Finite Group Databasebeta

← #496×D37 #18005026

Subgroups — #496×D37 #18005026

Order 4736.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D37C212yesmaximal, complemented
2C2×D8×D37C212yesmaximal, complemented
3#127×D37C211yesmaximal, characteristic, complemented
4#124×D37C211yesmaximal, characteristic, complemented
5#14557923C211yesmaximal, characteristic, complemented
6#14557922C211yesmaximal, characteristic, complemented
7#14557921C212yesmaximal, complemented
8#14557920C211yesmaximal, characteristic, complemented
9#14557919C212yesmaximal, complemented
10#97×D37C211yesmaximal, characteristic, complemented
11C37×#496C211yesmaximal, characteristic, complemented
12C2×Q8×D37C2211yescharacteristic
13C2×D4×D37—22no—
14C2×D4×D37C2211yescharacteristic
15C2×D4×D37C2212yes—
16C2×#10596930C2212yescomplemented
17C2×#10596940C2212yescomplemented
18C2×#10596941C2212yescomplemented
19SD16×D37—28no—
20C2×D296C2212yescomplemented
21C2×#10596939C2212yescomplemented
22C2×#10596939C2212yescomplemented
23D8×D37—28no—
24C2×C8×D37C2212yes—
25#11924533C2211yescharacteristic, complemented
26#11924532—22no—
27#11924532C2212yescomplemented
28#11924525C2211yescharacteristic, complemented
29#11924524C2211yescharacteristic, complemented
30#11924523C2211yescharacteristic, complemented
31#11924522—22no—
32C22⋊C4×D37—22no—
33C42×D37C2211yescharacteristic
34#11924447C2211yescharacteristic, complemented
35#11924446C2212yescomplemented
36C2×SD16×C37C2212yescomplemented
37C2×D8×C37C2212yescomplemented
38#127×C37C2211yescharacteristic, complemented
39#124×C37C2211yescharacteristic, complemented
40#97×C37C2211yescharacteristic, complemented
41C23×D37—21no—
42C23×D37—22no—
43C2×Q8×C37C2311yescharacteristic
44C2×D4×C37—22no—
45C2×D4×C37C2311yescharacteristic
46C2×D4×C37C2312yes—
47C2×C22⋊D37—21no—
48C2×C22⋊D37—22no—
49C2×C22⋊D37—24no—
50Q8×D37—24no—
51D4×D37—24no—
52D4×D37—24no—
53D4×D37—24no—
54D4×D37—48no—
55C2×D148—22no—
56C2×D148C2311yescharacteristic
57C2×D148C2312yes—
58C2×C4×D37—21no—
59C2×C4×D37D412yes—
60C2×C4×D37C2311yescharacteristic
61C2×Dic74—21no—
62C2×Dic74C2311yescharacteristic
63SD16×C37—24no—
64D8×C37—24no—
65C2×C8×C37C2312yes—
66C22⋊C4×C37—22no—
67C42×C37C2311yescharacteristic
68C22⋊Dic37—22no—
69#10596941—24no—
70#10596940—24no—
71#10596939—24no—
72#10596939—24no—
73#10596938—22no—
74#10596938—22no—
75C4×Dic37D412yes—
76C4×Dic37C2311yescharacteristic
77C2×#64436C2312yes—
78D296—24no—
79#10596930—24no—
80C8×D37D418yescomplemented
81C23×C37—21no—
82C23×C37—22no—
83C22×D37—21no—
84C22×D37—22no—
85C22×D37—42no—
86C22×D37—44no—
87C22×D37—48no—
88C22×D37—44no—
89C22×D37C2×D411yescharacteristic
90Q8×C37—22no—
91D4×C37—22no—
92D4×C37—22no—
93D4×C37—22no—
94D4×C37—44no—
95C2×C4×C37—21no—
96C2×C4×C37C2×D412yes—
97C2×C4×C37C2411yescharacteristic
98C22⋊D37—48no—
99C22⋊D37—42no—
100C22⋊D37—44no—
101C2×Dic37—21no—
102C2×Dic37C2×D412yes—
103C2×Dic37C2×D411yescharacteristic
104D148—22no—
105D148—22no—
106D148—22no—
107D148—44no—
108C4×D37—24no—
109C4×D37—42no—
110C4×D37C2×D414yes—
111Dic74—22no—
112Dic74—42no—
113C8×C37C2×D414yescomplemented
114#64436C2×D414yescomplemented
115C22×C37—41no—
116C22×C37—42no—
117C22×C37—44no—
118C22×C37—42no—
119C22×C37C22×D411yescharacteristic
120C2×D37—42no—
121C2×D37—41no—
122C2×D37—44no—
123C2×D37—42no—
124C2×D37—88no—
125C2×D37—84no—
126C2×D37#12712yes—
127C2×D37AGL(1,ℤ/8)14yes—
128C4×C37—22no—
129C4×C37—41no—
130C4×C37C22×D412yes—
131Dic37—22no—
132Dic37—41no—
133Dic37#12712yes—
134C2×#496—371nomaximal, Sylow
135C2×C37—82no—
136C2×C37—81no—
137C2×C37C2×#12711yescharacteristic
138C2×C37C2×AGL(1,ℤ/8)12yes—
139D37—82no—
140D37—81no—
141D37#49614yesdirect product factor
142C22×SD16—372no—
143C22×D8—372no—
144C2×#127—371no—
145C2×#124—371no—
146#496—372no—
147#496—372no—
148#496—371no—
149#496—371no—
150#496—371no—
151#496D3711yescharacteristic, direct product factor
152C2×#97—371no—
153C37C2×#49611yescharacteristic, complemented, Sylow
154C22×Q8—371no—
155C22×D4—371no—
156C22×D4—372no—
157C22×D4—742no—
158C2×SD16—372no—
159C2×SD16—372no—
160C2×SD16—372no—
161C2×SD16—748no—
162C2×SD16C2×D3712yescomplemented
163C2×D8—372no—
164C2×D8—372no—
165C2×D8—372no—
166C2×D8—748no—
167C2×D8C2×D3712yescomplemented
168C22×C8—372no—
169#127—372no—
170#127—371no—
171#127—742no—
172#127C2×D3711yescharacteristic, complemented
173#124—371no—
174#124—371no—
175#124—371no—
176#124—742no—
177#124C2×D3711yescharacteristic, complemented
178C2×C22⋊C4—742no—
179C2×C42—371no—
180#97—372no—
181#97—371no—
182#97C2×D3711yescharacteristic, complemented
183C24—741no—
184C24—742no—
185C2×Q8—371no—
186C2×Q8—741no—
187C2×Q8—744no—
188C2×Q8C22×D3711yescharacteristic
189C2×D4—22no—
190C2×D4—371no—
191C2×D4—372no—
192C2×D4—744no—
193C2×D4—744no—
194C2×D4—744no—
195C2×D4—741no—
196C2×D4—744no—
197C2×D4—742no—
198C2×D4—742no—
199C2×D4—1488no—
200C2×D4C22×D3711yescharacteristic
201C2×D4C22×D3712yes—
202C22×C4—371no—
203C22×C4—372no—
204C22×C4—741no—
205SD16—24no—
206SD16—744no—
207SD16—744no—
208SD16—744no—
209D8—24no—
210D8—744no—
211D8—744no—
212D8—744no—
213C2×C8—372no—
214C2×C8—378no—
215C2×C8C22×D3712yes—
216C22⋊C4—22no—
217C22⋊C4—742no—
218C22⋊C4—742no—
219C22⋊C4—742no—
220C42—372no—
221C42—371no—
222C42C22×D3711yescharacteristic
223C23—21no—
224C23—22no—
225C23—371no—
226C23—741no—
227C23—742no—
228C23—1482no—
229C23—1484no—
230C23—1488no—
231C23—1484no—
232Q8—22no—
233Q8—742no—
234Q8—1482no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—742no—
240D4—742no—
241D4—742no—
242D4—1484no—
243D4—1482no—
244D4—1484no—
245D4—1488no—
246C2×C4—21no—
247C2×C4—371no—
248C2×C4—374no—
249C2×C4—372no—
250C2×C4—744no—
251C2×C4—741no—
252C2×C4—1482no—
253C2×C4D4×D3712yes—
254C2×C4C23×D3711yescharacteristic
255C8—374no—
256C8D4×D3714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—374no—
262C22—372no—
263C22—2968no—
264C22—2964no—
265C22—1484no—
266C22—1482no—
267C22—1481no—
268C22—1482no—
269C22C2×D4×D3711yescharacteristic
270C4—22no—
271C4—41no—
272C4—372no—
273C4—742no—
274C4—1481no—
275C4C2×D4×D3712yes—
276C2—82no—
277C2—81no—
278C2—374no—
279C2—2962no—
280C2—2961no—
281C2#127×D3711yescharacteristic
282C2AGL(1,ℤ/8)×D3712yes—
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#496C2
socleC22×C37C22×D4
hypercenter#496D37
hypocenterC37C2×#496
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D37C2semidirect
2. C2×D8×D37C2semidirect
3. #127×D37C2semidirect
4. #124×D37C2semidirect
5. #14557923C2semidirect
6. #14557922C2semidirect
7. #14557921C2semidirect
8. #14557920C2semidirect
9. #14557919C2semidirect
10. #97×D37C2semidirect
11. C37×#496C2semidirect
16. C2×#10596930C22semidirect
17. C2×#10596940C22semidirect
18. C2×#10596941C22semidirect
20. C2×D296C22semidirect
21. C2×#10596939C22semidirect
22. C2×#10596939C22semidirect
25. #11924533C22semidirect
27. #11924532C22semidirect
28. #11924525C22semidirect
29. #11924524C22semidirect
30. #11924523C22semidirect
34. #11924447C22semidirect
35. #11924446C22semidirect
36. C2×SD16×C37C22semidirect
37. C2×D8×C37C22semidirect
38. #127×C37C22semidirect
39. #124×C37C22semidirect
40. #97×C37C22semidirect
80. C8×D37D4semidirect
113. C8×C37C2×D4semidirect
114. #64436C2×D4semidirect
141. D37#496direct (N × H)
151. #496D37direct (N × H)
153. C37C2×#496semidirect
162. C2×SD16C2×D37semidirect
167. C2×D8C2×D37semidirect
172. #127C2×D37semidirect
177. #124C2×D37semidirect
182. #97C2×D37semidirect
256. C8D4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D372
C2×D8×D372
#127×D372
#124×D372
#145579232
#145579222
#145579212
#145579202
#145579192
#97×D372
C37×#4962
C2×#49637