Finite Group Databasebeta

← C2×#22550043 #30648053

Subgroups — C2×#22550043 #30648053

Order 16256.

844 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861254C211yesmaximal, characteristic, complemented
2C2×#16861235C212yesmaximal, complemented
3#22550043C218yesmaximal, direct product factor
4C2×#16861137C212yesmaximal, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6C2×#127×C127C211yesmaximal, characteristic, complemented
7#16861254—24no—
8#16861254—22no—
9#16861254C2214yescomplemented
10C2×#14523248C2212yes—
11C2×#14523247—22no—
12C22×C22⋊D127—22no—
13C2×D4×D127—22no—
14#16861235—24no—
15#16861235C2218yescomplemented
16C2×#14523236—22no—
17C2×#14523233—22no—
18C2×#14523232C2212yescomplemented
19#16861137C2218yescomplemented
20#16861126C2214yescomplemented
21C22×D4×C127—22no—
22C22×D4×C127C2211yescharacteristic
23C22×D4×C127C2211yescharacteristic, complemented
24#127×C127—22no—
25#127×C127C2214yescomplemented
26C2×C42×C127C2211yescharacteristic, complemented
27C23×D127—22no—
28#14523248—22no—
29#14523248—22no—
30#14523248D414yescomplemented
31#14523248C2314yes—
32#14523247—28no—
33#14523247—28no—
34C2×Dic254D412yescomplemented
35C2×D508D412yescomplemented
36C2×C22⋊D127—24no—
37C2×C22⋊D127—28no—
38C2×C22⋊D127—22no—
39C2×C22⋊D127—28no—
40D4×D127—28no—
41D4×D127—28no—
42C22×Dic127—22no—
43C2×C4×D127—22no—
44#14523236—28no—
45#14523233—28no—
46#14523232—22no—
47#14523232C2314yescomplemented
48C2×C127⋊C8—22no—
49C24×C127—21no—
50C24×C127D412yes—
51C2×D4×C127—28no—
52C2×D4×C127—22no—
53C2×D4×C127—22no—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—21no—
57C2×D4×C127—48no—
58C2×D4×C127D412yes—
59C2×D4×C127C2312yes—
60C2×D4×C127C2312yescomplemented
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63C42×C127—21no—
64C42×C127C2312yescomplemented
65C22×D127—22no—
66C22×D127—28no—
67C22×D127—84no—
68Dic254—22no—
69Dic254C2×D414yescomplemented
70D508—22no—
71D508C2×D414yescomplemented
72C22⋊D127—216no—
73C22⋊D127—24no—
74C22⋊D127—24no—
75C2×Dic127—28no—
76C2×Dic127—22no—
77C4×D127—24no—
78C4×D127—24no—
79C127⋊C8—24no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—22no—
83C23×C127—44no—
84C23×C127—44no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127C2×D411yescharacteristic
88C23×C127C2×D414yes—
89D4×C127—24no—
90D4×C127—24no—
91D4×C127—24no—
92D4×C127—416no—
93D4×C127—42no—
94C2×C4×C127—24no—
95C2×C4×C127—22no—
96C2×C4×C127C2×D412yes—
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C22×C127—22no—
101C22×C127—24no—
102C22×C127—44no—
103C22×C127—44no—
104C22×C127—44no—
105C22×C127—42no—
106C22×C127—82no—
107C22×C127—82no—
108C22×C127—82no—
109C22×C127#12011yescharacteristic
110C22×C127#12011yescharacteristic
111C22×C127C22×D411yescharacteristic
112C2×D127—24no—
113C2×D127—44no—
114C2×D127—88no—
115C4×C127—24no—
116C4×C127#12012yes—
117C4×C127C22×D412yes—
118Dic127—24no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—44no—
122C2×C127—82no—
123C2×C127#45312yescomplemented
124C2×C127C2×#12011yescharacteristic
125D127—44no—
126C2×#453—1271nomaximal, Sylow
127C127C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)—1271no—
129C2×AGL(1,ℤ/8)—1272no—
130C2×#127D12711yescharacteristic, complemented
131#453—1278no—
132C2×#104—1272no—
133C2×#100—1271no—
134H(5,𝔽2)—1274no—
135H(5,𝔽2)—2544no—
136H(5,𝔽2)—2542no—
137C2×#41—1272no—
138C2×#41—2542no—
139C22×D4—22no—
140C22×D4—2542no—
141C22×D4—2542no—
142C22×D4C2×D12711yescharacteristic
143C22×D4C2×D12711yescharacteristic, complemented
144AGL(1,ℤ/8)—1278no—
145AGL(1,ℤ/8)—2544no—
146C2×SD16—2542no—
147C2×D8—2542no—
148C2×M16—1272no—
149#127—22no—
150#127C2×D12714yescomplemented
151C2×C42C2×D12711yescharacteristic, complemented
152#104—1278no—
153#100—1274no—
154C24—21no—
155C24—2542no—
156C24C22⋊D12712yes—
157#41—1274no—
158#41—1274no—
159#41—2548no—
160#41—2548no—
161#41—2542no—
162#41—2542no—
163C2×Q8—1272no—
164C2×D4—28no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—21no—
170C2×D4—48no—
171C2×D4—1272no—
172C2×D4—2548no—
173C2×D4—2548no—
174C2×D4—2544no—
175C2×D4—2548no—
176C2×D4—2542no—
177C2×D4—2548no—
178C2×D4C22⋊D12712yes—
179C2×D4C22×D12712yes—
180C2×D4C22×D12712yescomplemented
181C22×C4—21no—
182C22×C4—2542no—
183C22×C4—2542no—
184C22×C4C22×D12711yescharacteristic
185SD16—2548no—
186D8—2548no—
187M16—1274no—
188M16—2542no—
189C2×C8—2542no—
190C42—21no—
191C42C22×D12712yescomplemented
192C23—22no—
193C23—22no—
194C23—22no—
195C23—44no—
196C23—44no—
197C23—41no—
198C23—41no—
199C23—10164no—
200C23—2548no—
201C23—2542no—
202C23C2×C22⋊D12711yescharacteristic
203C23C2×C22⋊D12714yes—
204Q8—1274no—
205Q8—2542no—
206D4—24no—
207D4—24no—
208D4—24no—
209D4—416no—
210D4—42no—
211D4—1274no—
212D4—25416no—
213D4—2544no—
214D4—2544no—
215D4—2542no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—2548no—
219C2×C4—2542no—
220C2×C4—2544no—
221C2×C4—2544no—
222C2×C4D4×D12712yes—
223C2×C4D4×D12712yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—2544no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—42no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—10168no—
237C22—5084no—
238C22—2544no—
239C22#1686117711yescharacteristic
240C22#1686117711yescharacteristic
241C22C22×C22⋊D12711yescharacteristic
242C4—24no—
243C4—2544no—
244C4#1686117712yes—
245C4C2×D4×D12712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—82no—
250C2—5084no—
251C2#2255004312yesdirect product factor
252C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#127×C127C2
socleC22×C127#120
hypercenterC2×#127D127
hypocenterC127C2×#453
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. C2×#16861235C2semidirect
3. #22550043C2direct (N × H)
4. C2×#16861137C2semidirect
5. C2×#16861126C2semidirect
6. C2×#127×C127C2semidirect
9. #16861254C22semidirect
15. #16861235C22semidirect
18. C2×#14523232C22semidirect
19. #16861137C22semidirect
20. #16861126C22semidirect
23. C22×D4×C127C22semidirect
25. #127×C127C22semidirect
26. C2×C42×C127C22semidirect
30. #14523248D4semidirect
34. C2×Dic254D4semidirect
35. C2×D508D4semidirect
47. #14523232C23semidirect
60. C2×D4×C127C23semidirect
64. C42×C127C23semidirect
69. Dic254C2×D4semidirect
71. D508C2×D4semidirect
123. C2×C127#453semidirect
127. C127C2×#453semidirect
130. C2×#127D127semidirect
143. C22×D4C2×D127semidirect
150. #127C2×D127semidirect
151. C2×C42C2×D127semidirect
180. C2×D4C22×D127semidirect
191. C42C22×D127semidirect
251. C2#22550043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612352
#225500432
C2×#168611372
C2×#168611262
C2×#127×C1272
C2×#453127