Finite Group Databasebeta

← C127×#3019 #30635971

Subgroups — C127×#3019 #30635971

Order 16256.

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

normal subgroups only

This group has 762 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
1C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
2#575×C127C212yesmaximal, complemented
3#535×C127C211yesmaximal, characteristic, complemented
4#455×C127C214yesmaximal, complemented
5#453×C127C214yesmaximal, complemented
6#421×C127C212yesmaximal, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8H(5,𝔽2)×C127—24no—
9H(5,𝔽2)×C127—22no—
10H(5,𝔽2)×C127C2214yes—
11C2×#41×C127—22no—
12C2×#41×C127C2212yes—
13C2×#41×C127C2211yescharacteristic, complemented
14C22×D4×C127—22no—
15C22×D4×C127—22no—
16C22×D4×C127C2211yescharacteristic
17#137×C127—22no—
18#137×C127C2214yescomplemented
19AGL(1,ℤ/8)×C127—22no—
20AGL(1,ℤ/8)×C127C2214yescomplemented
21#135×C127—24no—
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#100×C127C2214yescomplemented
30C24×C127—22no—
31C24×C127D412yes—
32#41×C127—28no—
33#41×C127—28no—
34#41×C127—22no—
35#41×C127—22no—
36#41×C127—22no—
37#41×C127—22no—
38#41×C127D414yescomplemented
39#41×C127C2314yes—
40C2×Q8×C127D412yescomplemented
41C2×Q8×C127C2311yescharacteristic, complemented
42C2×D4×C127—24no—
43C2×D4×C127—28no—
44C2×D4×C127—28no—
45C2×D4×C127—22no—
46C2×D4×C127—28no—
47C2×D4×C127—28no—
48C2×D4×C127—21no—
49C2×D4×C127—24no—
50C2×D4×C127—22no—
51C2×D4×C127—22no—
52C2×D4×C127D412yes—
53C2×D4×C127D412yescomplemented
54C2×D4×C127C2312yes—
55C2×D4×C127C2311yescharacteristic, complemented
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60Q16×C127—24no—
61SD16×C127—24no—
62SD16×C127—24no—
63D8×C127—24no—
64M16×C127—22no—
65M16×C127C2314yescomplemented
66C2×C8×C127—22no—
67C4⋊C4×C127—21no—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—24no—
70C42×C127C2312yescomplemented
71C23×C127—28no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—21no—
76C23×C127—82no—
77C23×C127—84no—
78C23×C127C2×D414yes—
79C23×C127C2×D411yescharacteristic
80Q8×C127—22no—
81Q8×C127—22no—
82Q8×C127C2×D414yes—
83D4×C127—216no—
84D4×C127—24no—
85D4×C127—24no—
86D4×C127—24no—
87D4×C127—22no—
88D4×C127—24no—
89D4×C127—22no—
90D4×C127—41no—
91D4×C127—41no—
92D4×C127—84no—
93D4×C127C2×D414yes—
94C2×C4×C127—28no—
95C2×C4×C127—22no—
96C2×C4×C127—24no—
97C2×C4×C127—24no—
98C2×C4×C127—22no—
99C2×C4×C127—21no—
100C2×C4×C127—42no—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2411yescharacteristic
105C8×C127—24no—
106C22×C127—24no—
107C22×C127—24no—
108C22×C127—41no—
109C22×C127—44no—
110C22×C127—44no—
111C22×C127—44no—
112C22×C127—42no—
113C22×C127—41no—
114C22×C127—82no—
115C22×C127—88no—
116C22×C127#12011yescharacteristic
117C22×C127#12011yescharacteristic
118C22×C127C22×D411yescharacteristic
119C4×C127—24no—
120C4×C127—42no—
121C4×C127—41no—
122C4×C127#12012yes—
123C4×C127C22×D412yes—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—44no—
128C2×C127—44no—
129C2×C127—81no—
130C2×C127C2×#12011yescharacteristic
131#3019C12711yesmaximal, characteristic, direct product factor, Sylow
132C127#301911yescharacteristic, direct product factor, Sylow
133C2×H(5,𝔽2)C2×C12711yescharacteristic, complemented
134#575C2×C12712yescomplemented
135#535C2×C12711yescharacteristic, complemented
136#455C2×C12714yescomplemented
137#453C2×C12714yescomplemented
138#421C2×C12712yescomplemented
139C2×#100C2×C12711yescharacteristic, complemented
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C22×C12714yes—
143C2×#41—22no—
144C2×#41C22×C12712yes—
145C2×#41C22×C12711yescharacteristic, complemented
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22×C12711yescharacteristic
149#137—22no—
150#137C22×C12714yescomplemented
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)C22×C12714yescomplemented
153#135—24no—
154C2×M16C22×C12712yescomplemented
155#127C22×C12712yescomplemented
156#124C22×C12712yescomplemented
157#121—22no—
158#120—22no—
159#117C22×C12711yescharacteristic, complemented
160#104C22×C12718yescomplemented
161#100C22×C12714yescomplemented
162C24—22no—
163C24D4×C12712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—22no—
169#41—22no—
170#41D4×C12714yescomplemented
171#41C23×C12714yes—
172C2×Q8D4×C12712yescomplemented
173C2×Q8C23×C12711yescharacteristic, complemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—21no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4D4×C12712yes—
185C2×D4D4×C12712yescomplemented
186C2×D4C23×C12712yes—
187C2×D4C23×C12711yescharacteristic, complemented
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4C23×C12711yescharacteristic
192Q16—24no—
193SD16—24no—
194SD16—24no—
195D8—24no—
196M16—22no—
197M16C23×C12714yescomplemented
198C2×C8—22no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C23×C12712yescomplemented
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—82no—
209C23—84no—
210C23C2×D4×C12714yes—
211C23C2×D4×C12711yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8C2×D4×C12714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—22no—
222D4—41no—
223D4—41no—
224D4—84no—
225D4C2×D4×C12714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—42no—
233C2×C4C2×D4×C12711yescharacteristic
234C2×C4C2×D4×C12712yes—
235C2×C4C2×D4×C12712yes—
236C2×C4C24×C12711yescharacteristic
237C8—24no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—42no—
245C22—41no—
246C22—82no—
247C22—88no—
248C22#120×C12711yescharacteristic
249C22#120×C12711yescharacteristic
250C22C22×D4×C12711yescharacteristic
251C4—24no—
252C4—42no—
253C4—41no—
254C4#120×C12712yes—
255C4C22×D4×C12712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—81no—
262C2C2×#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. C2×H(5,𝔽2)×C127C2semidirect
2. #575×C127C2semidirect
3. #535×C127C2semidirect
4. #455×C127C2semidirect
5. #453×C127C2semidirect
6. #421×C127C2semidirect
7. C2×#100×C127C2semidirect
13. C2×#41×C127C22semidirect
18. #137×C127C22semidirect
20. AGL(1,ℤ/8)×C127C22semidirect
22. C2×M16×C127C22semidirect
23. #127×C127C22semidirect
24. #124×C127C22semidirect
27. #117×C127C22semidirect
28. #104×C127C22semidirect
29. #100×C127C22semidirect
38. #41×C127D4semidirect
40. C2×Q8×C127D4semidirect
41. C2×Q8×C127C23semidirect
53. C2×D4×C127D4semidirect
55. C2×D4×C127C23semidirect
65. M16×C127C23semidirect
70. C42×C127C23semidirect
131. #3019C127direct (N × H)
132. C127#3019direct (N × H)
133. C2×H(5,𝔽2)C2×C127semidirect
134. #575C2×C127semidirect
135. #535C2×C127semidirect
136. #455C2×C127semidirect
137. #453C2×C127semidirect
138. #421C2×C127semidirect
139. C2×#100C2×C127semidirect
145. C2×#41C22×C127semidirect
150. #137C22×C127semidirect
152. AGL(1,ℤ/8)C22×C127semidirect
154. C2×M16C22×C127semidirect
155. #127C22×C127semidirect
156. #124C22×C127semidirect
159. #117C22×C127semidirect
160. #104C22×C127semidirect
161. #100C22×C127semidirect
170. #41D4×C127semidirect
172. C2×Q8D4×C127semidirect
173. C2×Q8C23×C127semidirect
185. C2×D4D4×C127semidirect
187. C2×D4C23×C127semidirect
197. M16C23×C127semidirect
202. C42C23×C127semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C1272
#575×C1272
#535×C1272
#455×C1272
#453×C1272
#421×C1272
C2×#100×C1272
#3019127