Finite Group Databasebeta

← #18006196

Subgroups — #18006196

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×#11924561C212yesmaximal, complemented
2D4×D148C211yesmaximal, characteristic, complemented
3#14557907C211yesmaximal, characteristic, complemented
4#14557781C212yesmaximal, complemented
5#14557735C211yesmaximal, characteristic, complemented
6#14557728C211yesmaximal, characteristic, complemented
7#14557683C212yesmaximal, complemented
8#14557678C212yesmaximal, complemented
9C4×#10596939C211yesmaximal, characteristic, complemented
10#14557611C211yesmaximal, characteristic, complemented
11C37×#547C211yesmaximal, characteristic, complemented
12C22×D148C2212yes—
13C2×D4×D37—21no—
14#11924561—28no—
15C2×#10596941C2212yes—
16C2×#10596941C2212yescomplemented
17C2×#10596941C2212yescomplemented
18C2×#10596939C2211yescharacteristic
19C2×#10596939C2212yescomplemented
20C2×#10596934C2212yescomplemented
21#11924533C2211yescharacteristic
22#11924511—21no—
23#11924510C2212yes—
24#11924506—22no—
25C4×D148C2211yescharacteristic
26#11924469C2211yescharacteristic, complemented
27C4⋊#64436C2211yescharacteristic, complemented
28#11924460C2212yescomplemented
29#11924457C2211yescharacteristic, complemented
30#11924456C2211yescharacteristic
31#11924456C2212yescomplemented
32C22⋊#64436C2212yescomplemented
33C4×#64436C2211yescharacteristic, complemented
34C2×#41×C37C2212yescomplemented
35C4⋊Q8×C37C2211yescharacteristic, complemented
36#123×C37—22no—
37C22⋊Q8×C37C2212yescomplemented
38#121×C37—21no—
39C4×D4×C37C2211yescharacteristic
40C4×D4×C37C2211yescharacteristic, complemented
41C2×C4⋊C4×C37—21no—
42C23×D37—22no—
43#41×C37—24no—
44#41×C37—42no—
45C2×Q8×C37C2312yes—
46C2×D4×C37—22no—
47C2×D4×C37D411yescharacteristic
48C2×D4×C37C2311yescharacteristic
49C22×C4×C37—21no—
50C22×C4×C37—22no—
51C22×C4×C37C2312yes—
52C2×C22⋊D37—22no—
53D4×D37—44no—
54C2×D148—24no—
55C2×D148—22no—
56C2×D148—21no—
57C2×D148—21no—
58C2×D148—42no—
59C2×D148C2312yes—
60C2×D148C2311yescharacteristic
61C2×C4×D37—21no—
62C4⋊C4×C37—21no—
63C4⋊C4×C37—22no—
64C4⋊C4×C37—21no—
65C4⋊C4×C37D411yescharacteristic
66C4⋊C4×C37C2311yescharacteristic
67C22⋊C4×C37—22no—
68C22⋊C4×C37—21no—
69C22⋊C4×C37D412yes—
70C42×C37C2311yescharacteristic
71#10596941—24no—
72#10596941—24no—
73#10596941—42no—
74#10596939—24no—
75#10596939D414yescomplemented
76#10596938—21no—
77#10596938—22no—
78C4⋊Dic37C2311yescharacteristic
79#10596934—24no—
80C2×#64436C2312yes—
81C2×#64436C2311yescharacteristic
82C2×#64436C2311yescharacteristic
83C23×C37—21no—
84C23×C37C2×D412yes—
85C22×D37—21no—
86C22×D37—22no—
87C22×D37—44no—
88C22×D37—44no—
89C22×D37—82no—
90Q8×C37—22no—
91Q8×C37—22no—
92D4×C37—22no—
93D4×C37—21no—
94D4×C37—44no—
95D4×C37C2×D412yescomplemented
96C2×C4×C37—22no—
97C2×C4×C37—22no—
98C2×C4×C37—21no—
99C2×C4×C37—22no—
100C2×C4×C37—44no—
101C2×C4×C37—42no—
102C2×C4×C37C2×D411yescharacteristic
103C2×C4×C37C2×D412yes—
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37C2411yescharacteristic
106C22⋊D37—44no—
107C2×Dic37—21no—
108D148—22no—
109D148—21no—
110D148—22no—
111D148—44no—
112D148—42no—
113D148—81no—
114D148C2×D412yes—
115C4×D37—42no—
116#64436—22no—
117#64436—21no—
118#64436C2×D412yescomplemented
119C22×C37—22no—
120C22×C37—22no—
121C22×C37—22no—
122C22×C37—22no—
123C22×C37—41no—
124C22×C37—41no—
125C22×C37C22×D411yescharacteristic
126C2×D37—22no—
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×C37—42no—
136C4×C37—41no—
137C4×C37AGL(1,ℤ/8)12yescomplemented
138C4×C37C22×D411yescharacteristic
139C4×C37H(5,𝔽2)11yescharacteristic
140Dic37—41no—
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#547D3711yescharacteristic, complemented
151D42—371no—
152#492—371no—
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×#41C2×D3712yescomplemented
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×M16—372no—
171C4⋊Q8C2×D3711yescharacteristic, complemented
172#127—371no—
173#123—22no—
174C22⋊Q8C2×D3712yescomplemented
175#121—21no—
176#121—372no—
177#121—741no—
178#120—742no—
179C4×D4—371no—
180C4×D4C2×D3711yescharacteristic
181C4×D4C2×D3711yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—371no—
184C4⋊C8—371no—
185#103—372no—
186#102—371no—
187#102—371no—
188#102—372no—
189C22⋊C8—372no—
190C4×C8—371no—
191C24—742no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D3712yes—
195C2×D4—22no—
196C2×D4—371no—
197C2×D4—372no—
198C2×D4—744no—
199C2×D4—742no—
200C2×D4—741no—
201C2×D4—742no—
202C2×D4—741no—
203C2×D4—1482no—
204C2×D4—1484no—
205C2×D4C22⋊D3711yescharacteristic
206C2×D4C22×D3711yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—741no—
210C22×C4C22×D3712yes—
211SD16—744no—
212SD16—744no—
213SD16—1482no—
214D8—374no—
215D8—744no—
216M16—744no—
217C2×C8—371no—
218C2×C8—371no—
219C2×C8—372no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—371no—
224C4⋊C4C22⋊D3711yescharacteristic
225C4⋊C4C22×D3711yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—741no—
229C22⋊C4—742no—
230C22⋊C4C22⋊D3712yes—
231C42C22×D3711yescharacteristic
232C23—21no—
233C23—2962no—
234C23—741no—
235C23—742no—
236C23—1484no—
237C23—1484no—
238C23C2×C22⋊D3712yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—372no—
245D4—2961no—
246D4—742no—
247D4—741no—
248D4—742no—
249D4—1484no—
250D4—1484no—
251D4—1482no—
252D4D4×D3712yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—741no—
260C2×C4—1482no—
261C2×C4C2×C22⋊D3711yescharacteristic
262C2×C4C2×C22⋊D3712yes—
263C2×C4C2×C22⋊D3711yescharacteristic
264C2×C4C23×D3711yescharacteristic
265C8—372no—
266C8—742no—
267C8—741no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—2968no—
275C22—742no—
276C22—1482no—
277C22—1482no—
278C22—1482no—
279C22—1481no—
280C22—1481no—
281C22C22×C22⋊D3711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—1481no—
287C4#1192456112yescomplemented
288C4C2×D4×D3711yescharacteristic
289C4#1192458111yescharacteristic
290C2—42no—
291C2—42no—
292C2—2962no—
293C2—1482no—
294C2D4×C22⋊D3711yescharacteristic
295C2C2×#1192456111yescharacteristic
296C2#1455834111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#547C2
socleC22×C37C22×D4
hypercenter#547D37
hypocenterC37#3272
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#11924561C2semidirect
2. D4×D148C2semidirect
3. #14557907C2semidirect
4. #14557781C2semidirect
5. #14557735C2semidirect
6. #14557728C2semidirect
7. #14557683C2semidirect
8. #14557678C2semidirect
9. C4×#10596939C2semidirect
10. #14557611C2semidirect
11. C37×#547C2semidirect
16. C2×#10596941C22semidirect
17. C2×#10596941C22semidirect
19. C2×#10596939C22semidirect
20. C2×#10596934C22semidirect
26. #11924469C22semidirect
27. C4⋊#64436C22semidirect
28. #11924460C22semidirect
29. #11924457C22semidirect
31. #11924456C22semidirect
32. C22⋊#64436C22semidirect
33. C4×#64436C22semidirect
34. C2×#41×C37C22semidirect
35. C4⋊Q8×C37C22semidirect
37. C22⋊Q8×C37C22semidirect
40. C4×D4×C37C22semidirect
75. #10596939D4semidirect
95. D4×C37C2×D4semidirect
118. #64436C2×D4semidirect
137. C4×C37AGL(1,ℤ/8)semidirect
150. #547D37semidirect
160. C37#3272semidirect
161. C2×#41C2×D37semidirect
171. C4⋊Q8C2×D37semidirect
174. C22⋊Q8C2×D37semidirect
181. C4×D4C2×D37semidirect
252. D4D4×D37semidirect
287. C4#11924561semidirect

Maximal subgroups

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

subgroupindex
C2×#119245612
D4×D1482
#145579072
#145577812
#145577352
#145577282
#145576832
#145576782
C4×#105969392
#145576112
C37×#5472
#327237