Finite Group Databasebeta

← #18005364

Subgroups — #18005364

Order 4736.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#14558046C211yesmaximal, characteristic, complemented
2C2×#11924507C211yesmaximal, characteristic, complemented
3#14557866C211yesmaximal, characteristic, complemented
4#14557856C212yesmaximal, complemented
5#14557852C211yesmaximal, characteristic, complemented
6#14557768C212yesmaximal, complemented
7#14557759C212yesmaximal, complemented
8#14557677C212yesmaximal, complemented
9#14557667C211yesmaximal, characteristic, complemented
10C22⋊C8×D37C211yesmaximal, characteristic, complemented
11C37×#450C211yesmaximal, characteristic, complemented
12C2×#10596966C2211yescharacteristic, complemented
13C22×C22⋊D37—21no—
14C22×C4×D37C2211yescharacteristic
15C2×#10596939C2212yescomplemented
16C2×C8×D37C2212yescomplemented
17#11924521—21no—
18#11924512C2211yescharacteristic, complemented
19#11924511C2211yescharacteristic, complemented
20#11924510C2211yescharacteristic
21#11924507—21no—
22#11924507—21no—
23#11924507C2212yes—
24#11924507C2212yes—
25#11924506—21no—
26#11924498—21no—
27#11924496—21no—
28#11924494C2211yescharacteristic, complemented
29C2×C4⋊Dic37C2211yescharacteristic
30C2×C22⋊Dic37—21no—
31#11924469C2212yescomplemented
32#11924468C2212yescomplemented
33#11924460C2212yescomplemented
34#11924458C2212yescomplemented
35#11924457C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yescomplemented
38C22×D4×C37C2211yescharacteristic
39C2×SD16×C37C2212yescomplemented
40C22⋊Q8×C37C2211yescharacteristic
41#102×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C24×C37—21no—
44C23×D37D411yescharacteristic
45C2×Q8×C37C2311yescharacteristic
46C2×D4×C37—21no—
47C2×D4×C37—42no—
48C2×D4×C37C2312yes—
49C22×C4×C37C2311yescharacteristic
50C2×C22⋊D37—21no—
51C2×C22⋊D37—22no—
52C2×C22⋊D37—22no—
53C2×C22⋊D37—44no—
54C22×Dic37—21no—
55C22×Dic37D411yescharacteristic
56#10596966—42no—
57C2×D148—21no—
58C2×D148—21no—
59C2×D148C2311yescharacteristic
60C2×C4×D37—21no—
61C2×C4×D37—42no—
62C2×C4×D37D411yescharacteristic
63C2×C4×D37D411yescharacteristic
64C2×C4×D37C2311yescharacteristic
65C2×C4×D37C2311yescharacteristic
66SD16×C37—42no—
67C2×C8×C37C2312yes—
68C4⋊C4×C37—21no—
69C4⋊C4×C37C2311yescharacteristic
70C22⋊C4×C37—21no—
71C22⋊Dic37—22no—
72C22⋊Dic37—22no—
73#10596939—42no—
74#10596938—21no—
75#10596938—21no—
76#10596938—21no—
77#10596938—21no—
78C4⋊Dic37—21no—
79C4⋊Dic37C2312yes—
80C4⋊Dic37C2311yescharacteristic
81#10596936—21no—
82C4×Dic37—21no—
83C2×#64436C2312yes—
84C8×D37—42no—
85C23×C37—22no—
86C23×C37—42no—
87C23×C37—42no—
88C23×C37—42no—
89C23×C37C2×D411yescharacteristic
90C22×D37—22no—
91C22×D37—22no—
92C22×D37—21no—
93C22×D37—41no—
94C22×D37C2×D411yescharacteristic
95C22×D37C2×D411yescharacteristic
96Q8×C37—22no—
97D4×C37—22no—
98D4×C37—22no—
99D4×C37—42no—
100C2×C4×C37—22no—
101C2×C4×C37—21no—
102C2×C4×C37—21no—
103C2×C4×C37C2×D411yescharacteristic
104C2×C4×C37C2411yescharacteristic
105C22⋊D37—44no—
106C22⋊D37—42no—
107C22⋊D37—44no—
108C2×Dic37—21no—
109C2×Dic37—22no—
110C2×Dic37—41no—
111C2×Dic37—42no—
112C2×Dic37C2×D411yescharacteristic
113C2×Dic37C2×D411yescharacteristic
114D148—22no—
115D148—42no—
116D148—42no—
117C4×D37—22no—
118C4×D37—22no—
119C4×D37—41no—
120C4×D37—41no—
121C4×D37—42no—
122C8×C37—22no—
123#64436—22no—
124C22×C37—21no—
125C22×C37—21no—
126C22×C37—42no—
127C22×C37—42no—
128C22×C37—42no—
129C22×C37—84no—
130C22×C37#13512yes—
131C22×C37C22×D411yescharacteristic
132C2×D37—21no—
133C2×D37—21no—
134C2×D37—21no—
135C2×D37—21no—
136C2×D37—21no—
137C2×D37—44no—
138C2×D37—41no—
139C2×D37—41no—
140C2×D37—41no—
141C2×D37AGL(1,ℤ/8)12yes—
142C4×C37—21no—
143C4×C37—41no—
144C4×C37—41no—
145C4×C37H(5,𝔽2)12yes—
146Dic37—21no—
147Dic37—21no—
148Dic37—41no—
149Dic37—42no—
150#3188—371nomaximal, Sylow
151C2×C37—22no—
152C2×C37—82no—
153C2×C37#53411yescharacteristic
154C2×C37C2×#13511yescharacteristic
155C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
156D37—22no—
157D37—41no—
158D37—81no—
159#525—371no—
160C2×#121—371no—
161#484—371no—
162#482—372no—
163#481—371no—
164#466—372no—
165#465—372no—
166#450D3711yescharacteristic, complemented
167#449—372no—
168#447—371no—
169C2×C22⋊C8—371no—
170C37#318811yescharacteristic, complemented, Sylow
171C2×#41—371no—
172C22×D4—741no—
173C22×D4C2×D3711yescharacteristic
174C23×C4—371no—
175C2×SD16C2×D3712yescomplemented
176C2×D8—372no—
177C22×C8—372no—
178#123—741no—
179C22⋊Q8—371no—
180C22⋊Q8C2×D3711yescharacteristic
181#121—372no—
182#121—372no—
183#121—371no—
184#121—371no—
185#121—741no—
186#121—741no—
187#120—741no—
188C4×D4—741no—
189C4×D4—741no—
190#117—371no—
191C2×C4⋊C4—371no—
192C2×C22⋊C4—741no—
193#107—372no—
194#106—372no—
195#103—372no—
196#102—372no—
197#102—372no—
198#102C2×D3712yescomplemented
199C22⋊C8—371no—
200C22⋊C8—372no—
201C22⋊C8C2×D3711yescharacteristic
202C24—21no—
203C24—371no—
204#41—1482no—
205C2×Q8C22×D3711yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—371no—
209C2×D4—741no—
210C2×D4—741no—
211C2×D4—742no—
212C2×D4—742no—
213C2×D4—741no—
214C2×D4—1484no—
215C2×D4C22×D3712yes—
216C22×C4—371no—
217C22×C4—371no—
218C22×C4—371no—
219C22×C4—371no—
220C22×C4—371no—
221C22×C4—741no—
222C22×C4—741no—
223C22×C4—1482no—
224C22×C4C22×D3711yescharacteristic
225SD16—42no—
226D8—1482no—
227C2×C8—372no—
228C2×C8—1482no—
229C2×C8C22×D3712yes—
230C4⋊C4—21no—
231C4⋊C4—371no—
232C4⋊C4—372no—
233C4⋊C4—741no—
234C4⋊C4—741no—
235C4⋊C4C22×D3711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—741no—
238C22⋊C4—741no—
239C22⋊C4—741no—
240C22⋊C4—742no—
241C22⋊C4—742no—
242C22⋊C4—741no—
243C42—741no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—371no—
249C23—371no—
250C23—742no—
251C23—742no—
252C23—741no—
253C23—1481no—
254C23D4×D3711yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—742no—
260D4—1482no—
261D4—1482no—
262D4—1484no—
263D4—1484no—
264D4—1482no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—371no—
269C2×C4—371no—
270C2×C4—742no—
271C2×C4—742no—
272C2×C4—741no—
273C2×C4—742no—
274C2×C4—1481no—
275C2×C4—1481no—
276C2×C4—1482no—
277C2×C4—1481no—
278C2×C4—1482no—
279C2×C4D4×D3711yescharacteristic
280C2×C4C23×D3711yescharacteristic
281C8—22no—
282C8—742no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—372no—
290C22—741no—
291C22—741no—
292C22—741no—
293C22—741no—
294C22—741no—
295C22—1484no—
296C22—1481no—
297C22—1481no—
298C22—1481no—
299C22#1192455712yes—
300C22C2×D4×D3711yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—741no—
305C4—741no—
306C4—1481no—
307C4—1482no—
308C4#1192458112yes—
309C2—22no—
310C2—82no—
311C2—2961no—
312C2—742no—
313C2—1481no—
314C2#1455809311yescharacteristic
315C2C2×#1192455711yescharacteristic
316C2AGL(1,ℤ/8)×D3711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#450C2
socleC22×C37C22×D4
hypercenter#450D37
hypocenterC37#3188
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #14558046C2semidirect
2. C2×#11924507C2semidirect
3. #14557866C2semidirect
4. #14557856C2semidirect
5. #14557852C2semidirect
6. #14557768C2semidirect
7. #14557759C2semidirect
8. #14557677C2semidirect
9. #14557667C2semidirect
10. C22⋊C8×D37C2semidirect
11. C37×#450C2semidirect
12. C2×#10596966C22semidirect
15. C2×#10596939C22semidirect
16. C2×C8×D37C22semidirect
18. #11924512C22semidirect
19. #11924511C22semidirect
28. #11924494C22semidirect
31. #11924469C22semidirect
32. #11924468C22semidirect
33. #11924460C22semidirect
34. #11924458C22semidirect
35. #11924457C22semidirect
37. #11924448C22semidirect
39. C2×SD16×C37C22semidirect
41. #102×C37C22semidirect
166. #450D37semidirect
170. C37#3188semidirect
175. C2×SD16C2×D37semidirect
198. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580462
C2×#119245072
#145578662
#145578562
#145578522
#145577682
#145577592
#145576772
#145576672
C22⋊C8×D372
C37×#4502
#318837