Finite Group Databasebeta

← #469×D127 #30648322

Subgroups — #469×D127 #30648322

Order 16256.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D127C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3#121×D127C212yesmaximal, complemented
4#22550130C211yesmaximal, characteristic, complemented
5#22550129C211yesmaximal, characteristic, complemented
6#22550128C212yesmaximal, complemented
7#22550127C212yesmaximal, complemented
8#22550126C211yesmaximal, characteristic, complemented
9#106×D127C211yesmaximal, characteristic, complemented
10#102×D127C212yesmaximal, complemented
11#469×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127—22no—
13C2×D4×D127C2212yes—
14C22×C4×D127C2211yescharacteristic
15C2×D1016C2211yescharacteristic, complemented
16C2×#14523233C2212yescomplemented
17D8×D127—24no—
18C2×#14523232C2211yescharacteristic, complemented
19C2×#14523231C2211yescharacteristic, complemented
20C2×#14523231C2211yescharacteristic, complemented
21M16×D127—24no—
22C2×C8×D127C2211yescharacteristic
23C2×C8×D127C2211yescharacteristic
24#16861184C2212yescomplemented
25#16861183C2212yescomplemented
26#16861182—22no—
27#16861181—22no—
28#16861180C2212yescomplemented
29C4⋊C4×D127C2212yes—
30C22⋊C4×D127—22no—
31#16861141C2211yescharacteristic, complemented
32#16861140C2212yescomplemented
33#16861131C2212yescomplemented
34#16861130C2212yescomplemented
35#16861129C2212yescomplemented
36C2×D8×C127C2211yescharacteristic, complemented
37C2×M16×C127C2211yescharacteristic, complemented
38#121×C127C2212yescomplemented
39#106×C127C2211yescharacteristic, complemented
40#102×C127C2212yescomplemented
41C23×D127—22no—
42C23×D127D411yescharacteristic
43C2×D508—22no—
44C2×D508C2312yes—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48D4×D127—24no—
49D4×D127—24no—
50D4×D127—48no—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58C2×C4×D127C2311yescharacteristic
59D1016—22no—
60#14523233—24no—
61#14523232—22no—
62#14523231—22no—
63#14523231—22no—
64C2×C127⋊C8C2311yescharacteristic
65C2×C127⋊C8C2311yescharacteristic
66C8×D127—22no—
67C8×D127D414yescomplemented
68C4⋊Dic127C2312yes—
69Dic127⋊C4—22no—
70C22⋊Dic127—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—22no—
73C2×D4×C127—22no—
74C2×D4×C127C2312yes—
75C22×C4×C127C2311yescharacteristic
76D8×C127—22no—
77M16×C127—22no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—22no—
82C22×D127—22no—
83C22×D127—24no—
84C22×D127—22no—
85C22×D127—44no—
86C22×D127—44no—
87C22×D127—44no—
88C22×D127C2×D411yescharacteristic
89C22×D127C2×D411yescharacteristic
90D508—22no—
91D508—22no—
92D508—44no—
93C22⋊D127—44no—
94C22⋊D127—44no—
95C22⋊D127—44no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C4×D127—22no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—22no—
104C4×D127—44no—
105C4×D127C2×D412yes—
106C4×D127#4112yes—
107C127⋊C8—21no—
108C127⋊C8C2×D412yescomplemented
109C23×C127—22no—
110C23×C127C2×D411yescharacteristic
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—44no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2411yescharacteristic
119C8×C127—21no—
120C8×C127C2×D412yescomplemented
121C22×C127—21no—
122C22×C127—22no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127C22×D411yescharacteristic
127C2×D127—21no—
128C2×D127—22no—
129C2×D127—44no—
130C2×D127—42no—
131C2×D127—42no—
132C2×D127—42no—
133C2×D127—88no—
134C2×D127#12112yes—
135C2×D127AGL(1,ℤ/8)14yes—
136C4×C127—21no—
137C4×C127—42no—
138C4×C127C22×D411yescharacteristic
139C4×C127C2×#4111yescharacteristic
140Dic127—21no—
141Dic127—42no—
142Dic127#12112yes—
143C2×C127—41no—
144C2×C127—82no—
145C2×C127C2×#12111yescharacteristic
146C2×C127C2×AGL(1,ℤ/8)12yes—
147D127—41no—
148D127—82no—
149D127#46914yesdirect product factor
150C2×#469—1271nomaximal, Sylow
151C127C2×#46911yescharacteristic, complemented, Sylow
152C22×D8—1271no—
153C22×M16—1271no—
154C2×#121—1272no—
155#469—1271no—
156#469—1271no—
157#469—1272no—
158#469—1271no—
159#469—1272no—
160#469D12711yescharacteristic, direct product factor
161C2×#106—1271no—
162C2×#102—1272no—
163C22×D4—1272no—
164C22×D4—2542no—
165C23×C4—1271no—
166C2×D8—1272no—
167C2×D8—1271no—
168C2×D8—2544no—
169C2×D8C2×D12711yescharacteristic, complemented
170C2×M16—1271no—
171C2×M16—1271no—
172C2×M16—1271no—
173C2×M16—2544no—
174C2×M16C2×D12711yescharacteristic, complemented
175C22×C8—1271no—
176C22×C8—1271no—
177#121—1272no—
178#121—1272no—
179#121—1272no—
180#121—2542no—
181#121—2542no—
182#121C2×D12712yescomplemented
183C2×C4⋊C4—1272no—
184C2×C22⋊C4—2542no—
185#106—1271no—
186#106—1272no—
187#106C2×D12711yescharacteristic, complemented
188#102—1272no—
189#102—1272no—
190#102—1272no—
191#102C2×D12712yescomplemented
192C24—1271no—
193C24—2542no—
194C2×D4—22no—
195C2×D4—1272no—
196C2×D4—5088no—
197C2×D4—2542no—
198C2×D4—2544no—
199C2×D4—2544no—
200C2×D4—2542no—
201C2×D4—2542no—
202C2×D4—2542no—
203C2×D4C22×D12712yes—
204C22×C4—1271no—
205C22×C4—1271no—
206C22×C4—1271no—
207C22×C4—1271no—
208C22×C4—1271no—
209C22×C4—2542no—
210C22×C4—2542no—
211C22×C4—2542no—
212C22×C4C22×D12711yescharacteristic
213D8—22no—
214D8—2544no—
215D8—2542no—
216M16—22no—
217M16—2542no—
218M16—2542no—
219M16—2542no—
220C2×C8—1271no—
221C2×C8—1274no—
222C2×C8—1271no—
223C2×C8—2542no—
224C2×C8C22×D12711yescharacteristic
225C2×C8C22×D12711yescharacteristic
226C4⋊C4—1272no—
227C4⋊C4—2542no—
228C4⋊C4C22×D12712yes—
229C22⋊C4—22no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C22⋊C4—2542no—
233C23—22no—
234C23—1271no—
235C23—1271no—
236C23—5084no—
237C23—5084no—
238C23—5084no—
239C23—2542no—
240C23—2544no—
241C23—2542no—
242C23D4×D12711yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—5084no—
247D4—5084no—
248D4—5084no—
249D4—5084no—
250D4—2542no—
251D4—2542no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—1271no—
256C2×C4—1272no—
257C2×C4—1272no—
258C2×C4—1271no—
259C2×C4—5084no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4D4×D12711yescharacteristic
267C2×C4C23×D12711yescharacteristic
268C8—21no—
269C8—1272no—
270C8—2541no—
271C8D4×D12712yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—1272no—
278C22—1274no—
279C22—10168no—
280C22—5084no—
281C22—5082no—
282C22—5082no—
283C22—5082no—
284C22—2541no—
285C22—2542no—
286C22C2×D4×D12711yescharacteristic
287C4—21no—
288C4—42no—
289C4—1272no—
290C4—5082no—
291C4—2541no—
292C4C2×D4×D12711yescharacteristic
293C4#41×D12711yescharacteristic
294C2—41no—
295C2—82no—
296C2—1274no—
297C2—10162no—
298C2—5081no—
299C2#121×D12711yescharacteristic
300C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127C2×#469
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D127C2semidirect
2. C2×M16×D127C2semidirect
3. #121×D127C2semidirect
4. #22550130C2semidirect
5. #22550129C2semidirect
6. #22550128C2semidirect
7. #22550127C2semidirect
8. #22550126C2semidirect
9. #106×D127C2semidirect
10. #102×D127C2semidirect
11. #469×C127C2semidirect
15. C2×D1016C22semidirect
16. C2×#14523233C22semidirect
18. C2×#14523232C22semidirect
19. C2×#14523231C22semidirect
20. C2×#14523231C22semidirect
24. #16861184C22semidirect
25. #16861183C22semidirect
28. #16861180C22semidirect
31. #16861141C22semidirect
32. #16861140C22semidirect
33. #16861131C22semidirect
34. #16861130C22semidirect
35. #16861129C22semidirect
36. C2×D8×C127C22semidirect
37. C2×M16×C127C22semidirect
38. #121×C127C22semidirect
39. #106×C127C22semidirect
40. #102×C127C22semidirect
67. C8×D127D4semidirect
108. C127⋊C8C2×D4semidirect
120. C8×C127C2×D4semidirect
149. D127#469direct (N × H)
151. C127C2×#469semidirect
160. #469D127direct (N × H)
169. C2×D8C2×D127semidirect
174. C2×M16C2×D127semidirect
182. #121C2×D127semidirect
187. #106C2×D127semidirect
191. #102C2×D127semidirect
271. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1272
C2×M16×D1272
#121×D1272
#225501302
#225501292
#225501282
#225501272
#225501262
#106×D1272
#102×D1272
#469×C1272
C2×#469127