Finite Group Databasebeta

← #18004541

Subgroups — #18004541

Order 4736.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D37C211yesmaximal, characteristic
2C2×#11924494C211yesmaximal, characteristic
3#14557876C212yesmaximal, complemented
4#14557872C212yesmaximal, complemented
5#14557868C211yesmaximal, characteristic, complemented
6#14557864C211yesmaximal, characteristic, complemented
7#14557861C211yesmaximal, characteristic, complemented
8#14557574C212yesmaximal, complemented
9#14557540C212yesmaximal, complemented
10#14557495C211yesmaximal, characteristic
11C37×#484C211yesmaximal, characteristic, complemented
12C2×Q8×D37C2211yescharacteristic
13C22×C4×D37C2211yescharacteristic
14C2×#10596929C2212yes—
15#11924516C2211yescharacteristic
16#11924515C2211yescharacteristic
17#11924514—21no—
18#11924513—21no—
19#11924512C2211yescharacteristic
20#11924494—21no—
21#11924494—21no—
22#11924494C2212yes—
23#11924494C2212yes—
24C2×C4⋊Dic37C2211yescharacteristic
25C4⋊C4×D37—21no—
26C4⋊C4×D37C2211yescharacteristic
27C2×#10596938—21no—
28C22⋊C4×D37—21no—
29C2×C4×Dic37—21no—
30#11924469C2212yescomplemented
31#11924468C2212yescomplemented
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924460C2212yescomplemented
35#11924459C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yes—
38C22⋊Q8×C37C2211yescharacteristic
39C2×C4⋊C4×C37C2211yescharacteristic
40#107×C37C2212yescomplemented
41#103×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C23×D37D411yescharacteristic
44C2×Q8×C37C2311yescharacteristic
45C22×C4×C37—21no—
46C22×C4×C37C2311yescharacteristic
47C22×Dic37—21no—
48C22×Dic37D411yescharacteristic
49Q8×D37—24no—
50C2×C4×D37—24no—
51C2×C4×D37—21no—
52C2×C4×D37—21no—
53C2×C4×D37D411yescharacteristic
54C2×C4×D37D411yescharacteristic
55C2×C4×D37C2311yescharacteristic
56C2×C4×D37C2311yescharacteristic
57C2×Dic74—21no—
58C2×Dic74C2311yescharacteristic
59C2×C8×C37C2312yes—
60C4⋊C4×C37—21no—
61C4⋊C4×C37—21no—
62C4⋊C4×C37C2311yescharacteristic
63C4⋊C4×C37C2312yes—
64C22⋊C4×C37—21no—
65C22⋊Dic37—21no—
66#10596938—21no—
67#10596938—21no—
68#10596938—22no—
69#10596938—22no—
70C4⋊Dic37—21no—
71C4⋊Dic37—21no—
72C4⋊Dic37C2311yescharacteristic
73C4⋊Dic37C2312yes—
74#10596936—21no—
75#10596936—21no—
76#10596936—21no—
77C4×Dic37—22no—
78C4×Dic37—22no—
79C2×#64436C2312yes—
80#10596929—24no—
81C23×C37C2×D411yescharacteristic
82C22×D37—22no—
83C22×D37—22no—
84C22×D37—41no—
85C22×D37C2×D411yescharacteristic
86C22×D37C2×D411yescharacteristic
87Q8×C37—22no—
88C2×C4×C37—22no—
89C2×C4×C37—22no—
90C2×C4×C37—21no—
91C2×C4×C37—21no—
92C2×C4×C37—42no—
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2411yescharacteristic
95C2×Dic37—22no—
96C2×Dic37—22no—
97C2×Dic37—21no—
98C2×Dic37—21no—
99C2×Dic37—42no—
100C2×Dic37C2×D411yescharacteristic
101C2×Dic37C2×D411yescharacteristic
102C4×D37—22no—
103C4×D37—22no—
104C4×D37—22no—
105C4×D37—42no—
106C4×D37—42no—
107C4×D37#4114yes—
108Dic74—22no—
109Dic74—42no—
110C8×C37—22no—
111#64436—22no—
112C22×C37—21no—
113C22×C37—21no—
114C22×C37#13712yes—
115C22×C37C22×D411yescharacteristic
116C2×D37—21no—
117C2×D37—21no—
118C2×D37—21no—
119C2×D37—21no—
120C2×D37—21no—
121C2×D37—44no—
122C2×D37#12312yes—
123C4×C37—21no—
124C4×C37—41no—
125C4×C37—41no—
126C4×C37—42no—
127C4×C37C2×#4112yes—
128Dic37—21no—
129Dic37—41no—
130Dic37—42no—
131Dic37—41no—
132Dic37#12312yes—
133#3095—371nomaximal, Sylow
134C2×C37—22no—
135C2×C37C2×#12311yescharacteristic
136C2×C37C2×#13711yescharacteristic
137C2×C37#57511yescharacteristic
138D37—22no—
139D37—41no—
140C2×C22⋊Q8—371no—
141C2×#117—371no—
142#485—372no—
143#485—372no—
144#484—371no—
145#484D3711yescharacteristic, complemented
146#483—371no—
147#483—371no—
148#428—372no—
149#419—372no—
150#407—371no—
151C37#309511yescharacteristic, complemented, Sylow
152C22×Q8—371no—
153C23×C4—371no—
154C2×M16—372no—
155C22⋊Q8—371no—
156C22⋊Q8—371no—
157C22⋊Q8—371no—
158C22⋊Q8—741no—
159C22⋊Q8—741no—
160C22⋊Q8C2×D3711yescharacteristic
161#117—372no—
162#117—372no—
163#117—741no—
164#117—741no—
165C2×C4⋊C4—371no—
166C2×C4⋊C4—371no—
167C2×C4⋊C4—741no—
168C2×C4⋊C4C2×D3711yescharacteristic
169C2×C22⋊C4—741no—
170C2×C22⋊C4—741no—
171C2×C42—741no—
172#107—372no—
173#107C2×D3712yescomplemented
174#106—372no—
175#106—372no—
176#103—372no—
177#103—372no—
178#103—372no—
179#103C2×D3712yescomplemented
180C22⋊C8—371no—
181C22⋊C8—372no—
182C22⋊C8C2×D3711yescharacteristic
183C24—371no—
184C2×Q8—371no—
185C2×Q8—741no—
186C2×Q8—744no—
187C2×Q8C22×D3711yescharacteristic
188C22×C4—21no—
189C22×C4—371no—
190C22×C4—371no—
191C22×C4—371no—
192C22×C4—371no—
193C22×C4—371no—
194C22×C4—744no—
195C22×C4—741no—
196C22×C4—741no—
197C22×C4—741no—
198C22×C4C22×D3711yescharacteristic
199M16—744no—
200C2×C8—372no—
201C2×C8C22×D3712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—372no—
205C4⋊C4—371no—
206C4⋊C4—741no—
207C4⋊C4—741no—
208C4⋊C4—741no—
209C4⋊C4—741no—
210C4⋊C4—741no—
211C4⋊C4C22×D3712yes—
212C4⋊C4C22×D3711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—742no—
215C22⋊C4—742no—
216C22⋊C4—741no—
217C22⋊C4—741no—
218C22⋊C4—741no—
219C42—742no—
220C42—742no—
221C23—371no—
222C23—371no—
223C23—742no—
224C23—742no—
225C23—1481no—
226C23D4×D3711yescharacteristic
227Q8—22no—
228Q8—742no—
229Q8—1482no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—371no—
236C2×C4—374no—
237C2×C4—371no—
238C2×C4—742no—
239C2×C4—742no—
240C2×C4—742no—
241C2×C4—742no—
242C2×C4—742no—
243C2×C4—741no—
244C2×C4—741no—
245C2×C4—1482no—
246C2×C4—1482no—
247C2×C4—1482no—
248C2×C4D4×D3711yescharacteristic
249C2×C4C23×D3711yescharacteristic
250C8—22no—
251C8—742no—
252C22—21no—
253C22—21no—
254C22—372no—
255C22—741no—
256C22—741no—
257C22—741no—
258C22—741no—
259C22—741no—
260C22—1484no—
261C22#1192456812yes—
262C22C2×D4×D3711yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—372no—
268C4—741no—
269C4—1481no—
270C4—1482no—
271C4—1481no—
272C4#41×D3712yes—
273C2—22no—
274C2—742no—
275C2—1481no—
276C2#123×D3711yescharacteristic
277C2C2×#1192456811yescharacteristic
278C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#484C2
socleC22×C37C22×D4
prime radicalC23×D37D4
hypercenter#484D37
hypocenterC37#3095
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #14557876C2semidirect
4. #14557872C2semidirect
5. #14557868C2semidirect
6. #14557864C2semidirect
7. #14557861C2semidirect
8. #14557574C2semidirect
9. #14557540C2semidirect
11. C37×#484C2semidirect
30. #11924469C22semidirect
31. #11924468C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924460C22semidirect
35. #11924459C22semidirect
40. #107×C37C22semidirect
41. #103×C37C22semidirect
145. #484D37semidirect
151. C37#3095semidirect
173. #107C2×D37semidirect
179. #103C2×D37semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D372
C2×#119244942
#145578762
#145578722
#145578682
#145578642
#145578612
#145575742
#145575402
#145574952
C37×#4842
#309537