Finite Group Databasebeta

← C127×#3021 #30635973

Subgroups — C127×#3021 #30635973

Order 16256.

736 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 212 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
1#585×C127C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)×C127C212yesmaximal, complemented
3#535×C127C211yesmaximal, characteristic, complemented
4#454×C127C214yesmaximal, complemented
5#453×C127C214yesmaximal, complemented
6#421×C127C212yesmaximal, complemented
7#413×C127C211yesmaximal, characteristic, complemented
82−1+4×C127—22no—
92−1+4×C127C2212yescomplemented
10H(5,𝔽2)×C127—22no—
11H(5,𝔽2)×C127—22no—
12H(5,𝔽2)×C127C2212yescomplemented
13C2×#41×C127—24no—
14C2×#41×C127—22no—
15C2×#41×C127C2212yes—
16C2×#41×C127C2211yescharacteristic
17C22×D4×C127C2211yescharacteristic, complemented
18AGL(1,ℤ/8)×C127—24no—
19AGL(1,ℤ/8)×C127C2218yescomplemented
20C2×SD16×C127—22no—
21C2×D8×C127—22no—
22C2×M16×C127C2212yescomplemented
23#127×C127C2212yescomplemented
24#124×C127C2212yescomplemented
25#121×C127—22no—
26#120×C127—22no—
27#117×C127C2211yescharacteristic, complemented
28#104×C127C2218yescomplemented
29#101×C127C2212yescomplemented
30#100×C127C2212yescomplemented
31C24×C127—21no—
32#41×C127—24no—
33#41×C127—28no—
34#41×C127—24no—
35#41×C127—24no—
36#41×C127—24no—
37#41×C127—22no—
38#41×C127—22no—
39#41×C127—24no—
40#41×C127—22no—
41#41×C127—22no—
42#41×C127D414yescomplemented
43#41×C127C2314yes—
44C2×Q8×C127—22no—
45C2×Q8×C127—24no—
46C2×Q8×C127D412yescomplemented
47C2×Q8×C127C2311yescharacteristic
48C2×D4×C127—28no—
49C2×D4×C127—22no—
50C2×D4×C127—24no—
51C2×D4×C127—22no—
52C2×D4×C127—24no—
53C2×D4×C127—24no—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—21no—
57C2×D4×C127D412yes—
58C2×D4×C127D412yescomplemented
59C2×D4×C127C2312yes—
60C2×D4×C127C2311yescharacteristic
61C22×C4×C127—24no—
62C22×C4×C127—22no—
63C22×C4×C127D412yes—
64C22×C4×C127C2311yescharacteristic
65SD16×C127—28no—
66D8×C127—28no—
67M16×C127—22no—
68M16×C127C2314yes—
69C2×C8×C127—22no—
70C4⋊C4×C127—21no—
71C22⋊C4×C127—21no—
72C22⋊C4×C127—24no—
73C42×C127C2312yes—
74C23×C127—24no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—42no—
78C23×C127—82no—
79C23×C127C2×D412yes—
80C23×C127C2×D411yescharacteristic
81Q8×C127—24no—
82Q8×C127—22no—
83Q8×C127—22no—
84Q8×C127C2×D414yes—
85D4×C127—28no—
86D4×C127—24no—
87D4×C127—24no—
88D4×C127—24no—
89D4×C127—24no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—24no—
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—84no—
96D4×C127C2×D414yes—
97C2×C4×C127—24no—
98C2×C4×C127—28no—
99C2×C4×C127—22no—
100C2×C4×C127—24no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—24no—
104C2×C4×C127—22no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2411yescharacteristic
110C8×C127—24no—
111C22×C127—22no—
112C22×C127—24no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—44no—
116C22×C127—42no—
117C22×C127—82no—
118C22×C127—84no—
119C22×C127—84no—
120C22×C127#12012yes—
121C22×C127C22×D411yescharacteristic
122C4×C127—24no—
123C4×C127—22no—
124C4×C127—42no—
125C4×C127#12011yescharacteristic
126C4×C127#12011yescharacteristic
127C4×C127C22×D412yes—
128C2×C127—21no—
129C2×C127—22no—
130C2×C127—42no—
131C2×C127—44no—
132C2×C127—82no—
133C2×C127C2×#12011yescharacteristic
134#3021C12711yesmaximal, characteristic, direct product factor, Sylow
135C127#302111yescharacteristic, direct product factor, Sylow
136#585C2×C12711yescharacteristic, complemented
137C2×AGL(1,ℤ/8)C2×C12712yescomplemented
138#535C2×C12711yescharacteristic, complemented
139#454C2×C12714yescomplemented
140#453C2×C12714yescomplemented
141#421C2×C12712yescomplemented
142#413C2×C12711yescharacteristic, complemented
1432−1+4—22no—
1442−1+4C22×C12712yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C22×C12712yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C22×C12712yes—
151C2×#41C22×C12711yescharacteristic
152C22×D4C22×C12711yescharacteristic, complemented
153AGL(1,ℤ/8)—24no—
154AGL(1,ℤ/8)C22×C12718yescomplemented
155C2×SD16—22no—
156C2×D8—22no—
157C2×M16C22×C12712yescomplemented
158#127C22×C12712yescomplemented
159#124C22×C12712yescomplemented
160#121—22no—
161#120—22no—
162#117C22×C12711yescharacteristic, complemented
163#104C22×C12718yescomplemented
164#101C22×C12712yescomplemented
165#100C22×C12712yescomplemented
166C24—21no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41D4×C12714yescomplemented
178#41C23×C12714yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8D4×C12712yescomplemented
182C2×Q8C23×C12711yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4D4×C12712yes—
193C2×D4D4×C12712yescomplemented
194C2×D4C23×C12712yes—
195C2×D4C23×C12711yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4D4×C12712yes—
199C22×C4C23×C12711yescharacteristic
200SD16—28no—
201D8—28no—
202M16—22no—
203M16C23×C12714yes—
204C2×C8—22no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C23×C12712yes—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—82no—
214C23C2×D4×C12712yes—
215C23C2×D4×C12711yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×D4×C12714yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—24no—
228D4—41no—
229D4—41no—
230D4—84no—
231D4C2×D4×C12714yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4C2×D4×C12712yes—
241C2×C4C2×D4×C12711yescharacteristic
242C2×C4C2×D4×C12712yes—
243C2×C4C2×D4×C12712yes—
244C2×C4C24×C12711yescharacteristic
245C8—24no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—44no—
251C22—42no—
252C22—82no—
253C22—84no—
254C22—84no—
255C22#120×C12712yes—
256C22C22×D4×C12711yescharacteristic
257C4—24no—
258C4—22no—
259C4—42no—
260C4#120×C12711yescharacteristic
261C4#120×C12711yescharacteristic
262C4C22×D4×C12712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—82no—
268C2C2×#120×C12711yescharacteristic
subgroupG/H
center, socleC2×C127C2×#120
commutator, Frattini, cosocleC2×C4C24×C127
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C127C2semidirect
2. C2×AGL(1,ℤ/8)×C127C2semidirect
3. #535×C127C2semidirect
4. #454×C127C2semidirect
5. #453×C127C2semidirect
6. #421×C127C2semidirect
7. #413×C127C2semidirect
9. 2−1+4×C127C22semidirect
12. H(5,𝔽2)×C127C22semidirect
17. C22×D4×C127C22semidirect
19. AGL(1,ℤ/8)×C127C22semidirect
22. C2×M16×C127C22semidirect
23. #127×C127C22semidirect
24. #124×C127C22semidirect
27. #117×C127C22semidirect
28. #104×C127C22semidirect
29. #101×C127C22semidirect
30. #100×C127C22semidirect
42. #41×C127D4semidirect
46. C2×Q8×C127D4semidirect
58. C2×D4×C127D4semidirect
134. #3021C127direct (N × H)
135. C127#3021direct (N × H)
136. #585C2×C127semidirect
137. C2×AGL(1,ℤ/8)C2×C127semidirect
138. #535C2×C127semidirect
139. #454C2×C127semidirect
140. #453C2×C127semidirect
141. #421C2×C127semidirect
142. #413C2×C127semidirect
144. 2−1+4C22×C127semidirect
147. H(5,𝔽2)C22×C127semidirect
152. C22×D4C22×C127semidirect
154. AGL(1,ℤ/8)C22×C127semidirect
157. C2×M16C22×C127semidirect
158. #127C22×C127semidirect
159. #124C22×C127semidirect
162. #117C22×C127semidirect
163. #104C22×C127semidirect
164. #101C22×C127semidirect
165. #100C22×C127semidirect
177. #41D4×C127semidirect
181. C2×Q8D4×C127semidirect
193. C2×D4D4×C127semidirect

Maximal subgroups

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

subgroupindex
#585×C1272
C2×AGL(1,ℤ/8)×C1272
#535×C1272
#454×C1272
#453×C1272
#421×C1272
#413×C1272
#3021127