Finite Group Databasebeta

← #18006422

Subgroups — #18006422

Order 4736.

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

normal subgroups only

This group has 462 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
1#14558243C211yesmaximal, characteristic, complemented
2D4×D148C211yesmaximal, characteristic
3#14557880C211yesmaximal, characteristic, complemented
4#14557853C212yesmaximal, complemented
5#14557823C211yesmaximal, characteristic, complemented
6#14557765C212yesmaximal, complemented
7#14557733C211yesmaximal, characteristic, complemented
8#14557682C212yesmaximal, complemented
9C8×D148C211yesmaximal, characteristic, complemented
10#102×D37C212yesmaximal, complemented
11C4×SD16×C37C211yesmaximal, characteristic, complemented
12C22×D148—21no—
13C2×D4×D37C2212yes—
14C2×#10596941C2212yescomplemented
15C2×C8×D37C2212yescomplemented
16#11924536C2211yescharacteristic, complemented
17#11924533C2211yescharacteristic
18#11924528—21no—
19#11924514—22no—
20#11924512C2212yescomplemented
21#11924511C2212yes—
22#11924510—21no—
23#11924506—22no—
24C4×D148—21no—
25C4×D148C2211yescharacteristic
26C4⋊C4×D37C2212yescomplemented
27#11924467C2212yescomplemented
28C4⋊#64436C2211yescharacteristic, complemented
29#11924458C2212yescomplemented
30#11924458C2212yescomplemented
31#11924457C2212yes—
32#11924456C2212yescomplemented
33#11924448C2212yescomplemented
34C2×SD16×C37C2211yescharacteristic
35C4×Q8×C37C2211yescharacteristic, complemented
36C4×D4×C37C2211yescharacteristic
37#106×C37C2211yescharacteristic, complemented
38#103×C37C2211yescharacteristic, complemented
39#102×C37C2211yescharacteristic
40C4×C8×C37C2211yescharacteristic, complemented
41C23×D37—22no—
42C2×Q8×C37C2311yescharacteristic
43C2×D4×C37C2311yescharacteristic
44C22×C4×C37—21no—
45C2×C22⋊D37—22no—
46D4×D37—24no—
47D4×D37—42no—
48C2×D148—22no—
49C2×D148—21no—
50C2×D148—21no—
51C2×D148—44no—
52C2×D148D411yescharacteristic
53C2×D148C2312yes—
54C2×C4×D37—22no—
55C2×C4×D37C2312yes—
56C2×Dic74—21no—
57SD16×C37—22no—
58C2×C8×C37C2311yescharacteristic
59C2×C8×C37C2311yescharacteristic
60C4⋊C4×C37—21no—
61C4⋊C4×C37C2311yescharacteristic
62C4⋊C4×C37C2311yescharacteristic
63C22⋊C4×C37—21no—
64C42×C37—21no—
65C42×C37C2311yescharacteristic
66#10596941—42no—
67#10596938—22no—
68#10596938—22no—
69#10596938D412yes—
70C4⋊Dic37—21no—
71C4⋊Dic37D411yescharacteristic
72C4⋊Dic37C2312yes—
73#10596936—22no—
74#10596936—22no—
75C2×#64436C2312yes—
76C8×D37—42no—
77C23×C37—21no—
78C22×D37—22no—
79C22×D37—44no—
80C22×D37—44no—
81C22×D37—82no—
82C22×D37C2×D412yes—
83Q8×C37—21no—
84Q8×C37—21no—
85D4×C37—21no—
86D4×C37#4112yes—
87C2×C4×C37—21no—
88C2×C4×C37—21no—
89C2×C4×C37—21no—
90C2×C4×C37—42no—
91C2×C4×C37C2×D411yescharacteristic
92C2×C4×C37C2×D411yescharacteristic
93C2×C4×C37C2411yescharacteristic
94C22⋊D37—44no—
95C2×Dic37—22no—
96C2×Dic37C2×D412yes—
97D148—22no—
98D148—22no—
99D148—44no—
100D148—42no—
101D148—81no—
102D148SD1614yescomplemented
103C4×D37—22no—
104C4×D37—22no—
105C4×D37—82no—
106Dic74—81no—
107C8×C37—21no—
108C8×C37—21no—
109#64436—22no—
110C22×C37—22no—
111C22×C37—41no—
112C22×C37—41no—
113C22×C37C22×D411yescharacteristic
114C2×D37—22no—
115C2×D37—22no—
116C2×D37—42no—
117C2×D37—42no—
118C2×D37—42no—
119C2×D37—88no—
120C2×D37C2×SD1614yes—
121C4×C37—21no—
122C4×C37—41no—
123C4×C37—41no—
124C4×C37—41no—
125C4×C37C2×SD1612yes—
126C4×C37C2×#4111yescharacteristic
127C4×C37H(5,𝔽2)11yescharacteristic
128Dic37—22no—
129Dic37—42no—
130#3295—371nomaximal, Sylow
131C2×C37—42no—
132C2×C37#54611yescharacteristic
133C2×C37C22×SD1611yescharacteristic
134C2×C37#57611yescharacteristic
135D37—24no—
136D37—82no—
137#554—371no—
138D42—371no—
139#486—371no—
140#481—372no—
141#476—371no—
142#465—372no—
143#460—371no—
144#450—372no—
145C4×SD16D3711yescharacteristic, complemented
146C8×D4—371no—
147C2×#102—372no—
148C37#329511yescharacteristic, complemented, Sylow
149C22×D4—372no—
150C22×D4—741no—
151C2×SD16—372no—
152C2×SD16C2×D3711yescharacteristic
153C22×C8—372no—
154C4⋊Q8—371no—
155#127—371no—
156#125—741no—
157C22⋊Q8—372no—
158C22⋊Q8—742no—
159#121—372no—
160#121—741no—
161#120—742no—
162C4×Q8C2×D3711yescharacteristic, complemented
163C4×D4—371no—
164C4×D4—741no—
165C4×D4C2×D3711yescharacteristic
166C2×C4⋊C4—372no—
167#106—372no—
168#106C2×D3711yescharacteristic, complemented
169C4⋊C8—371no—
170#103C2×D3711yescharacteristic, complemented
171#102—372no—
172#102—372no—
173#102—372no—
174#102—372no—
175#102C2×D3711yescharacteristic
176C22⋊C8—372no—
177C4×C8C2×D3711yescharacteristic, complemented
178C24—742no—
179C2×Q8—741no—
180C2×Q8C22×D3711yescharacteristic
181C2×D4—371no—
182C2×D4—372no—
183C2×D4—742no—
184C2×D4—742no—
185C2×D4—744no—
186C2×D4—741no—
187C2×D4—741no—
188C2×D4—1484no—
189C2×D4—1482no—
190C2×D4C22×D3711yescharacteristic
191C22×C4—21no—
192C22×C4—372no—
193C22×C4—742no—
194SD16—22no—
195SD16—1482no—
196C2×C8—372no—
197C2×C8—1482no—
198C2×C8C22×D3711yescharacteristic
199C2×C8C22×D3711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—371no—
202C4⋊C4—372no—
203C4⋊C4—742no—
204C4⋊C4—742no—
205C4⋊C4—741no—
206C4⋊C4C22×D3711yescharacteristic
207C4⋊C4C22×D3711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—372no—
210C22⋊C4—742no—
211C22⋊C4—742no—
212C42—21no—
213C42C22×D3711yescharacteristic
214C23—21no—
215C23—372no—
216C23—2962no—
217C23—742no—
218C23—1484no—
219C23—1484no—
220Q8—21no—
221Q8—21no—
222Q8—2961no—
223D4—21no—
224D4—374no—
225D4—2961no—
226D4—742no—
227D4—742no—
228D4—1484no—
229D4—1484no—
230D4—1482no—
231D4#1059696212yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—372no—
237C2×C4—2962no—
238C2×C4—742no—
239C2×C4—742no—
240C2×C4—742no—
241C2×C4D4×D3711yescharacteristic
242C2×C4D4×D3711yescharacteristic
243C2×C4C23×D3711yescharacteristic
244C8—21no—
245C8—21no—
246C8—742no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—374no—
251C22—2968no—
252C22—742no—
253C22—742no—
254C22—1482no—
255C22—1482no—
256C22—1482no—
257C22C2×D4×D3711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—742no—
263C4—1482no—
264C4SD16×D3712yes—
265C4C2×#1059696211yescharacteristic
266C4#1192458211yescharacteristic
267C2—42no—
268C2—2962no—
269C2—744no—
270C2#1455817911yescharacteristic
271C2C2×SD16×D3711yescharacteristic
272C2#1455833511yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC4×SD16×C37C2
socleC22×C37C22×D4
prime radicalD4×D148C2
hypercenterC4×SD16D37
hypocenterC37#3295
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #14558243C2semidirect
3. #14557880C2semidirect
4. #14557853C2semidirect
5. #14557823C2semidirect
6. #14557765C2semidirect
7. #14557733C2semidirect
8. #14557682C2semidirect
9. C8×D148C2semidirect
10. #102×D37C2semidirect
11. C4×SD16×C37C2semidirect
14. C2×#10596941C22semidirect
15. C2×C8×D37C22semidirect
16. #11924536C22semidirect
20. #11924512C22semidirect
26. C4⋊C4×D37C22semidirect
27. #11924467C22semidirect
28. C4⋊#64436C22semidirect
29. #11924458C22semidirect
30. #11924458C22semidirect
32. #11924456C22semidirect
33. #11924448C22semidirect
35. C4×Q8×C37C22semidirect
37. #106×C37C22semidirect
38. #103×C37C22semidirect
40. C4×C8×C37C22semidirect
102. D148SD16semidirect
145. C4×SD16D37semidirect
148. C37#3295semidirect
162. C4×Q8C2×D37semidirect
168. #106C2×D37semidirect
170. #103C2×D37semidirect
177. C4×C8C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145582432
D4×D1482
#145578802
#145578532
#145578232
#145577652
#145577332
#145576822
C8×D1482
#102×D372
C4×SD16×C372
#329537