Finite Group Databasebeta

← #496×D101 #27181562

Subgroups — #496×D101 #27181562

Order 12928.

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×D101C212yesmaximal, complemented
2C2×D8×D101C212yesmaximal, complemented
3#127×D101C211yesmaximal, characteristic, complemented
4#124×D101C211yesmaximal, characteristic, complemented
5#20265567C211yesmaximal, characteristic, complemented
6#20265566C211yesmaximal, characteristic, complemented
7#20265565C212yesmaximal, complemented
8#20265564C211yesmaximal, characteristic, complemented
9#20265563C212yesmaximal, complemented
10#97×D101C211yesmaximal, characteristic, complemented
11#496×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101C2211yescharacteristic
13C2×D4×D101—22no—
14C2×D4×D101C2211yescharacteristic
15C2×D4×D101C2212yes—
16C2×#13108063C2212yescomplemented
17C2×#13108062C2212yescomplemented
18C2×#13108061C2212yescomplemented
19SD16×D101—28no—
20C2×D808C2212yescomplemented
21C2×#13108059C2212yescomplemented
22C2×#13108059C2212yescomplemented
23D8×D101—28no—
24C2×C8×D101C2212yes—
25#15692252C2211yescharacteristic, complemented
26#15692251—22no—
27#15692251C2212yescomplemented
28#15692244C2211yescharacteristic, complemented
29#15692243C2211yescharacteristic, complemented
30#15692242C2211yescharacteristic, complemented
31#15692241—22no—
32C22⋊C4×D101—22no—
33C42×D101C2211yescharacteristic
34#15692166C2211yescharacteristic, complemented
35#15692165C2212yescomplemented
36C2×SD16×C101C2212yescomplemented
37C2×D8×C101C2212yescomplemented
38#127×C101C2211yescharacteristic, complemented
39#124×C101C2211yescharacteristic, complemented
40#97×C101C2211yescharacteristic, complemented
41C23×D101—22no—
42C23×D101—21no—
43C2×Dic202—21no—
44C2×Dic202C2311yescharacteristic
45Q8×D101—24no—
46C2×D404—22no—
47C2×D404C2311yescharacteristic
48C2×D404C2312yes—
49C2×C22⋊D101—24no—
50C2×C22⋊D101—21no—
51C2×C22⋊D101—22no—
52D4×D101—24no—
53D4×D101—24no—
54D4×D101—24no—
55D4×D101—48no—
56C2×C4×D101—21no—
57C2×C4×D101D412yes—
58C2×C4×D101C2311yescharacteristic
59#13108063—24no—
60#13108062—24no—
61#13108061—24no—
62D808—24no—
63#13108059—24no—
64#13108059—24no—
65C2×#11717579C2312yes—
66C8×D101D418yescomplemented
67C22⋊Dic101—22no—
68#13108051—22no—
69#13108051—22no—
70C4×Dic101D412yes—
71C4×Dic101C2311yescharacteristic
72C2×Q8×C101C2311yescharacteristic
73C2×D4×C101—22no—
74C2×D4×C101C2311yescharacteristic
75C2×D4×C101C2312yes—
76SD16×C101—24no—
77D8×C101—24no—
78C2×C8×C101C2312yes—
79C22⋊C4×C101—22no—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C23×C101—22no—
83C22×D101—21no—
84C22×D101—22no—
85C22×D101—42no—
86C22×D101—44no—
87C22×D101—48no—
88C22×D101—44no—
89C22×D101C2×D411yescharacteristic
90Q8×C101—22no—
91D4×C101—22no—
92D4×C101—22no—
93D4×C101—22no—
94D4×C101—44no—
95C2×C4×C101—21no—
96C2×C4×C101C2×D412yes—
97C2×C4×C101C2411yescharacteristic
98C22⋊D101—48no—
99C22⋊D101—42no—
100C22⋊D101—44no—
101C2×Dic101—21no—
102C2×Dic101C2×D412yes—
103C2×Dic101C2×D411yescharacteristic
104D404—22no—
105D404—22no—
106D404—22no—
107D404—44no—
108C4×D101—24no—
109C4×D101—42no—
110C4×D101C2×D414yes—
111Dic202—22no—
112Dic202—42no—
113C8×C101C2×D414yescomplemented
114#11717579C2×D414yescomplemented
115C22×C101—41no—
116C22×C101—42no—
117C22×C101—44no—
118C22×C101—42no—
119C22×C101C22×D411yescharacteristic
120C2×D101—41no—
121C2×D101—42no—
122C2×D101—44no—
123C2×D101—42no—
124C2×D101—88no—
125C2×D101—84no—
126C2×D101#12712yes—
127C2×D101AGL(1,ℤ/8)14yes—
128C4×C101—22no—
129C4×C101—41no—
130C4×C101C22×D412yes—
131Dic101—22no—
132Dic101—41no—
133Dic101#12712yes—
134C2×C101—82no—
135C2×C101—81no—
136C2×C101C2×#12711yescharacteristic
137C2×C101C2×AGL(1,ℤ/8)12yes—
138D101—82no—
139D101—81no—
140D101#49614yesdirect product factor
141C2×#496—1011nomaximal, Sylow
142C101C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—1012no—
144C22×D8—1012no—
145C2×#127—1011no—
146C2×#124—1011no—
147#496—1012no—
148#496—1012no—
149#496—1011no—
150#496—1011no—
151#496—1011no—
152#496D10111yescharacteristic, direct product factor
153C2×#97—1011no—
154C22×Q8—1011no—
155C22×D4—1011no—
156C22×D4—1012no—
157C22×D4—2022no—
158C2×SD16—1012no—
159C2×SD16—1012no—
160C2×SD16—1012no—
161C2×SD16—2028no—
162C2×SD16C2×D10112yescomplemented
163C2×D8—1012no—
164C2×D8—1012no—
165C2×D8—1012no—
166C2×D8—2028no—
167C2×D8C2×D10112yescomplemented
168C22×C8—1012no—
169#127—1012no—
170#127—1011no—
171#127—2022no—
172#127C2×D10111yescharacteristic, complemented
173#124—1011no—
174#124—1011no—
175#124—1011no—
176#124—2022no—
177#124C2×D10111yescharacteristic, complemented
178C2×C22⋊C4—2022no—
179C2×C42—1011no—
180#97—1012no—
181#97—1011no—
182#97C2×D10111yescharacteristic, complemented
183C24—2021no—
184C24—2022no—
185C2×Q8—1011no—
186C2×Q8—2021no—
187C2×Q8—2024no—
188C2×Q8C22×D10111yescharacteristic
189C2×D4—22no—
190C2×D4—1011no—
191C2×D4—1012no—
192C2×D4—4048no—
193C2×D4—2024no—
194C2×D4—2024no—
195C2×D4—2024no—
196C2×D4—2021no—
197C2×D4—2024no—
198C2×D4—2022no—
199C2×D4—2022no—
200C2×D4C22×D10111yescharacteristic
201C2×D4C22×D10112yes—
202C22×C4—1011no—
203C22×C4—1012no—
204C22×C4—2021no—
205SD16—24no—
206SD16—2024no—
207SD16—2024no—
208SD16—2024no—
209D8—24no—
210D8—2024no—
211D8—2024no—
212D8—2024no—
213C2×C8—1012no—
214C2×C8—1018no—
215C2×C8C22×D10112yes—
216C22⋊C4—22no—
217C22⋊C4—2022no—
218C22⋊C4—2022no—
219C22⋊C4—2022no—
220C42—1012no—
221C42—1011no—
222C42C22×D10111yescharacteristic
223C23—21no—
224C23—22no—
225C23—1011no—
226C23—4048no—
227C23—4042no—
228C23—4044no—
229C23—4044no—
230C23—2021no—
231C23—2022no—
232Q8—22no—
233Q8—4042no—
234Q8—2022no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—4044no—
240D4—4042no—
241D4—4044no—
242D4—4048no—
243D4—2022no—
244D4—2022no—
245D4—2022no—
246C2×C4—21no—
247C2×C4—1011no—
248C2×C4—1014no—
249C2×C4—1012no—
250C2×C4—4042no—
251C2×C4—2024no—
252C2×C4—2021no—
253C2×C4D4×D10112yes—
254C2×C4C23×D10111yescharacteristic
255C8—1014no—
256C8D4×D10114yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—8088no—
262C22—8084no—
263C22—1014no—
264C22—1012no—
265C22—4044no—
266C22—4042no—
267C22—4041no—
268C22—4042no—
269C22C2×D4×D10111yescharacteristic
270C4—22no—
271C4—41no—
272C4—1012no—
273C4—4041no—
274C4—2022no—
275C4C2×D4×D10112yes—
276C2—82no—
277C2—81no—
278C2—8082no—
279C2—8081no—
280C2—1014no—
281C2#127×D10111yescharacteristic
282C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101C2×#496
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D101C2semidirect
2. C2×D8×D101C2semidirect
3. #127×D101C2semidirect
4. #124×D101C2semidirect
5. #20265567C2semidirect
6. #20265566C2semidirect
7. #20265565C2semidirect
8. #20265564C2semidirect
9. #20265563C2semidirect
10. #97×D101C2semidirect
11. #496×C101C2semidirect
16. C2×#13108063C22semidirect
17. C2×#13108062C22semidirect
18. C2×#13108061C22semidirect
20. C2×D808C22semidirect
21. C2×#13108059C22semidirect
22. C2×#13108059C22semidirect
25. #15692252C22semidirect
27. #15692251C22semidirect
28. #15692244C22semidirect
29. #15692243C22semidirect
30. #15692242C22semidirect
34. #15692166C22semidirect
35. #15692165C22semidirect
36. C2×SD16×C101C22semidirect
37. C2×D8×C101C22semidirect
38. #127×C101C22semidirect
39. #124×C101C22semidirect
40. #97×C101C22semidirect
66. C8×D101D4semidirect
113. C8×C101C2×D4semidirect
114. #11717579C2×D4semidirect
140. D101#496direct (N × H)
142. C101C2×#496semidirect
152. #496D101direct (N × H)
162. C2×SD16C2×D101semidirect
167. C2×D8C2×D101semidirect
172. #127C2×D101semidirect
177. #124C2×D101semidirect
182. #97C2×D101semidirect
256. C8D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1012
C2×D8×D1012
#127×D1012
#124×D1012
#202655672
#202655662
#202655652
#202655642
#202655632
#97×D1012
#496×C1012
C2×#496101