Finite Group Databasebeta

← #496×D127 #30649160

Subgroups — #496×D127 #30649160

Order 16256.

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×D127C212yesmaximal, complemented
2C2×D8×D127C212yesmaximal, complemented
3#127×D127C211yesmaximal, characteristic, complemented
4#124×D127C211yesmaximal, characteristic, complemented
5#22550266C211yesmaximal, characteristic, complemented
6#22550265C211yesmaximal, characteristic, complemented
7#22550264C212yesmaximal, complemented
8#22550263C211yesmaximal, characteristic, complemented
9#22550262C212yesmaximal, complemented
10#97×D127C211yesmaximal, characteristic, complemented
11#496×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C2×D4×D127—22no—
14C2×D4×D127C2211yescharacteristic
15C2×D4×D127C2212yes—
16C2×#14523237C2212yescomplemented
17C2×#14523236C2212yescomplemented
18C2×#14523235C2212yescomplemented
19SD16×D127—28no—
20C2×D1016C2212yescomplemented
21C2×#14523233C2212yescomplemented
22C2×#14523233C2212yescomplemented
23D8×D127—28no—
24C2×C8×D127C2212yes—
25#16861206C2211yescharacteristic, complemented
26#16861205—22no—
27#16861205C2212yescomplemented
28#16861198C2211yescharacteristic, complemented
29#16861197C2211yescharacteristic, complemented
30#16861196C2211yescharacteristic, complemented
31#16861195—22no—
32C22⋊C4×D127—22no—
33C42×D127C2211yescharacteristic
34#16861120C2211yescharacteristic, complemented
35#16861119C2212yescomplemented
36C2×SD16×C127C2212yescomplemented
37C2×D8×C127C2212yescomplemented
38#127×C127C2211yescharacteristic, complemented
39#124×C127C2211yescharacteristic, complemented
40#97×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42C23×D127—21no—
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45Q8×D127—24no—
46C2×D508—22no—
47C2×D508C2311yescharacteristic
48C2×D508C2312yes—
49C2×C22⋊D127—21no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—24no—
52D4×D127—24no—
53D4×D127—24no—
54D4×D127—24no—
55D4×D127—48no—
56C2×C4×D127—21no—
57C2×C4×D127D412yes—
58C2×C4×D127C2311yescharacteristic
59#14523237—24no—
60#14523236—24no—
61#14523235—24no—
62D1016—24no—
63#14523233—24no—
64#14523233—24no—
65C2×C127⋊C8C2312yes—
66C8×D127D418yescomplemented
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70C4×Dic127D412yes—
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127C2311yescharacteristic
73C2×D4×C127—22no—
74C2×D4×C127C2311yescharacteristic
75C2×D4×C127C2312yes—
76SD16×C127—24no—
77D8×C127—24no—
78C2×C8×C127C2312yes—
79C22⋊C4×C127—22no—
80C42×C127C2311yescharacteristic
81C22×D127—21no—
82C22×D127—22no—
83C22×D127—42no—
84C22×D127—44no—
85C22×D127—48no—
86C22×D127—44no—
87C22×D127C2×D411yescharacteristic
88Dic254—22no—
89Dic254—42no—
90D508—22no—
91D508—22no—
92D508—22no—
93D508—44no—
94C22⋊D127—48no—
95C22⋊D127—42no—
96C22⋊D127—44no—
97C2×Dic127—21no—
98C2×Dic127C2×D412yes—
99C2×Dic127C2×D411yescharacteristic
100C4×D127—24no—
101C4×D127—42no—
102C4×D127C2×D414yes—
103C127⋊C8C2×D414yescomplemented
104C23×C127—21no—
105C23×C127—22no—
106Q8×C127—22no—
107D4×C127—22no—
108D4×C127—22no—
109D4×C127—22no—
110D4×C127—44no—
111C2×C4×C127—21no—
112C2×C4×C127C2×D412yes—
113C2×C4×C127C2411yescharacteristic
114C8×C127C2×D414yescomplemented
115C22×C127—41no—
116C22×C127—42no—
117C22×C127—44no—
118C22×C127—42no—
119C22×C127C22×D411yescharacteristic
120C2×D127—41no—
121C2×D127—42no—
122C2×D127—44no—
123C2×D127—42no—
124C2×D127—88no—
125C2×D127—84no—
126C2×D127#12712yes—
127C2×D127AGL(1,ℤ/8)14yes—
128C4×C127—22no—
129C4×C127—41no—
130C4×C127C22×D412yes—
131Dic127—22no—
132Dic127—41no—
133Dic127#12712yes—
134C2×C127—82no—
135C2×C127—81no—
136C2×C127C2×#12711yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)12yes—
138D127—82no—
139D127—81no—
140D127#49614yesdirect product factor
141C2×#496—1271nomaximal, Sylow
142C127C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—1272no—
144C22×D8—1272no—
145C2×#127—1271no—
146C2×#124—1271no—
147#496—1272no—
148#496—1272no—
149#496—1271no—
150#496—1271no—
151#496—1271no—
152#496D12711yescharacteristic, direct product factor
153C2×#97—1271no—
154C22×Q8—1271no—
155C22×D4—1271no—
156C22×D4—1272no—
157C22×D4—2542no—
158C2×SD16—1272no—
159C2×SD16—1272no—
160C2×SD16—1272no—
161C2×SD16—2548no—
162C2×SD16C2×D12712yescomplemented
163C2×D8—1272no—
164C2×D8—1272no—
165C2×D8—1272no—
166C2×D8—2548no—
167C2×D8C2×D12712yescomplemented
168C22×C8—1272no—
169#127—1272no—
170#127—1271no—
171#127—2542no—
172#127C2×D12711yescharacteristic, complemented
173#124—1271no—
174#124—1271no—
175#124—1271no—
176#124—2542no—
177#124C2×D12711yescharacteristic, complemented
178C2×C22⋊C4—2542no—
179C2×C42—1271no—
180#97—1272no—
181#97—1271no—
182#97C2×D12711yescharacteristic, complemented
183C24—2541no—
184C24—2542no—
185C2×Q8—1271no—
186C2×Q8—2541no—
187C2×Q8—2544no—
188C2×Q8C22×D12711yescharacteristic
189C2×D4—22no—
190C2×D4—1271no—
191C2×D4—1272no—
192C2×D4—5088no—
193C2×D4—2544no—
194C2×D4—2544no—
195C2×D4—2544no—
196C2×D4—2541no—
197C2×D4—2544no—
198C2×D4—2542no—
199C2×D4—2542no—
200C2×D4C22×D12711yescharacteristic
201C2×D4C22×D12712yes—
202C22×C4—1271no—
203C22×C4—1272no—
204C22×C4—2541no—
205SD16—24no—
206SD16—2544no—
207SD16—2544no—
208SD16—2544no—
209D8—24no—
210D8—2544no—
211D8—2544no—
212D8—2544no—
213C2×C8—1272no—
214C2×C8—1278no—
215C2×C8C22×D12712yes—
216C22⋊C4—22no—
217C22⋊C4—2542no—
218C22⋊C4—2542no—
219C22⋊C4—2542no—
220C42—1272no—
221C42—1271no—
222C42C22×D12711yescharacteristic
223C23—21no—
224C23—22no—
225C23—1271no—
226C23—5088no—
227C23—5082no—
228C23—5084no—
229C23—5084no—
230C23—2541no—
231C23—2542no—
232Q8—22no—
233Q8—5082no—
234Q8—2542no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—5084no—
240D4—5082no—
241D4—5084no—
242D4—5088no—
243D4—2542no—
244D4—2542no—
245D4—2542no—
246C2×C4—21no—
247C2×C4—1271no—
248C2×C4—1274no—
249C2×C4—1272no—
250C2×C4—5082no—
251C2×C4—2544no—
252C2×C4—2541no—
253C2×C4D4×D12712yes—
254C2×C4C23×D12711yescharacteristic
255C8—1274no—
256C8D4×D12714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—1274no—
262C22—1272no—
263C22—10168no—
264C22—10164no—
265C22—5084no—
266C22—5082no—
267C22—5081no—
268C22—5082no—
269C22C2×D4×D12711yescharacteristic
270C4—22no—
271C4—41no—
272C4—1272no—
273C4—5081no—
274C4—2542no—
275C4C2×D4×D12712yes—
276C2—82no—
277C2—81no—
278C2—1274no—
279C2—10162no—
280C2—10161no—
281C2#127×D12711yescharacteristic
282C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#496×C127C2
socleC22×C127C22×D4
hypercenter#496D127
hypocenterC127C2×#496
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D127C2semidirect
2. C2×D8×D127C2semidirect
3. #127×D127C2semidirect
4. #124×D127C2semidirect
5. #22550266C2semidirect
6. #22550265C2semidirect
7. #22550264C2semidirect
8. #22550263C2semidirect
9. #22550262C2semidirect
10. #97×D127C2semidirect
11. #496×C127C2semidirect
16. C2×#14523237C22semidirect
17. C2×#14523236C22semidirect
18. C2×#14523235C22semidirect
20. C2×D1016C22semidirect
21. C2×#14523233C22semidirect
22. C2×#14523233C22semidirect
25. #16861206C22semidirect
27. #16861205C22semidirect
28. #16861198C22semidirect
29. #16861197C22semidirect
30. #16861196C22semidirect
34. #16861120C22semidirect
35. #16861119C22semidirect
36. C2×SD16×C127C22semidirect
37. C2×D8×C127C22semidirect
38. #127×C127C22semidirect
39. #124×C127C22semidirect
40. #97×C127C22semidirect
66. C8×D127D4semidirect
103. C127⋊C8C2×D4semidirect
114. C8×C127C2×D4semidirect
140. D127#496direct (N × H)
142. C127C2×#496semidirect
152. #496D127direct (N × H)
162. C2×SD16C2×D127semidirect
167. C2×D8C2×D127semidirect
172. #127C2×D127semidirect
177. #124C2×D127semidirect
182. #97C2×D127semidirect
256. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1272
C2×D8×D1272
#127×D1272
#124×D1272
#225502662
#225502652
#225502642
#225502632
#225502622
#97×D1272
#496×C1272
C2×#496127