Finite Group Databasebeta

← #18006195

Subgroups — #18006195

Order 4736.

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

normal subgroups only

This group has 546 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×#11924563C212yesmaximal, complemented
2#14558186C211yesmaximal, characteristic, complemented
3#14558168C211yesmaximal, characteristic, complemented
4#14557776C212yesmaximal, complemented
5#14557735C211yesmaximal, characteristic, complemented
6#14557725C211yesmaximal, characteristic, complemented
7#14557682C212yesmaximal, complemented
8#14557677C212yesmaximal, complemented
9C4×D296C211yesmaximal, characteristic, complemented
10#14557612C211yesmaximal, characteristic, complemented
11C37×#492C211yesmaximal, characteristic, complemented
12C2×#10596966C2212yescomplemented
13C2×D4×D37—21no—
14C2×D4×D37C2212yes—
15#11924563—28no—
16C2×#10596941C2212yes—
17C2×D296C2211yescharacteristic
18C2×#10596939C2212yescomplemented
19C2×#10596929C2212yescomplemented
20#11924521—22no—
21#11924512C2212yescomplemented
22#11924511—21no—
23#11924507—21no—
24#11924507C2212yes—
25#11924504—22no—
26C4×D148C2211yescharacteristic
27C4×D148C2211yescharacteristic
28C4×D148C2211yescharacteristic, complemented
29C4⋊C4×D37—21no—
30#11924470C2211yescharacteristic, complemented
31C4⋊#64436C2211yescharacteristic, complemented
32#11924460C2212yescomplemented
33#11924458C2211yescharacteristic
34#11924458C2211yescharacteristic, complemented
35#11924457C2212yescomplemented
36#11924448C2212yescomplemented
37C2×SD16×C37C2212yescomplemented
38C2×SD16×C37C2212yescomplemented
39C4⋊Q8×C37C2211yescharacteristic, complemented
40#127×C37C2211yescharacteristic
41C4×C8×C37C2211yescharacteristic, complemented
42C23×D37—22no—
43C2×Q8×C37C2312yes—
44C2×D4×C37—21no—
45C2×D4×C37—21no—
46C2×D4×C37C2312yes—
47C2×C22⋊D37—22no—
48C2×C22⋊D37—22no—
49#10596966—24no—
50#10596966—42no—
51D4×D37—24no—
52D4×D37—42no—
53D4×D37—44no—
54C2×D148—22no—
55C2×D148D411yescharacteristic
56C2×D148C2311yescharacteristic
57C2×D148C2311yescharacteristic
58C2×C4×D37—21no—
59C2×C4×D37—21no—
60C2×C4×D37—22no—
61C2×C4×D37C2312yes—
62SD16×C37—24no—
63SD16×C37—42no—
64C2×C8×C37C2311yescharacteristic
65C2×C8×C37C2311yescharacteristic
66C4⋊C4×C37—21no—
67C4⋊C4×C37—21no—
68C42×C37C2311yescharacteristic
69C22⋊Dic37—22no—
70#10596941—24no—
71#10596939—24no—
72#10596938—22no—
73#10596938—21no—
74#10596938—21no—
75#10596938D412yes—
76C4⋊Dic37D411yescharacteristic
77C4⋊Dic37C2311yescharacteristic
78C4⋊Dic37C2311yescharacteristic
79#10596936—22no—
80C2×#64436C2312yes—
81D296D414yescomplemented
82#10596929—24no—
83C23×C37—22no—
84C22×D37—21no—
85C22×D37—21no—
86C22×D37—44no—
87C22×D37—44no—
88C22×D37—82no—
89C22×D37C2×D412yes—
90Q8×C37—22no—
91Q8×C37—22no—
92D4×C37—22no—
93D4×C37—22no—
94D4×C37—42no—
95D4×C37—81no—
96C2×C4×C37—22no—
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37C2×D411yescharacteristic
99C2×C4×C37C2411yescharacteristic
100C22⋊D37—44no—
101C22⋊D37—44no—
102C2×Dic37—21no—
103C2×Dic37—21no—
104C2×Dic37C2×D412yes—
105D148—22no—
106D148—21no—
107D148—21no—
108D148—44no—
109D148C2×D412yes—
110D148C2×D412yescomplemented
111C4×D37—22no—
112C4×D37—22no—
113C4×D37—44no—
114C4×D37—42no—
115C4×D37—42no—
116C8×C37—21no—
117C8×C37C2×D412yescomplemented
118#64436—22no—
119C22×C37—42no—
120C22×C37—42no—
121C22×C37—42no—
122C22×C37C22×D411yescharacteristic
123C2×D37—22no—
124C2×D37—22no—
125C2×D37—22no—
126C2×D37—22no—
127C2×D37—22no—
128C2×D37—41no—
129C2×D37—41no—
130C2×D37—41no—
131C2×D37—41no—
132C2×D37—88no—
133C4×C37—21no—
134C4×C37—42no—
135C4×C37AGL(1,ℤ/8)12yescomplemented
136C4×C37C22×D411yescharacteristic
137C4×C37H(5,𝔽2)11yescharacteristic
138Dic37—22no—
139Dic37—41no—
140Dic37—41no—
141#3272—371nomaximal, Sylow
142C2×C37—82no—
143C2×C37D4211yescharacteristic
144C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C37#57711yescharacteristic
146D37—42no—
147D37—42no—
148D37—42no—
149C2×AGL(1,ℤ/8)—372no—
150#547—371no—
151D42—371no—
152#492D3711yescharacteristic, complemented
153#468—372no—
154#461—371no—
155#459—371no—
156#450—372no—
157#449—372no—
158C4×D8—371no—
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×SD16C2×D3712yescomplemented
167C2×SD16C2×D3712yescomplemented
168C2×D8—372no—
169C2×D8—371no—
170C2×M16—372no—
171C4⋊Q8C2×D3711yescharacteristic, complemented
172#127C2×D3711yescharacteristic
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×D4—371no—
182C2×C4⋊C4—741no—
183#107—371no—
184C4⋊C8—371no—
185#103—372no—
186#102—372no—
187#102—371no—
188#102—371no—
189C22⋊C8—372no—
190C4×C8C2×D3711yescharacteristic, complemented
191C24—742no—
192#41—744no—
193#41—1482no—
194C2×Q8C22×D3712yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—371no—
198C2×D4—371no—
199C2×D4—371no—
200C2×D4—742no—
201C2×D4—742no—
202C2×D4—744no—
203C2×D4—742no—
204C2×D4—1482no—
205C2×D4—1484no—
206C2×D4C22×D3712yes—
207C22×C4—372no—
208C22×C4—742no—
209C22×C4—741no—
210C22×C4—741no—
211SD16—24no—
212SD16—42no—
213SD16—744no—
214D8—374no—
215D8—744no—
216M16—744no—
217C2×C8—372no—
218C2×C8C22×D3711yescharacteristic
219C2×C8C22×D3711yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—371no—
223C4⋊C4—371no—
224C4⋊C4—371no—
225C4⋊C4—742no—
226C22⋊C4—372no—
227C22⋊C4—742no—
228C22⋊C4—742no—
229C22⋊C4—741no—
230C22⋊C4—741no—
231C42C22×D3711yescharacteristic
232C23—22no—
233C23—372no—
234C23—2962no—
235C23—741no—
236C23—741no—
237C23—1484no—
238C23—1484no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—372no—
246D4—372no—
247D4—742no—
248D4—741no—
249D4—741no—
250D4—1484no—
251D4—1484no—
252D4—1484no—
253C2×C4—22no—
254C2×C4—372no—
255C2×C4—742no—
256C2×C4—742no—
257C2×C4—741no—
258C2×C4—741no—
259C2×C4—1484no—
260C2×C4—1482no—
261C2×C4—1482no—
262C2×C4D4×D3711yescharacteristic
263C2×C4D4×D3711yescharacteristic
264C2×C4C23×D3711yescharacteristic
265C8—21no—
266C8—742no—
267C8D4×D3712yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—2968no—
272C22—742no—
273C22—742no—
274C22—742no—
275C22—742no—
276C22—742no—
277C22—1481no—
278C22—1481no—
279C22—1481no—
280C22—1481no—
281C22C2×D4×D3711yescharacteristic
282C4—21no—
283C4—42no—
284C4—742no—
285C4—1481no—
286C4—1481no—
287C4#1192456312yescomplemented
288C4C2×D4×D3711yescharacteristic
289C4#1192458111yescharacteristic
290C2—82no—
291C2—1482no—
292C2—1482no—
293C2—1482no—
294C2#1455816811yescharacteristic
295C2C2×#1192456311yescharacteristic
296C2#1455833911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#492C2
socleC22×C37C22×D4
hypercenter#492D37
hypocenterC37#3272
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#11924563C2semidirect
2. #14558186C2semidirect
3. #14558168C2semidirect
4. #14557776C2semidirect
5. #14557735C2semidirect
6. #14557725C2semidirect
7. #14557682C2semidirect
8. #14557677C2semidirect
9. C4×D296C2semidirect
10. #14557612C2semidirect
11. C37×#492C2semidirect
12. C2×#10596966C22semidirect
18. C2×#10596939C22semidirect
19. C2×#10596929C22semidirect
21. #11924512C22semidirect
28. C4×D148C22semidirect
30. #11924470C22semidirect
31. C4⋊#64436C22semidirect
32. #11924460C22semidirect
34. #11924458C22semidirect
35. #11924457C22semidirect
36. #11924448C22semidirect
37. C2×SD16×C37C22semidirect
38. C2×SD16×C37C22semidirect
39. C4⋊Q8×C37C22semidirect
41. C4×C8×C37C22semidirect
81. D296D4semidirect
110. D148C2×D4semidirect
117. C8×C37C2×D4semidirect
135. C4×C37AGL(1,ℤ/8)semidirect
152. #492D37semidirect
160. C37#3272semidirect
166. C2×SD16C2×D37semidirect
167. C2×SD16C2×D37semidirect
171. C4⋊Q8C2×D37semidirect
190. C4×C8C2×D37semidirect
267. C8D4×D37semidirect
287. C4#11924563semidirect

Maximal subgroups

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

subgroupindex
C2×#119245632
#145581862
#145581682
#145577762
#145577352
#145577252
#145576822
#145576772
C4×D2962
#145576122
C37×#4922
#327237