Finite Group Databasebeta

← #470×D127 #30648328

Subgroups — #470×D127 #30648328

Order 16256.

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

normal subgroups only

This group has 492 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×Q16×D127C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic
3C22⋊Q8×D127C212yesmaximal
4#22550135C211yesmaximal, characteristic, complemented
5#22550134C211yesmaximal, characteristic, complemented
6#22550133C212yesmaximal, complemented
7#22550132C212yesmaximal, complemented
8#22550131C211yesmaximal, characteristic, complemented
9#106×D127C211yesmaximal, characteristic, complemented
10#103×D127C212yesmaximal, complemented
11#470×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2212yes—
13C22×C4×D127C2211yescharacteristic
14C2×Dic508C2211yescharacteristic, complemented
15C2×#14523238C2212yescomplemented
16Q16×D127—24no—
17C2×#14523232C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20M16×D127—24no—
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861189C2212yes—
24#16861188C2212yes—
25#16861187—22no—
26#16861186—22no—
27#16861185C2212yes—
28C4⋊C4×D127—22no—
29C4⋊C4×D127C2212yes—
30C22⋊C4×D127—22no—
31#16861141C2211yescharacteristic, complemented
32#16861140C2212yescomplemented
33#16861134C2212yescomplemented
34#16861133C2212yescomplemented
35#16861132C2212yescomplemented
36C2×Q16×C127C2211yescharacteristic, complemented
37C2×M16×C127C2211yescharacteristic
38C22⋊Q8×C127C2212yes—
39#106×C127C2211yescharacteristic, complemented
40#103×C127C2212yescomplemented
41C23×D127D411yescharacteristic
42C2×Dic254—22no—
43C2×Dic254C2312yes—
44Q8×D127—24no—
45Q8×D127—24no—
46C22×Dic127D411yescharacteristic
47C2×C4×D127—22no—
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127D411yescharacteristic
52C2×C4×D127D411yescharacteristic
53C2×C4×D127C2311yescharacteristic
54C2×C4×D127C2311yescharacteristic
55Dic508—22no—
56#14523238—24no—
57#14523232—22no—
58#14523231—22no—
59#14523231—22no—
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2311yescharacteristic
62C8×D127—22no—
63C8×D127D414yes—
64C4⋊Dic127—22no—
65C4⋊Dic127C2312yes—
66Dic127⋊C4—22no—
67Dic127⋊C4—22no—
68Dic127⋊C4—22no—
69C22⋊Dic127—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C2×Q8×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—22no—
75M16×C127—22no—
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127—22no—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127—22no—
82C22×D127—24no—
83C22×D127C2×D411yescharacteristic
84C22×D127C2×D411yescharacteristic
85Dic254—22no—
86Dic254—22no—
87Dic254—44no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C4×D127—22no—
94C4×D127—21no—
95C4×D127—21no—
96C4×D127—22no—
97C4×D127—44no—
98C4×D127—44no—
99C4×D127C2×D412yes—
100C4×D127#4112yes—
101C127⋊C8—21no—
102C127⋊C8C2×D412yes—
103C23×C127C2×D411yescharacteristic
104Q8×C127—22no—
105Q8×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—21no—
113C8×C127C2×D412yes—
114C22×C127—21no—
115C22×C127—22no—
116C22×C127C22×D411yescharacteristic
117C2×D127—21no—
118C2×D127—22no—
119C2×D127—44no—
120C2×D127#12112yes—
121C2×D127#13714yes—
122C4×C127—21no—
123C4×C127—42no—
124C4×C127—42no—
125C4×C127C22×D411yescharacteristic
126C4×C127C2×#4111yescharacteristic
127Dic127—21no—
128Dic127—42no—
129Dic127—42no—
130Dic127#12112yes—
131C2×C127—41no—
132C2×C127C2×#12111yescharacteristic
133C2×C127C2×#13712yes—
134D127—41no—
135D127#47014yesdirect product factor
136C2×#470—1271nomaximal, Sylow
137C127C2×#47011yescharacteristic, complemented, Sylow
138C22×Q16—1271no—
139C22×M16—1271no—
140C2×C22⋊Q8—1272no—
141#470—1271no—
142#470—1271no—
143#470—1272no—
144#470—1271no—
145#470—1272no—
146#470D12711yescharacteristic, direct product factor
147C2×#106—1271no—
148C2×#103—1272no—
149C22×Q8—1272no—
150C23×C4—1271no—
151C2×Q16—1272no—
152C2×Q16—1271no—
153C2×Q16—2544no—
154C2×Q16C2×D12711yescharacteristic, complemented
155C2×M16—1271no—
156C2×M16—1271no—
157C2×M16—1271no—
158C2×M16—2544no—
159C2×M16C2×D12711yescharacteristic
160C22×C8—1271no—
161C22×C8—1271no—
162C22⋊Q8—1272no—
163C22⋊Q8—1272no—
164C22⋊Q8—1272no—
165C22⋊Q8—2542no—
166C22⋊Q8—2542no—
167C22⋊Q8C2×D12712yes—
168C2×C4⋊C4—1272no—
169C2×C4⋊C4—2542no—
170C2×C22⋊C4—2542no—
171#106—1271no—
172#106—1272no—
173#106C2×D12711yescharacteristic, complemented
174#103—1272no—
175#103—1272no—
176#103—1272no—
177#103C2×D12712yescomplemented
178C24—1271no—
179C2×Q8—1272no—
180C2×Q8—2542no—
181C2×Q8—2544no—
182C2×Q8—2544no—
183C2×Q8C22×D12712yes—
184C22×C4—1271no—
185C22×C4—1271no—
186C22×C4—1271no—
187C22×C4—1271no—
188C22×C4—1271no—
189C22×C4—2542no—
190C22×C4—2542no—
191C22×C4—2542no—
192C22×C4—2542no—
193C22×C4C22×D12711yescharacteristic
194Q16—22no—
195Q16—2544no—
196Q16—2542no—
197M16—22no—
198M16—2542no—
199M16—2542no—
200M16—2542no—
201C2×C8—1271no—
202C2×C8—1274no—
203C2×C8—1271no—
204C2×C8—2542no—
205C2×C8C22×D12711yescharacteristic
206C2×C8C22×D12711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—1272no—
209C4⋊C4—2542no—
210C4⋊C4—2542no—
211C4⋊C4—2542no—
212C4⋊C4—2542no—
213C4⋊C4C22×D12712yes—
214C22⋊C4—22no—
215C22⋊C4—2542no—
216C22⋊C4—2542no—
217C22⋊C4—2542no—
218C23—1271no—
219C23—1271no—
220C23—2542no—
221C23—2544no—
222C23D4×D12711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—5084no—
226Q8—2542no—
227Q8—2542no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—1271no—
233C2×C4—1272no—
234C2×C4—1272no—
235C2×C4—1271no—
236C2×C4—5084no—
237C2×C4—5084no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4D4×D12711yescharacteristic
246C2×C4C23×D12711yescharacteristic
247C8—21no—
248C8—1272no—
249C8—2541no—
250C8D4×D12712yes—
251C22—21no—
252C22—22no—
253C22—1272no—
254C22—1274no—
255C22—5084no—
256C22—2541no—
257C22—2542no—
258C22C2×D4×D12711yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—1272no—
263C4—5082no—
264C4—5082no—
265C4—2541no—
266C4C2×D4×D12711yescharacteristic
267C4#41×D12711yescharacteristic
268C2—41no—
269C2—1274no—
270C2—5081no—
271C2#121×D12711yescharacteristic
272C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
prime radicalC23×D127D4
hypercenter#470D127
hypocenterC127C2×#470
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D127C2semidirect
4. #22550135C2semidirect
5. #22550134C2semidirect
6. #22550133C2semidirect
7. #22550132C2semidirect
8. #22550131C2semidirect
9. #106×D127C2semidirect
10. #103×D127C2semidirect
11. #470×C127C2semidirect
14. C2×Dic508C22semidirect
15. C2×#14523238C22semidirect
31. #16861141C22semidirect
32. #16861140C22semidirect
33. #16861134C22semidirect
34. #16861133C22semidirect
35. #16861132C22semidirect
36. C2×Q16×C127C22semidirect
39. #106×C127C22semidirect
40. #103×C127C22semidirect
135. D127#470direct (N × H)
137. C127C2×#470semidirect
146. #470D127direct (N × H)
154. C2×Q16C2×D127semidirect
173. #106C2×D127semidirect
177. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1272
C2×M16×D1272
C22⋊Q8×D1272
#225501352
#225501342
#225501332
#225501322
#225501312
#106×D1272
#103×D1272
#470×C1272
C2×#470127