Finite Group Databasebeta

← #470×D37 #18004194

Subgroups — #470×D37 #18004194

Order 4736.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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×Q16×D37C211yesmaximal, characteristic, complemented
2C2×M16×D37C211yesmaximal, characteristic
3C22⋊Q8×D37C212yesmaximal
4#14557792C211yesmaximal, characteristic, complemented
5#14557791C211yesmaximal, characteristic, complemented
6#14557790C212yesmaximal, complemented
7#14557789C212yesmaximal, complemented
8#14557788C211yesmaximal, characteristic, complemented
9#106×D37C211yesmaximal, characteristic, complemented
10#103×D37C212yesmaximal, complemented
11C37×#470C211yesmaximal, characteristic, complemented
12C2×Q8×D37C2212yes—
13C22×C4×D37C2211yescharacteristic
14C2×Dic148C2211yescharacteristic, complemented
15C2×#10596942C2212yescomplemented
16Q16×D37—24no—
17C2×#10596934C2211yescharacteristic
18C2×#10596929C2211yescharacteristic
19C2×#10596929C2211yescharacteristic
20M16×D37—24no—
21C2×C8×D37C2211yescharacteristic
22C2×C8×D37C2211yescharacteristic
23#11924516C2212yes—
24#11924515C2212yes—
25#11924514—22no—
26#11924513—22no—
27#11924512C2212yes—
28C4⋊C4×D37—22no—
29C4⋊C4×D37C2212yes—
30C22⋊C4×D37—22no—
31#11924468C2211yescharacteristic, complemented
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924460C2212yescomplemented
35#11924459C2212yescomplemented
36C2×Q16×C37C2211yescharacteristic, complemented
37C2×M16×C37C2211yescharacteristic
38C22⋊Q8×C37C2212yes—
39#106×C37C2211yescharacteristic, complemented
40#103×C37C2212yescomplemented
41C23×D37D411yescharacteristic
42C2×Q8×C37C2312yes—
43C22×C4×C37C2311yescharacteristic
44C22×Dic37D411yescharacteristic
45Q8×D37—24no—
46Q8×D37—24no—
47C2×C4×D37—22no—
48C2×C4×D37—22no—
49C2×C4×D37—22no—
50C2×C4×D37—22no—
51C2×C4×D37D411yescharacteristic
52C2×C4×D37D411yescharacteristic
53C2×C4×D37C2311yescharacteristic
54C2×C4×D37C2311yescharacteristic
55C2×Dic74—22no—
56C2×Dic74C2312yes—
57Q16×C37—22no—
58M16×C37—22no—
59C2×C8×C37C2311yescharacteristic
60C2×C8×C37C2311yescharacteristic
61C4⋊C4×C37—22no—
62C4⋊C4×C37C2312yes—
63C22⋊C4×C37—22no—
64C22⋊Dic37—22no—
65#10596942—24no—
66#10596938—22no—
67#10596938—22no—
68C4⋊Dic37—22no—
69C4⋊Dic37C2312yes—
70#10596936—22no—
71#10596936—22no—
72#10596936—22no—
73#10596934—22no—
74C2×#64436C2311yescharacteristic
75C2×#64436C2311yescharacteristic
76Dic148—22no—
77#10596929—22no—
78#10596929—22no—
79C8×D37—22no—
80C8×D37D414yes—
81C23×C37C2×D411yescharacteristic
82C22×D37—22no—
83C22×D37—24no—
84C22×D37C2×D411yescharacteristic
85C22×D37C2×D411yescharacteristic
86Q8×C37—22no—
87Q8×C37—22no—
88C2×C4×C37—21no—
89C2×C4×C37—21no—
90C2×C4×C37—22no—
91C2×C4×C37—22no—
92C2×C4×C37C2×D411yescharacteristic
93C2×C4×C37C2411yescharacteristic
94C2×Dic37—22no—
95C2×Dic37—22no—
96C2×Dic37—22no—
97C2×Dic37C2×D411yescharacteristic
98C2×Dic37C2×D411yescharacteristic
99C4×D37—22no—
100C4×D37—21no—
101C4×D37—21no—
102C4×D37—22no—
103C4×D37—44no—
104C4×D37—44no—
105C4×D37C2×D412yes—
106C4×D37#4112yes—
107Dic74—22no—
108Dic74—22no—
109Dic74—44no—
110C8×C37—21no—
111C8×C37C2×D412yes—
112#64436—21no—
113#64436C2×D412yes—
114C22×C37—21no—
115C22×C37—22no—
116C22×C37C22×D411yescharacteristic
117C2×D37—21no—
118C2×D37—22no—
119C2×D37—44no—
120C2×D37#12112yes—
121C2×D37#13714yes—
122C4×C37—21no—
123C4×C37—42no—
124C4×C37—42no—
125C4×C37C22×D411yescharacteristic
126C4×C37C2×#4111yescharacteristic
127Dic37—21no—
128Dic37—42no—
129Dic37—42no—
130Dic37#12112yes—
131C2×#470—371nomaximal, Sylow
132C2×C37—41no—
133C2×C37C2×#12111yescharacteristic
134C2×C37C2×#13712yes—
135D37—41no—
136D37#47014yesdirect product factor
137C22×Q16—371no—
138C22×M16—371no—
139C2×C22⋊Q8—372no—
140#470—371no—
141#470—371no—
142#470—372no—
143#470—371no—
144#470—372no—
145#470D3711yescharacteristic, direct product factor
146C2×#106—371no—
147C2×#103—372no—
148C37C2×#47011yescharacteristic, complemented, Sylow
149C22×Q8—372no—
150C23×C4—371no—
151C2×Q16—372no—
152C2×Q16—371no—
153C2×Q16—744no—
154C2×Q16C2×D3711yescharacteristic, complemented
155C2×M16—371no—
156C2×M16—371no—
157C2×M16—371no—
158C2×M16—744no—
159C2×M16C2×D3711yescharacteristic
160C22×C8—371no—
161C22×C8—371no—
162C22⋊Q8—372no—
163C22⋊Q8—372no—
164C22⋊Q8—372no—
165C22⋊Q8—742no—
166C22⋊Q8—742no—
167C22⋊Q8C2×D3712yes—
168C2×C4⋊C4—372no—
169C2×C4⋊C4—742no—
170C2×C22⋊C4—742no—
171#106—371no—
172#106—372no—
173#106C2×D3711yescharacteristic, complemented
174#103—372no—
175#103—372no—
176#103—372no—
177#103C2×D3712yescomplemented
178C24—371no—
179C2×Q8—372no—
180C2×Q8—742no—
181C2×Q8—744no—
182C2×Q8—744no—
183C2×Q8C22×D3712yes—
184C22×C4—371no—
185C22×C4—371no—
186C22×C4—371no—
187C22×C4—371no—
188C22×C4—371no—
189C22×C4—742no—
190C22×C4—742no—
191C22×C4—742no—
192C22×C4—742no—
193C22×C4C22×D3711yescharacteristic
194Q16—22no—
195Q16—744no—
196Q16—742no—
197M16—22no—
198M16—742no—
199M16—742no—
200M16—742no—
201C2×C8—371no—
202C2×C8—374no—
203C2×C8—371no—
204C2×C8—742no—
205C2×C8C22×D3711yescharacteristic
206C2×C8C22×D3711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—372no—
209C4⋊C4—742no—
210C4⋊C4—742no—
211C4⋊C4—742no—
212C4⋊C4—742no—
213C4⋊C4C22×D3712yes—
214C22⋊C4—22no—
215C22⋊C4—742no—
216C22⋊C4—742no—
217C22⋊C4—742no—
218C23—371no—
219C23—371no—
220C23—742no—
221C23—744no—
222C23D4×D3711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—742no—
226Q8—742no—
227Q8—1484no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—371no—
233C2×C4—372no—
234C2×C4—372no—
235C2×C4—371no—
236C2×C4—742no—
237C2×C4—742no—
238C2×C4—742no—
239C2×C4—741no—
240C2×C4—741no—
241C2×C4—742no—
242C2×C4—742no—
243C2×C4—1484no—
244C2×C4—1484no—
245C2×C4D4×D3711yescharacteristic
246C2×C4C23×D3711yescharacteristic
247C8—21no—
248C8—372no—
249C8—741no—
250C8D4×D3712yes—
251C22—21no—
252C22—22no—
253C22—372no—
254C22—374no—
255C22—741no—
256C22—742no—
257C22—1484no—
258C22C2×D4×D3711yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—372no—
263C4—741no—
264C4—1482no—
265C4—1482no—
266C4C2×D4×D3711yescharacteristic
267C4#41×D3711yescharacteristic
268C2—41no—
269C2—374no—
270C2—1481no—
271C2#121×D3711yescharacteristic
272C2#137×D3712yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#470C2
socleC22×C37C22×D4
prime radicalC23×D37D4
hypercenter#470D37
hypocenterC37C2×#470
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D37C2semidirect
4. #14557792C2semidirect
5. #14557791C2semidirect
6. #14557790C2semidirect
7. #14557789C2semidirect
8. #14557788C2semidirect
9. #106×D37C2semidirect
10. #103×D37C2semidirect
11. C37×#470C2semidirect
14. C2×Dic148C22semidirect
15. C2×#10596942C22semidirect
31. #11924468C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924460C22semidirect
35. #11924459C22semidirect
36. C2×Q16×C37C22semidirect
39. #106×C37C22semidirect
40. #103×C37C22semidirect
136. D37#470direct (N × H)
145. #470D37direct (N × H)
148. C37C2×#470semidirect
154. C2×Q16C2×D37semidirect
173. #106C2×D37semidirect
177. #103C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D372
C2×M16×D372
C22⋊Q8×D372
#145577922
#145577912
#145577902
#145577892
#145577882
#106×D372
#103×D372
C37×#4702
C2×#47037