Finite Group Databasebeta

← C2×#22550044 #30648054

Subgroups — C2×#22550044 #30648054

Order 16256.

844 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861234C212yesmaximal, complemented
2C2×#16861205C211yesmaximal, characteristic, complemented
3#22550044C218yesmaximal, direct product factor
4C2×#16861136C212yesmaximal, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
7C22×D508C2211yescharacteristic, complemented
8C22×C22⋊D127—22no—
9#16861234—24no—
10#16861234C2218yescomplemented
11C2×#14523235—22no—
12C2×#14523233—22no—
13C2×#14523232C2212yescomplemented
14#16861205—22no—
15#16861205C2214yescomplemented
16C2×C4×Dic127C2211yescharacteristic, complemented
17#16861136C2218yescomplemented
18#16861126C2214yescomplemented
19H(5,𝔽2)×C127—24no—
20H(5,𝔽2)×C127—22no—
21H(5,𝔽2)×C127C2214yescomplemented
22C2×#41×C127—22no—
23C2×#41×C127C2212yes—
24C22×D4×C127—22no—
25C22×D4×C127—22no—
26C22×D4×C127C2211yescharacteristic
27C23×D127—21no—
28C2×D508—22no—
29C2×D508—22no—
30C2×D508—21no—
31C2×D508C2312yescomplemented
32C2×C22⋊D127—28no—
33C2×C22⋊D127—48no—
34C22×Dic127—21no—
35#14523235—28no—
36#14523233—28no—
37#14523232—22no—
38#14523232C2314yescomplemented
39C2×C127⋊C8—22no—
40C4×Dic127—21no—
41C4×Dic127C2312yescomplemented
42C24×C127—22no—
43C24×C127D412yes—
44#41×C127—28no—
45#41×C127—28no—
46#41×C127—22no—
47#41×C127—22no—
48#41×C127D414yescomplemented
49#41×C127C2314yes—
50C2×Q8×C127D412yescomplemented
51C2×D4×C127—24no—
52C2×D4×C127—28no—
53C2×D4×C127—28no—
54C2×D4×C127—22no—
55C2×D4×C127—28no—
56C2×D4×C127—28no—
57C2×D4×C127—22no—
58C2×D4×C127—22no—
59C2×D4×C127D412yes—
60C2×D4×C127D412yescomplemented
61C2×D4×C127C2312yes—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127C2311yescharacteristic
65C22×D127—22no—
66C22×D127—44no—
67C22×D127—41no—
68C22×D127—41no—
69D508—24no—
70D508—42no—
71C22⋊D127—416no—
72C2×Dic127—24no—
73C2×Dic127—22no—
74C127⋊C8—24no—
75C23×C127—28no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—22no—
79C23×C127—44no—
80C23×C127—84no—
81C23×C127C2×D411yescharacteristic
82C23×C127C2×D414yes—
83Q8×C127—22no—
84Q8×C127C2×D414yescomplemented
85D4×C127—216no—
86D4×C127—24no—
87D4×C127—24no—
88D4×C127—24no—
89D4×C127—22no—
90D4×C127—24no—
91D4×C127C2×D414yescomplemented
92C2×C4×C127—28no—
93C2×C4×C127—22no—
94C2×C4×C127—24no—
95C2×C4×C127—24no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2411yescharacteristic
100C22×C127—22no—
101C22×C127—24no—
102C22×C127—24no—
103C22×C127—44no—
104C22×C127—44no—
105C22×C127—44no—
106C22×C127—82no—
107C22×C127—88no—
108C22×C127#12011yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127C22×D411yescharacteristic
111C2×D127—44no—
112C2×D127—42no—
113C2×D127—82no—
114C2×D127—82no—
115C4×C127—24no—
116C4×C127#12012yes—
117C4×C127C22×D412yes—
118Dic127—24no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—44no—
122C2×C127—44no—
123C2×C127#45312yescomplemented
124C2×C127C2×#12011yescharacteristic
125D127—82no—
126C2×#453—1271nomaximal, Sylow
127C127C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)D12711yescharacteristic, complemented
129C2×AGL(1,ℤ/8)—1272no—
130C2×#127—1271no—
131#453—1278no—
132C2×#104—1272no—
133C2×#100—1271no—
134H(5,𝔽2)—24no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)C2×D12714yescomplemented
137C2×#41—22no—
138C2×#41C2×D12712yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—1271no—
142C22×D4—2542no—
143C22×D4C2×D12711yescharacteristic
144AGL(1,ℤ/8)—1278no—
145AGL(1,ℤ/8)—2544no—
146C2×SD16—2542no—
147C2×D8—2542no—
148C2×M16—1272no—
149#127—1274no—
150#127—2542no—
151C2×C42—1271no—
152#104—1278no—
153#100—1274no—
154C24—22no—
155C24—2541no—
156C24C22⋊D12712yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41C22⋊D12714yescomplemented
162#41C22×D12714yes—
163C2×Q8C22⋊D12712yescomplemented
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—1272no—
173C2×D4—5088no—
174C2×D4—2548no—
175C2×D4—2541no—
176C2×D4—2542no—
177C2×D4—2542no—
178C2×D4C22⋊D12712yes—
179C2×D4C22⋊D12712yescomplemented
180C2×D4C22×D12712yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—2541no—
184C22×C4C22×D12711yescharacteristic
185SD16—2548no—
186D8—2548no—
187M16—1274no—
188M16—2542no—
189C2×C8—2542no—
190C42—1272no—
191C42—2541no—
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—44no—
197C23—84no—
198C23—5084no—
199C23—5081no—
200C23—5081no—
201C23—2542no—
202C23C2×C22⋊D12711yescharacteristic
203C23C2×C22⋊D12714yes—
204Q8—22no—
205Q8C2×C22⋊D12714yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—24no—
211D4—22no—
212D4—50816no—
213D4—5082no—
214D4—2544no—
215D4C2×C22⋊D12714yescomplemented
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—2544no—
221C2×C4—2542no—
222C2×C4C2×C22⋊D12712yes—
223C2×C4C2×C22⋊D12712yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—2544no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—82no—
234C22—88no—
235C22—10162no—
236C22—10162no—
237C22—5084no—
238C22—5082no—
239C22#1686117811yescharacteristic
240C22#1686117811yescharacteristic
241C22C22×C22⋊D12711yescharacteristic
242C4—24no—
243C4—2544no—
244C4#1686117812yes—
245C4C22×C22⋊D12712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—10162no—
251C2#2255004412yesdirect product factor
252C2C2×#1686117811yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×H(5,𝔽2)×C127C2
socleC22×C127#120
hypercenterC2×H(5,𝔽2)D127
hypocenterC127C2×#453
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861205C2semidirect
3. #22550044C2direct (N × H)
4. C2×#16861136C2semidirect
5. C2×#16861126C2semidirect
6. C2×H(5,𝔽2)×C127C2semidirect
7. C22×D508C22semidirect
10. #16861234C22semidirect
13. C2×#14523232C22semidirect
15. #16861205C22semidirect
16. C2×C4×Dic127C22semidirect
17. #16861136C22semidirect
18. #16861126C22semidirect
21. H(5,𝔽2)×C127C22semidirect
31. C2×D508C23semidirect
38. #14523232C23semidirect
41. C4×Dic127C23semidirect
48. #41×C127D4semidirect
50. C2×Q8×C127D4semidirect
60. C2×D4×C127D4semidirect
84. Q8×C127C2×D4semidirect
91. D4×C127C2×D4semidirect
123. C2×C127#453semidirect
127. C127C2×#453semidirect
128. C2×H(5,𝔽2)D127semidirect
136. H(5,𝔽2)C2×D127semidirect
161. #41C22⋊D127semidirect
163. C2×Q8C22⋊D127semidirect
179. C2×D4C22⋊D127semidirect
205. Q8C2×C22⋊D127semidirect
215. D4C2×C22⋊D127semidirect
251. C2#22550044direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168612052
#225500442
C2×#168611362
C2×#168611262
C2×H(5,𝔽2)×C1272
C2×#453127