Finite Group Databasebeta

← #18004203

Subgroups — #18004203

Order 4736.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#11924555C211yesmaximal, characteristic, complemented
2C2×M16×D37C211yesmaximal, characteristic, complemented
3#14558043C212yesmaximal, complemented
4#14557792C211yesmaximal, characteristic, complemented
5#14557791C211yesmaximal, characteristic, complemented
6#14557783C211yesmaximal, characteristic, complemented
7#14557778C212yesmaximal, complemented
8#14557777C212yesmaximal, complemented
9#14557568C211yesmaximal, characteristic, complemented
10#14557526C212yesmaximal, complemented
11C37×#469C211yesmaximal, characteristic, complemented
12C2×#10596964C2212yes—
13C22×C4×D37C2211yescharacteristic
14#11924555—24no—
15C2×Dic148C2211yescharacteristic, complemented
16C2×#10596940C2212yescomplemented
17C2×#10596934C2211yescharacteristic
18C2×#10596929C2211yescharacteristic
19C2×#10596929C2211yescharacteristic
20M16×D37—24no—
21C2×C8×D37C2211yescharacteristic
22C2×C8×D37C2211yescharacteristic
23#11924518—22no—
24#11924516C2212yes—
25#11924515C2212yes—
26#11924507C2212yes—
27#11924504—22no—
28D4×Dic37—22no—
29#11924496—22no—
30#11924494C2212yes—
31#11924469C2212yescomplemented
32#11924468C2211yescharacteristic, complemented
33#11924461C2212yescomplemented
34#11924459C2212yescomplemented
35#11924457C2212yescomplemented
36C2×D8×C37C2211yescharacteristic, complemented
37C2×M16×C37C2211yescharacteristic
38#121×C37C2212yes—
39#106×C37C2211yescharacteristic, complemented
40#102×C37C2212yescomplemented
41C23×D37D411yescharacteristic
42C2×D4×C37—22no—
43C2×D4×C37C2312yes—
44C22×C4×C37C2311yescharacteristic
45C2×C22⋊D37—22no—
46C2×C22⋊D37—22no—
47C22×Dic37—22no—
48C22×Dic37D411yescharacteristic
49#10596964—24no—
50#10596964—24no—
51C2×C4×D37—22no—
52C2×C4×D37—22no—
53C2×C4×D37D411yescharacteristic
54C2×C4×D37D411yescharacteristic
55C2×C4×D37C2311yescharacteristic
56C2×C4×D37C2311yescharacteristic
57C2×Dic74C2312yes—
58D8×C37—22no—
59M16×C37—22no—
60C2×C8×C37C2311yescharacteristic
61C2×C8×C37C2311yescharacteristic
62C4⋊C4×C37C2312yes—
63C22⋊C4×C37—22no—
64C22⋊Dic37—22no—
65C22⋊Dic37—22no—
66#10596940—24no—
67#10596938—22no—
68#10596938—22no—
69C4⋊Dic37—22no—
70C4⋊Dic37C2312yes—
71#10596936—22no—
72C4×Dic37—22no—
73#10596934—22no—
74C2×#64436C2311yescharacteristic
75C2×#64436C2311yescharacteristic
76Dic148—22no—
77#10596929—22no—
78#10596929—22no—
79C8×D37—22no—
80C8×D37D414yescomplemented
81C23×C37—22no—
82C23×C37C2×D411yescharacteristic
83C22×D37—22no—
84C22×D37—42no—
85C22×D37C2×D411yescharacteristic
86C22×D37C2×D411yescharacteristic
87D4×C37—22no—
88D4×C37—22no—
89D4×C37—44no—
90C2×C4×C37—21no—
91C2×C4×C37—21no—
92C2×C4×C37—22no—
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2411yescharacteristic
95C22⋊D37—44no—
96C22⋊D37—44no—
97C2×Dic37—22no—
98C2×Dic37—22no—
99C2×Dic37—22no—
100C2×Dic37—44no—
101C2×Dic37C2×D411yescharacteristic
102C2×Dic37C2×D411yescharacteristic
103C4×D37—22no—
104C4×D37—21no—
105C4×D37—21no—
106C4×D37—22no—
107C4×D37C2×D412yes—
108C4×D37#4112yes—
109Dic74—22no—
110Dic74—22no—
111C8×C37—21no—
112C8×C37C2×D412yes—
113#64436—21no—
114#64436C2×D412yes—
115C22×C37—21no—
116C22×C37—22no—
117C22×C37—42no—
118C22×C37—42no—
119C22×C37—42no—
120C22×C37C22×D411yescharacteristic
121C2×D37—22no—
122C2×D37—21no—
123C2×D37—22no—
124C2×D37—44no—
125C2×D37#12112yes—
126C4×C37—21no—
127C4×C37—42no—
128C4×C37C22×D411yescharacteristic
129C4×C37C2×#4111yescharacteristic
130Dic37—21no—
131Dic37—42no—
132Dic37—42no—
133Dic37#12112yes—
134#3054—371nomaximal, Sylow
135C2×C37—41no—
136C2×C37—82no—
137C2×C37C2×#12111yescharacteristic
138C2×C37#57512yes—
139D37—22no—
140D37—41no—
141C2×#135—371no—
142C22×M16—371no—
143#525—372no—
144#470—371no—
145#470—371no—
146#469—371no—
147#469D3711yescharacteristic, complemented
148#468—372no—
149#468—372no—
150#427—371no—
151#416—372no—
152C37#305411yescharacteristic, complemented, Sylow
153C2×#41—372no—
154C23×C4—371no—
155#135—744no—
156C2×Q16—371no—
157C2×SD16—372no—
158C2×D8C2×D3711yescharacteristic, complemented
159C2×M16—371no—
160C2×M16—371no—
161C2×M16—371no—
162C2×M16—744no—
163C2×M16C2×D3711yescharacteristic
164C22×C8—371no—
165C22×C8—371no—
166#123—742no—
167C22⋊Q8—372no—
168C22⋊Q8—372no—
169#121—372no—
170#121C2×D3712yes—
171#120—742no—
172C4×D4—742no—
173C4×D4—742no—
174#117—372no—
175#107—372no—
176#106—371no—
177#106C2×D3711yescharacteristic, complemented
178#103—372no—
179#103—372no—
180#102—372no—
181#102C2×D3712yescomplemented
182C24—371no—
183#41—744no—
184#41—744no—
185C2×Q8—372no—
186C2×D4—22no—
187C2×D4—742no—
188C2×D4—742no—
189C2×D4C22×D3712yes—
190C22×C4—371no—
191C22×C4—371no—
192C22×C4—371no—
193C22×C4—371no—
194C22×C4—371no—
195C22×C4—742no—
196C22×C4—742no—
197C22×C4—742no—
198C22×C4C22×D3711yescharacteristic
199Q16—742no—
200SD16—744no—
201D8—22no—
202M16—22no—
203M16—742no—
204M16—742no—
205M16—742no—
206C2×C8—371no—
207C2×C8—374no—
208C2×C8—371no—
209C2×C8—742no—
210C2×C8C22×D3711yescharacteristic
211C2×C8C22×D3711yescharacteristic
212C4⋊C4—372no—
213C4⋊C4—742no—
214C4⋊C4—742no—
215C4⋊C4C22×D3712yes—
216C22⋊C4—22no—
217C22⋊C4—742no—
218C22⋊C4—742no—
219C22⋊C4—742no—
220C22⋊C4—742no—
221C42—742no—
222C23—22no—
223C23—371no—
224C23—371no—
225C23—742no—
226C23—1482no—
227C23D4×D3711yescharacteristic
228Q8—742no—
229Q8—742no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—1484no—
234D4—1484no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—371no—
239C2×C4—372no—
240C2×C4—372no—
241C2×C4—371no—
242C2×C4—742no—
243C2×C4—742no—
244C2×C4—742no—
245C2×C4—741no—
246C2×C4—741no—
247C2×C4—742no—
248C2×C4—742no—
249C2×C4—1484no—
250C2×C4D4×D3711yescharacteristic
251C2×C4C23×D3711yescharacteristic
252C8—21no—
253C8—372no—
254C8—741no—
255C8D4×D3712yes—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—372no—
262C22—742no—
263C22—741no—
264C22—742no—
265C22—1484no—
266C22C2×D4×D3711yescharacteristic
267C4—21no—
268C4—42no—
269C4—372no—
270C4—741no—
271C4—1482no—
272C4—1482no—
273C4C2×D4×D3711yescharacteristic
274C4#41×D3711yescharacteristic
275C2—41no—
276C2—82no—
277C2—742no—
278C2—1481no—
279C2#121×D3711yescharacteristic
280C2#1455832912yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#469C2
socleC22×C37C22×D4
hypercenter#469D37
hypocenterC37#3054
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#11924555C2semidirect
2. C2×M16×D37C2semidirect
3. #14558043C2semidirect
4. #14557792C2semidirect
5. #14557791C2semidirect
6. #14557783C2semidirect
7. #14557778C2semidirect
8. #14557777C2semidirect
9. #14557568C2semidirect
10. #14557526C2semidirect
11. C37×#469C2semidirect
15. C2×Dic148C22semidirect
16. C2×#10596940C22semidirect
31. #11924469C22semidirect
32. #11924468C22semidirect
33. #11924461C22semidirect
34. #11924459C22semidirect
35. #11924457C22semidirect
36. C2×D8×C37C22semidirect
39. #106×C37C22semidirect
40. #102×C37C22semidirect
80. C8×D37D4semidirect
147. #469D37semidirect
152. C37#3054semidirect
158. C2×D8C2×D37semidirect
177. #106C2×D37semidirect
181. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245552
C2×M16×D372
#145580432
#145577922
#145577912
#145577832
#145577782
#145577772
#145575682
#145575262
C37×#4692
#305437