Finite Group Databasebeta

← #18004546

Subgroups — #18004546

Order 4736.

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

normal subgroups only

This group has 410 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×#11924516C211yesmaximal, characteristic
2#117×D37C211yesmaximal, characteristic
3#14557877C212yesmaximal, complemented
4#14557876C212yesmaximal, complemented
5#14557870C212yesmaximal, complemented
6#14557865C212yesmaximal, complemented
7#14557573C211yesmaximal, characteristic, complemented
8#14557543C211yesmaximal, characteristic, complemented
9#14557541C211yesmaximal, characteristic, complemented
10#14557496C211yesmaximal, characteristic
11C37×#428C211yesmaximal, characteristic, complemented
12C22×Dic74C2211yescharacteristic
13C22×C4×D37C2211yescharacteristic
14C2×#10596934C2211yescharacteristic
15#11924516—21no—
16#11924516—21no—
17#11924516C2212yes—
18#11924516C2212yes—
19#11924495C2211yescharacteristic
20#11924494C2211yescharacteristic
21#11924494C2211yescharacteristic
22#11924493—21no—
23#11924492—21no—
24C2×C4⋊Dic37—21no—
25C4⋊C4×D37C2211yescharacteristic
26C4⋊C4×D37C2211yescharacteristic
27C2×#10596938—21no—
28C22⋊C4×D37—21no—
29C42×D37—21no—
30#11924469C2212yescomplemented
31#11924467C2212yescomplemented
32#11924461C2212yescomplemented
33#11924461C2212yescomplemented
34#11924459C2212yescomplemented
35#11924459C2212yescomplemented
36#11924448C2212yes—
37#11924448C2212yes—
38C2×M16×C37C2211yescharacteristic
39#117×C37C2211yescharacteristic
40C2×C4⋊C4×C37C2211yescharacteristic
41#107×C37C2212yescomplemented
42#106×C37C2212yescomplemented
43C23×D37D411yescharacteristic
44C22×C4×C37—21no—
45C22×C4×C37C2311yescharacteristic
46C22×Dic37—21no—
47C22×Dic37D411yescharacteristic
48C2×C4×D37—22no—
49C2×C4×D37—22no—
50C2×C4×D37—21no—
51C2×C4×D37—21no—
52C2×C4×D37D411yescharacteristic
53C2×C4×D37D411yescharacteristic
54C2×C4×D37C2311yescharacteristic
55C2×C4×D37C2311yescharacteristic
56C2×Dic74—22no—
57C2×Dic74—22no—
58C2×Dic74—21no—
59C2×Dic74C2312yes—
60M16×C37—22no—
61C2×C8×C37C2312yes—
62C4⋊C4×C37—21no—
63C4⋊C4×C37C2311yescharacteristic
64C4⋊C4×C37C2311yescharacteristic
65C4⋊C4×C37C2312yes—
66C22⋊C4×C37—21no—
67C42×C37—21no—
68C22⋊Dic37—21no—
69#10596938—21no—
70#10596938—21no—
71#10596938—22no—
72#10596938—22no—
73C4⋊Dic37—22no—
74C4⋊Dic37—22no—
75C4⋊Dic37C2311yescharacteristic
76C4⋊Dic37C2311yescharacteristic
77#10596936—21no—
78#10596936—21no—
79C4×Dic37—21no—
80C4×Dic37—21no—
81C4×Dic37—21no—
82#10596934—22no—
83C2×#64436C2312yes—
84C23×C37C2×D411yescharacteristic
85C22×D37—22no—
86C22×D37—22no—
87C22×D37—41no—
88C22×D37C2×D411yescharacteristic
89C22×D37C2×D411yescharacteristic
90C2×C4×C37—21no—
91C2×C4×C37—21no—
92C2×C4×C37—22no—
93C2×C4×C37—42no—
94C2×C4×C37C2×D411yescharacteristic
95C2×C4×C37#4112yes—
96C2×C4×C37#4112yes—
97C2×C4×C37C2411yescharacteristic
98C2×Dic37—22no—
99C2×Dic37—21no—
100C2×Dic37—21no—
101C2×Dic37—22no—
102C2×Dic37—42no—
103C2×Dic37C2×D411yescharacteristic
104C2×Dic37C2×D411yescharacteristic
105C4×D37—22no—
106C4×D37—22no—
107C4×D37—21no—
108C4×D37—21no—
109C4×D37—21no—
110C4×D37—21no—
111C4×D37—42no—
112C4×D37—42no—
113Dic74—22no—
114Dic74—22no—
115Dic74—42no—
116C8×C37—22no—
117#64436—22no—
118C22×C37—21no—
119C22×C37—21no—
120C22×C37#12312yes—
121C22×C37C22×D411yescharacteristic
122C2×D37—21no—
123C2×D37—21no—
124C2×D37—21no—
125C2×D37—21no—
126C2×D37—21no—
127C2×D37—44no—
128C2×D37#13712yes—
129C4×C37—41no—
130C4×C37—41no—
131C4×C37—42no—
132C4×C37#12312yes—
133C4×C37C2×#4111yescharacteristic
134C4×C37C2×#4111yescharacteristic
135Dic37—21no—
136Dic37—21no—
137Dic37—42no—
138Dic37—41no—
139Dic37—41no—
140#3095—371nomaximal, Sylow
141C2×C37—22no—
142C2×C37C2×#12311yescharacteristic
143C2×C37C2×#13711yescharacteristic
144C2×C37#57511yescharacteristic
145D37—22no—
146D37—41no—
147C2×C22⋊Q8—371no—
148C2×#117—371no—
149#485—372no—
150#485—372no—
151#484—372no—
152#483—372no—
153#428—371no—
154#428D3711yescharacteristic, complemented
155#419—371no—
156#419—371no—
157#407—371no—
158C37#309511yescharacteristic, complemented, Sylow
159C22×Q8—371no—
160C23×C4—371no—
161C2×M16—371no—
162C2×M16C2×D3711yescharacteristic
163C22⋊Q8—372no—
164C22⋊Q8—372no—
165C22⋊Q8—741no—
166C22⋊Q8—741no—
167#117—371no—
168#117—371no—
169#117—371no—
170#117—741no—
171#117—741no—
172#117C2×D3711yescharacteristic
173C2×C4⋊C4—371no—
174C2×C4⋊C4—371no—
175C2×C4⋊C4—741no—
176C2×C4⋊C4C2×D3711yescharacteristic
177C2×C22⋊C4—741no—
178C2×C22⋊C4—741no—
179C2×C42—741no—
180#107—372no—
181#107C2×D3712yescomplemented
182#106—372no—
183#106C2×D3712yescomplemented
184#103—372no—
185#103—372no—
186#103—372no—
187#103—372no—
188C22⋊C8—372no—
189C22⋊C8—372no—
190C24—371no—
191C2×Q8—372no—
192C2×Q8—741no—
193C2×Q8—742no—
194C2×Q8—742no—
195C22×C4—21no—
196C22×C4—371no—
197C22×C4—371no—
198C22×C4—371no—
199C22×C4—371no—
200C22×C4—371no—
201C22×C4—741no—
202C22×C4—741no—
203C22×C4—742no—
204C22×C4—742no—
205C22×C4—741no—
206C22×C4C22×D3711yescharacteristic
207M16—22no—
208M16—742no—
209C2×C8—372no—
210C2×C8C22×D3712yes—
211C4⋊C4—21no—
212C4⋊C4—371no—
213C4⋊C4—371no—
214C4⋊C4—741no—
215C4⋊C4—741no—
216C4⋊C4—742no—
217C4⋊C4—742no—
218C4⋊C4C22×D3711yescharacteristic
219C4⋊C4C22×D3711yescharacteristic
220C4⋊C4C22×D3712yes—
221C22⋊C4—21no—
222C22⋊C4—741no—
223C22⋊C4—741no—
224C22⋊C4—742no—
225C22⋊C4—742no—
226C22⋊C4—741no—
227C42—21no—
228C42—741no—
229C42—741no—
230C42—741no—
231C23—371no—
232C23—371no—
233C23—742no—
234C23—742no—
235C23—1481no—
236C23D4×D3711yescharacteristic
237Q8—742no—
238Q8—742no—
239Q8—1482no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—371no—
245C2×C4—371no—
246C2×C4—742no—
247C2×C4—742no—
248C2×C4—742no—
249C2×C4—741no—
250C2×C4—741no—
251C2×C4—741no—
252C2×C4—741no—
253C2×C4—741no—
254C2×C4—741no—
255C2×C4—742no—
256C2×C4—1482no—
257C2×C4—1482no—
258C2×C4—1482no—
259C2×C4#1059696212yes—
260C2×C4D4×D3711yescharacteristic
261C2×C4#1059696612yes—
262C2×C4C23×D3711yescharacteristic
263C8—22no—
264C8—742no—
265C22—21no—
266C22—21no—
267C22—372no—
268C22—741no—
269C22—741no—
270C22—741no—
271C22—741no—
272C22—741no—
273C22—1484no—
274C22#1192452112yes—
275C22C2×D4×D3711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—741no—
280C4—741no—
281C4—1482no—
282C4—1481no—
283C4—1481no—
284C4#1192452112yes—
285C4C2×#1059696211yescharacteristic
286C4C2×#1059696611yescharacteristic
287C2—22no—
288C2—742no—
289C2—1481no—
290C2C2×#1192452111yescharacteristic
291C2#137×D3711yescharacteristic
292C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#428C2
socleC22×C37C22×D4
prime radicalC23×D37D4
hypercenter#428D37
hypocenterC37#3095
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #14557877C2semidirect
4. #14557876C2semidirect
5. #14557870C2semidirect
6. #14557865C2semidirect
7. #14557573C2semidirect
8. #14557543C2semidirect
9. #14557541C2semidirect
11. C37×#428C2semidirect
30. #11924469C22semidirect
31. #11924467C22semidirect
32. #11924461C22semidirect
33. #11924461C22semidirect
34. #11924459C22semidirect
35. #11924459C22semidirect
41. #107×C37C22semidirect
42. #106×C37C22semidirect
154. #428D37semidirect
158. C37#3095semidirect
181. #107C2×D37semidirect
183. #106C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245162
#117×D372
#145578772
#145578762
#145578702
#145578652
#145575732
#145575432
#145575412
#145574962
C37×#4282
#309537