Finite Group Databasebeta

← #18005376

Subgroups — #18005376

Order 4736.

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

normal subgroups only

This group has 420 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#14558043C211yesmaximal, characteristic
2C2×#11924516C211yesmaximal, characteristic, complemented
3#14557876C212yesmaximal, complemented
4#14557863C211yesmaximal, characteristic, complemented
5#14557775C212yesmaximal, complemented
6#14557761C212yesmaximal, complemented
7#14557696C212yesmaximal, complemented
8#14557691C211yesmaximal, characteristic, complemented
9#14557671C211yesmaximal, characteristic, complemented
10C22⋊C8×D37C211yesmaximal, characteristic, complemented
11C37×#480C211yesmaximal, characteristic, complemented
12C2×#10596964C2211yescharacteristic
13C22×Dic74C2211yescharacteristic
14C22×C4×D37C2211yescharacteristic
15C2×Dic148C2212yescomplemented
16C2×#10596940C2212yescomplemented
17C2×C8×D37C2212yes—
18#11924518—21no—
19#11924516—21no—
20#11924516—21no—
21#11924516C2212yes—
22#11924516C2212yes—
23#11924516C2211yescharacteristic
24#11924515C2211yescharacteristic
25#11924507C2211yescharacteristic
26#11924504—21no—
27D4×Dic37—21no—
28#11924496—21no—
29#11924494C2211yescharacteristic
30C2×C4⋊Dic37—21no—
31C2×#10596938—21no—
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924459C2212yescomplemented
35#11924459C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yes—
38#121×C37C2211yescharacteristic
39C2×C4⋊C4×C37C2211yescharacteristic
40#107×C37C2212yescomplemented
41#102×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C23×D37D411yescharacteristic
44C2×D4×C37—21no—
45C2×D4×C37C2311yescharacteristic
46C22×C4×C37—21no—
47C22×C4×C37C2311yescharacteristic
48C2×C22⋊D37—21no—
49C2×C22⋊D37—21no—
50C22×Dic37—21no—
51C22×Dic37—21no—
52C22×Dic37D411yescharacteristic
53#10596964—42no—
54C2×C4×D37—42no—
55C2×C4×D37D411yescharacteristic
56C2×C4×D37D411yescharacteristic
57C2×C4×D37C2311yescharacteristic
58C2×C4×D37C2311yescharacteristic
59C2×Dic74—21no—
60C2×Dic74—42no—
61C2×Dic74C2311yescharacteristic
62C2×Dic74C2312yes—
63C2×C8×C37C2312yes—
64C4⋊C4×C37—21no—
65C4⋊C4×C37C2311yescharacteristic
66C4⋊C4×C37C2312yes—
67C22⋊C4×C37—21no—
68C22⋊Dic37—21no—
69C22⋊Dic37—21no—
70#10596940—42no—
71#10596938—22no—
72#10596938—22no—
73#10596938—21no—
74#10596938—21no—
75C4⋊Dic37—22no—
76C4⋊Dic37—22no—
77C4⋊Dic37—21no—
78C4⋊Dic37C2311yescharacteristic
79#10596936—21no—
80C4×Dic37—21no—
81C2×#64436C2312yes—
82Dic148—42no—
83C8×D37—42no—
84C23×C37—21no—
85C23×C37C2×D411yescharacteristic
86C22×D37—22no—
87C22×D37—22no—
88C22×D37—41no—
89C22×D37C2×D411yescharacteristic
90C22×D37C2×D411yescharacteristic
91D4×C37—22no—
92D4×C37—42no—
93C2×C4×C37—22no—
94C2×C4×C37—21no—
95C2×C4×C37—22no—
96C2×C4×C37—42no—
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37C2411yescharacteristic
99C22⋊D37—42no—
100C22⋊D37—42no—
101C2×Dic37—21no—
102C2×Dic37—21no—
103C2×Dic37—22no—
104C2×Dic37—41no—
105C2×Dic37—42no—
106C2×Dic37—42no—
107C2×Dic37C2×D411yescharacteristic
108C2×Dic37C2×D411yescharacteristic
109C4×D37—22no—
110C4×D37—22no—
111C4×D37—41no—
112C4×D37—41no—
113Dic74—22no—
114Dic74—22no—
115Dic74—22no—
116Dic74—42no—
117C8×C37—22no—
118#64436—22no—
119C22×C37—21no—
120C22×C37—21no—
121C22×C37—41no—
122C22×C37—41no—
123C22×C37—41no—
124C22×C37#13512yes—
125C22×C37C22×D411yescharacteristic
126C2×D37—21no—
127C2×D37—21no—
128C2×D37—21no—
129C2×D37—21no—
130C2×D37—21no—
131C2×D37—44no—
132C2×D37#13712yes—
133C4×C37—21no—
134C4×C37—42no—
135C4×C37—41no—
136C4×C37H(5,𝔽2)12yes—
137Dic37—21no—
138Dic37—21no—
139Dic37—42no—
140Dic37—41no—
141Dic37—41no—
142#3190—371nomaximal, Sylow
143C2×C37—22no—
144C2×C37—81no—
145C2×C37#53411yescharacteristic
146C2×C37C2×#13511yescharacteristic
147C2×C37C2×#13711yescharacteristic
148D37—22no—
149D37—41no—
150#525—371no—
151C2×C22⋊Q8—371no—
152#485—372no—
153#483—371no—
154#480D3711yescharacteristic, complemented
155#467—372no—
156#465—372no—
157#452—372no—
158#451—371no—
159#448—371no—
160C2×C22⋊C8—371no—
161C37#319011yescharacteristic, complemented, Sylow
162C2×#41—371no—
163C22×Q8—371no—
164C23×C4—371no—
165C2×Q16—372no—
166C2×SD16—372no—
167C22×C8—372no—
168#123—741no—
169C22⋊Q8—371no—
170C22⋊Q8—372no—
171C22⋊Q8—372no—
172C22⋊Q8—371no—
173C22⋊Q8—741no—
174C22⋊Q8—741no—
175#121—371no—
176#121C2×D3711yescharacteristic
177#120—741no—
178C4×D4—741no—
179C4×D4—741no—
180#117—371no—
181C2×C4⋊C4—741no—
182C2×C4⋊C4C2×D3711yescharacteristic
183C2×C22⋊C4—741no—
184#107C2×D3712yescomplemented
185#106—372no—
186#103—372no—
187#103—372no—
188#103—372no—
189#102C2×D3712yescomplemented
190C22⋊C8—371no—
191C22⋊C8—372no—
192C22⋊C8C2×D3711yescharacteristic
193C24—371no—
194#41—1482no—
195C2×Q8—372no—
196C2×Q8—371no—
197C2×Q8—741no—
198C2×Q8—1482no—
199C2×D4—21no—
200C2×D4—741no—
201C2×D4—741no—
202C2×D4C22×D3711yescharacteristic
203C22×C4—21no—
204C22×C4—371no—
205C22×C4—371no—
206C22×C4—371no—
207C22×C4—371no—
208C22×C4—371no—
209C22×C4—741no—
210C22×C4—741no—
211C22×C4—1482no—
212C22×C4C22×D3711yescharacteristic
213Q16—1482no—
214SD16—1482no—
215C2×C8—372no—
216C2×C8—1482no—
217C2×C8C22×D3712yes—
218C4⋊C4—21no—
219C4⋊C4—371no—
220C4⋊C4—742no—
221C4⋊C4—742no—
222C4⋊C4—741no—
223C4⋊C4—741no—
224C4⋊C4C22×D3712yes—
225C4⋊C4C22×D3711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—742no—
228C22⋊C4—742no—
229C22⋊C4—741no—
230C22⋊C4—741no—
231C22⋊C4—741no—
232C22⋊C4—741no—
233C42—741no—
234C23—21no—
235C23—371no—
236C23—371no—
237C23—742no—
238C23—742no—
239C23—1481no—
240C23D4×D3711yescharacteristic
241Q8—742no—
242Q8—742no—
243Q8—742no—
244Q8—1482no—
245D4—22no—
246D4—42no—
247D4—1482no—
248D4—1482no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—371no—
254C2×C4—371no—
255C2×C4—742no—
256C2×C4—742no—
257C2×C4—741no—
258C2×C4—741no—
259C2×C4—742no—
260C2×C4—1481no—
261C2×C4—1481no—
262C2×C4—1481no—
263C2×C4—1482no—
264C2×C4—1482no—
265C2×C4D4×D3711yescharacteristic
266C2×C4C23×D3711yescharacteristic
267C8—22no—
268C8—742no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—372no—
275C22—741no—
276C22—741no—
277C22—741no—
278C22—741no—
279C22—741no—
280C22—1484no—
281C22#1192455512yes—
282C22C2×D4×D3711yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—741no—
287C4—741no—
288C4—1481no—
289C4—1481no—
290C4—1482no—
291C4#1192458212yes—
292C2—22no—
293C2—81no—
294C2—742no—
295C2—1481no—
296C2#1455809411yescharacteristic
297C2C2×#1192455511yescharacteristic
298C2#137×D3711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#480C2
socleC22×C37C22×D4
prime radical#14558043C2
hypercenter#480D37
hypocenterC37#3190
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#11924516C2semidirect
3. #14557876C2semidirect
4. #14557863C2semidirect
5. #14557775C2semidirect
6. #14557761C2semidirect
7. #14557696C2semidirect
8. #14557691C2semidirect
9. #14557671C2semidirect
10. C22⋊C8×D37C2semidirect
11. C37×#480C2semidirect
15. C2×Dic148C22semidirect
16. C2×#10596940C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924459C22semidirect
35. #11924459C22semidirect
40. #107×C37C22semidirect
41. #102×C37C22semidirect
154. #480D37semidirect
161. C37#3190semidirect
184. #107C2×D37semidirect
189. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580432
C2×#119245162
#145578762
#145578632
#145577752
#145577612
#145576962
#145576912
#145576712
C22⋊C8×D372
C37×#4802
#319037