41no—
127C2×D37—41no—
128C2×D37—41no—
129C2×D37—42no—
130C2×D37—42no—
131C2×D37—42no—
132C2×D37—88no—
133C4×C37—22no—
134C4×C37—21no—
135C4×C37AGL(1,ℤ/8)12yescomplemented
136C4×C37C22×D411yescharacteristic
137C4×C37H(5,𝔽2)11yescharacteristic
138Dic37—42no—
139Dic37—41no—
140Dic37—41no—
141#3272—371nomaximal, Sylow
142C2×C37—42no—
143C2×C37D4211yescharacteristic
144C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C37#57711yescharacteristic
146D37—42no—
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×D8—371no—
159#436D3711yescharacteristic, complemented
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—371no—
169C2×D8—372no—
170C2×M16C2×D3712yescomplemented
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×D4—371no—
181C4×D4C2×D3711yescharacteristic
182C2×C4⋊C4—741no—
183#107—371no—
184C4⋊C8C2×D3711yescharacteristic, complemented
185#103—372no—
186#102—371no—
187#102—371no—
188#102—372no—
189C22⋊C8C2×D3712yescomplemented
190C4×C8C2×D3711yescharacteristic, complemented
191C24—742no—
192#41—744no—
193#41—1482no—
194C2×Q8—372no—
195C2×D4—371no—
196C2×D4—371no—
197C2×D4—372no—
198C2×D4—744no—
199C2×D4—742no—
200C2×D4—742no—
201C2×D4—741no—
202C2×D4—742no—
203C2×D4—741no—
204C2×D4—1482no—
205C2×D4—1484no—
206C2×D4D14811yescharacteristic
207C22×C4—741no—
208C22×C4—742no—
209C22×C4—741no—
210C22×C4C22×D3712yes—
211SD16—744no—
212SD16—744no—
213SD16—1482no—
214D8—374no—
215D8—744no—
216M16—24no—
217C2×C8C22×D3711yescharacteristic
218C2×C8C22×D3712yes—
219C2×C8C22×D3711yescharacteristic
220C4⋊C4—371no—
221C4⋊C4—371no—
222C4⋊C4—742no—
223C4⋊C4—741no—
224C4⋊C4—741no—
225C4⋊C4D14811yescharacteristic
226C22⋊C4—742no—
227C22⋊C4—741no—
228C22⋊C4—741no—
229C22⋊C4—742no—
230C22⋊C4D14812yes—
231C42C22×D3711yescharacteristic
232C23—2962no—
233C23—741no—
234C23—741no—
235C23—742no—
236C23—1484no—
237C23—1484no—
238C23C2×D14812yes—
239Q8—742no—
240Q8—742no—
241D4—22no—
242D4—372no—
243D4—372no—
244D4—2961no—
245D4—742no—
246D4—741no—
247D4—741no—
248D4—742no—
249D4—1484no—
250D4—1484no—
251D4—1484no—
252D4—1482no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—741no—
256C2×C4—741no—
257C2×C4—742no—
258C2×C4—1484no—
259C2×C4—1482no—
260C2×C4—1482no—
261C2×C4C2×D14811yescharacteristic
262C2×C4C2×D14812yes—
263C2×C4C2×D14811yescharacteristic
264C2×C4C23×D3711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D3712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—2968no—
272C22—742no—
273C22—742no—
274C22—1482no—
275C22—1482no—
276C22—1482no—
277C22—1481no—
278C22—1481no—
279C22—1481no—
280C22—1481no—
281C22C22×D14811yescharacteristic
282C4—21no—
283C4—22no—
284C4—1481no—
285C4—1481no—
286C4—1482no—
287C4#1192456412yescomplemented
288C4C2×D4×D3711yescharacteristic
289C4#1192458211yescharacteristic
290C2—42no—
291C2—2962no—
292C2—1482no—
293C2—1482no—
294C2D4×D14811yescharacteristic
295C2C2×#1192456411yescharacteristic
296C2#1455834511yescharacteristic
subgroupG/H
centerC22C22×D148
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#436C2
socleC22×C37C22×D4
hypercenter#436D37
hypocenterC37#3272
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#11924564C2semidirect
2. #14558188C2semidirect
3. D4×D148C2semidirect
4. #14557906C2semidirect
5. #14557779C2semidirect
6. #14557739C2semidirect
7. #14557729C2semidirect
8. #14557686C2semidirect
9. #14557680C2semidirect
10. C4×D296C2semidirect
11. C37×#436C2semidirect
12. C2×#10596962C22semidirect
17. C2×#10596930C22semidirect
18. C2×#10596930C22semidirect
20. C2×D296C22semidirect
21. #11924536C22semidirect
24. #11924515C22semidirect
30. C4×D148C22semidirect
32. #11924470C22semidirect
33. #11924461C22semidirect
35. #11924458C22semidirect
36. #11924458C22semidirect
37. C2×M16×C37C22semidirect
39. C4⋊C8×C37C22semidirect
40. C22⋊C8×C37C22semidirect
41. C4×C8×C37C22semidirect
79. D296D4semidirect
110. D148C2×D4semidirect
118. C8×C37C2×D4semidirect
135. C4×C37AGL(1,ℤ/8)semidirect
159. #436D37semidirect
160. C37#3272semidirect
170. C2×M16C2×D37semidirect
184. C4⋊C8C2×D37semidirect
189. C22⋊C8C2×D37semidirect
190. C4×C8C2×D37semidirect
267. C8D4×D37semidirect
287. C4#11924564semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
#145581882
D4×D1482
#145579062
#145577792
#145577392
#145577292
#145576862
#145576802
C4×D2962
C37×#4362
#327237