Finite Group Databasebeta

← #496×D113 #28401004

Subgroups — #496×D113 #28401004

Order 14464.

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×D113C212yesmaximal, complemented
2C2×D8×D113C212yesmaximal, complemented
3#127×D113C211yesmaximal, characteristic, complemented
4#124×D113C211yesmaximal, characteristic, complemented
5#20298988C211yesmaximal, characteristic, complemented
6#20298987C211yesmaximal, characteristic, complemented
7#20298986C212yesmaximal, complemented
8#20298985C211yesmaximal, characteristic, complemented
9#20298984C212yesmaximal, complemented
10#97×D113C211yesmaximal, characteristic, complemented
11#496×C113C211yesmaximal, characteristic, complemented
12C2×Q8×D113C2211yescharacteristic
13C2×D4×D113—22no—
14C2×D4×D113C2211yescharacteristic
15C2×D4×D113C2212yes—
16C2×#14268636C2212yescomplemented
17C2×#14268635C2212yescomplemented
18C2×#14268634C2212yescomplemented
19SD16×D113—28no—
20C2×D904C2212yescomplemented
21C2×#14268632C2212yescomplemented
22C2×#14268632C2212yescomplemented
23D8×D113—28no—
24C2×C8×D113C2212yes—
25#16813619C2211yescharacteristic, complemented
26#16813618—22no—
27#16813618C2212yescomplemented
28#16813611C2211yescharacteristic, complemented
29#16813610C2211yescharacteristic, complemented
30#16813609C2211yescharacteristic, complemented
31#16813608—22no—
32C22⋊C4×D113—22no—
33C42×D113C2211yescharacteristic
34#16813533C2211yescharacteristic, complemented
35#16813532C2212yescomplemented
36C2×SD16×C113C2212yescomplemented
37C2×D8×C113C2212yescomplemented
38#127×C113C2211yescharacteristic, complemented
39#124×C113C2211yescharacteristic, complemented
40#97×C113C2211yescharacteristic, complemented
41C23×D113—21no—
42C23×D113—22no—
43C2×Dic226—21no—
44C2×Dic226C2311yescharacteristic
45Q8×D113—24no—
46C2×D452—22no—
47C2×D452C2311yescharacteristic
48C2×D452C2312yes—
49C2×C22⋊D113—21no—
50C2×C22⋊D113—22no—
51C2×C22⋊D113—24no—
52D4×D113—24no—
53D4×D113—24no—
54D4×D113—24no—
55D4×D113—48no—
56C2×C4×D113—21no—
57C2×C4×D113D412yes—
58C2×C4×D113C2311yescharacteristic
59#14268636—24no—
60#14268635—24no—
61#14268634—24no—
62D904—24no—
63#14268632—24no—
64#14268632—24no—
65C2×#11744751C2312yes—
66C8×D113D418yescomplemented
67C22⋊Dic113—22no—
68#14268624—22no—
69#14268624—22no—
70C4×Dic113D412yes—
71C4×Dic113C2311yescharacteristic
72C2×Q8×C113C2311yescharacteristic
73C2×D4×C113—22no—
74C2×D4×C113C2311yescharacteristic
75C2×D4×C113C2312yes—
76SD16×C113—24no—
77D8×C113—24no—
78C2×C8×C113C2312yes—
79C22⋊C4×C113—22no—
80C42×C113C2311yescharacteristic
81C23×C113—21no—
82C23×C113—22no—
83C22×D113—21no—
84C22×D113—22no—
85C22×D113—42no—
86C22×D113—44no—
87C22×D113—48no—
88C22×D113—44no—
89C22×D113C2×D411yescharacteristic
90Q8×C113—22no—
91D4×C113—22no—
92D4×C113—22no—
93D4×C113—22no—
94D4×C113—44no—
95C2×C4×C113—21no—
96C2×C4×C113C2×D412yes—
97C2×C4×C113C2411yescharacteristic
98C22⋊D113—48no—
99C22⋊D113—42no—
100C22⋊D113—44no—
101C2×Dic113—21no—
102C2×Dic113C2×D412yes—
103C2×Dic113C2×D411yescharacteristic
104D452—22no—
105D452—22no—
106D452—22no—
107D452—44no—
108C4×D113—24no—
109C4×D113—42no—
110C4×D113C2×D414yes—
111Dic226—22no—
112Dic226—42no—
113C8×C113C2×D414yescomplemented
114#11744751C2×D414yescomplemented
115C22×C113—41no—
116C22×C113—42no—
117C22×C113—44no—
118C22×C113—42no—
119C22×C113C22×D411yescharacteristic
120C2×D113—41no—
121C2×D113—42no—
122C2×D113—44no—
123C2×D113—42no—
124C2×D113—88no—
125C2×D113—84no—
126C2×D113#12712yes—
127C2×D113AGL(1,ℤ/8)14yes—
128C4×C113—22no—
129C4×C113—41no—
130C4×C113C22×D412yes—
131Dic113—22no—
132Dic113—41no—
133Dic113#12712yes—
134C2×C113—82no—
135C2×C113—81no—
136C2×C113C2×#12711yescharacteristic
137C2×C113C2×AGL(1,ℤ/8)12yes—
138D113—82no—
139D113—81no—
140D113#49614yesdirect product factor
141C2×#496—1131nomaximal, Sylow
142C113C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—1132no—
144C22×D8—1132no—
145C2×#127—1131no—
146C2×#124—1131no—
147#496—1132no—
148#496—1132no—
149#496—1131no—
150#496—1131no—
151#496—1131no—
152#496D11311yescharacteristic, direct product factor
153C2×#97—1131no—
154C22×Q8—1131no—
155C22×D4—1131no—
156C22×D4—1132no—
157C22×D4—2262no—
158C2×SD16—1132no—
159C2×SD16—1132no—
160C2×SD16—1132no—
161C2×SD16—2268no—
162C2×SD16C2×D11312yescomplemented
163C2×D8—1132no—
164C2×D8—1132no—
165C2×D8—1132no—
166C2×D8—2268no—
167C2×D8C2×D11312yescomplemented
168C22×C8—1132no—
169#127—1132no—
170#127—1131no—
171#127—2262no—
172#127C2×D11311yescharacteristic, complemented
173#124—1131no—
174#124—1131no—
175#124—1131no—
176#124—2262no—
177#124C2×D11311yescharacteristic, complemented
178C2×C22⋊C4—2262no—
179C2×C42—1131no—
180#97—1132no—
181#97—1131no—
182#97C2×D11311yescharacteristic, complemented
183C24—2261no—
184C24—2262no—
185C2×Q8—1131no—
186C2×Q8—2261no—
187C2×Q8—2264no—
188C2×Q8C22×D11311yescharacteristic
189C2×D4—22no—
190C2×D4—1131no—
191C2×D4—1132no—
192C2×D4—4528no—
193C2×D4—2264no—
194C2×D4—2264no—
195C2×D4—2264no—
196C2×D4—2261no—
197C2×D4—2264no—
198C2×D4—2262no—
199C2×D4—2262no—
200C2×D4C22×D11311yescharacteristic
201C2×D4C22×D11312yes—
202C22×C4—1131no—
203C22×C4—1132no—
204C22×C4—2261no—
205SD16—24no—
206SD16—2264no—
207SD16—2264no—
208SD16—2264no—
209D8—24no—
210D8—2264no—
211D8—2264no—
212D8—2264no—
213C2×C8—1132no—
214C2×C8—1138no—
215C2×C8C22×D11312yes—
216C22⋊C4—22no—
217C22⋊C4—2262no—
218C22⋊C4—2262no—
219C22⋊C4—2262no—
220C42—1132no—
221C42—1131no—
222C42C22×D11311yescharacteristic
223C23—21no—
224C23—22no—
225C23—1131no—
226C23—4528no—
227C23—4522no—
228C23—4524no—
229C23—4524no—
230C23—2261no—
231C23—2262no—
232Q8—22no—
233Q8—4522no—
234Q8—2262no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—4524no—
240D4—4522no—
241D4—4524no—
242D4—4528no—
243D4—2262no—
244D4—2262no—
245D4—2262no—
246C2×C4—21no—
247C2×C4—1131no—
248C2×C4—1134no—
249C2×C4—1132no—
250C2×C4—4522no—
251C2×C4—2264no—
252C2×C4—2261no—
253C2×C4D4×D11312yes—
254C2×C4C23×D11311yescharacteristic
255C8—1134no—
256C8D4×D11314yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—1134no—
262C22—1132no—
263C22—9048no—
264C22—9044no—
265C22—4524no—
266C22—4522no—
267C22—4521no—
268C22—4522no—
269C22C2×D4×D11311yescharacteristic
270C4—22no—
271C4—41no—
272C4—1132no—
273C4—4521no—
274C4—2262no—
275C4C2×D4×D11312yes—
276C2—82no—
277C2—81no—
278C2—1134no—
279C2—9042no—
280C2—9041no—
281C2#127×D11311yescharacteristic
282C2AGL(1,ℤ/8)×D11312yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#496×C113C2
socleC22×C113C22×D4
hypercenter#496D113
hypocenterC113C2×#496
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D113C2semidirect
2. C2×D8×D113C2semidirect
3. #127×D113C2semidirect
4. #124×D113C2semidirect
5. #20298988C2semidirect
6. #20298987C2semidirect
7. #20298986C2semidirect
8. #20298985C2semidirect
9. #20298984C2semidirect
10. #97×D113C2semidirect
11. #496×C113C2semidirect
16. C2×#14268636C22semidirect
17. C2×#14268635C22semidirect
18. C2×#14268634C22semidirect
20. C2×D904C22semidirect
21. C2×#14268632C22semidirect
22. C2×#14268632C22semidirect
25. #16813619C22semidirect
27. #16813618C22semidirect
28. #16813611C22semidirect
29. #16813610C22semidirect
30. #16813609C22semidirect
34. #16813533C22semidirect
35. #16813532C22semidirect
36. C2×SD16×C113C22semidirect
37. C2×D8×C113C22semidirect
38. #127×C113C22semidirect
39. #124×C113C22semidirect
40. #97×C113C22semidirect
66. C8×D113D4semidirect
113. C8×C113C2×D4semidirect
114. #11744751C2×D4semidirect
140. D113#496direct (N × H)
142. C113C2×#496semidirect
152. #496D113direct (N × H)
162. C2×SD16C2×D113semidirect
167. C2×D8C2×D113semidirect
172. #127C2×D113semidirect
177. #124C2×D113semidirect
182. #97C2×D113semidirect
256. C8D4×D113semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1132
C2×D8×D1132
#127×D1132
#124×D1132
#202989882
#202989872
#202989862
#202989852
#202989842
#97×D1132
#496×C1132
C2×#496113