Finite Group Databasebeta

← C2×#22550010 #30647906

Subgroups — C2×#22550010 #30647906

Order 16256.

1048 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C22×#14523233C212yesmaximal, complemented
2C2×#16861183C211yesmaximal, characteristic, complemented
3#22550010C218yesmaximal, direct product factor
4C2×#16861130C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6C23×D4×C127C211yesmaximal, characteristic, complemented
7C22×D508C2211yescharacteristic, complemented
8C22×C22⋊D127—21no—
9C2×#14523233—28no—
10C2×#14523233C2218yescomplemented
11C22×C127⋊C8C2212yescomplemented
12#16861183—22no—
13#16861183C2214yescomplemented
14C2×C4⋊Dic127C2211yescharacteristic, complemented
15C2×(C2×D127)⋊C4—21no—
16#16861130C2218yescomplemented
17C22⋊(C127⋊C8)C2214yescomplemented
18C25×C127—21no—
19C22×D4×C127—28no—
20C22×D4×C127—21no—
21C22×D4×C127—22no—
22C22×D4×C127C2212yes—
23C22×D4×C127C2214yescomplemented
24C23×C4×C127C2211yescharacteristic
25C23×D127—21no—
26C2×D508—24no—
27C2×D508—21no—
28C2×D508C2312yescomplemented
29C2×C22⋊D127—24no—
30C2×C22⋊D127—44no—
31C22×Dic127—21no—
32#14523233—216no—
33C2×C127⋊C8—24no—
34C2×C127⋊C8C2314yescomplemented
35C4⋊Dic127—21no—
36C4⋊Dic127C2312yescomplemented
37(C2×D127)⋊C4—24no—
38C24×C127—28no—
39C24×C127—22no—
40C24×C127—24no—
41C24×C127—44no—
42C24×C127—44no—
43C24×C127D411yescharacteristic
44C2×D4×C127—28no—
45C2×D4×C127—216no—
46C2×D4×C127—216no—
47C2×D4×C127—22no—
48C2×D4×C127—24no—
49C2×D4×C127—24no—
50C2×D4×C127D418yescomplemented
51C2×D4×C127C2314yes—
52C22×C4×C127—24no—
53C22×C4×C127D411yescharacteristic
54C22×C4×C127D412yes—
55C22×C4×C127C2312yes—
56C22×C4×C127C2311yescharacteristic
57C22×D127—22no—
58C22×D127—42no—
59C22×D127—42no—
60C22×D127—42no—
61D508—24no—
62D508—42no—
63C22⋊D127—48no—
64C2×Dic127—22no—
65C2×Dic127—42no—
66C127⋊C8—24no—
67C23×C127—22no—
68C23×C127—22no—
69C23×C127—216no—
70C23×C127—28no—
71C23×C127—24no—
72C23×C127—48no—
73C23×C127—432no—
74C23×C127—44no—
75C23×C127—48no—
76C23×C127—44no—
77C23×C127D814yescomplemented
78C23×C127C2×D412yes—
79C23×C127C2×D411yescharacteristic
80D4×C127—216no—
81D4×C127—28no—
82D4×C127—24no—
83D4×C127C2×D418yescomplemented
84C2×C4×C127—22no—
85C2×C4×C127—24no—
86C2×C4×C127—24no—
87C2×C4×C127C2×D412yes—
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D414yes—
90C2×C4×C127C2411yescharacteristic
91C22×C127—22no—
92C22×C127—22no—
93C22×C127—28no—
94C22×C127—28no—
95C22×C127—432no—
96C22×C127—44no—
97C22×C127—44no—
98C22×C127—416no—
99C22×C127#12012yes—
100C22×C127C2×D812yes—
101C22×C127C2×D814yescomplemented
102C22×C127AGL(1,ℤ/8)12yes—
103C22×C127C22×D411yescharacteristic
104C2×D127—42no—
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—84no—
108C4×C127—22no—
109C4×C127#12012yes—
110C4×C127C22×D412yes—
111Dic127—42no—
112C2×C127—24no—
113C2×C127—48no—
114C2×C127#44714yescomplemented
115C2×C127C2×#12011yescharacteristic
116C2×C127C22×D811yescharacteristic
117C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
118D127—82no—
119C2×#447—1271nomaximal, Sylow
120C127C2×#44711yescharacteristic, complemented, Sylow
121C23×D4D12711yescharacteristic, complemented
122C22×D8—1272no—
123C2×#121—1271no—
124#447—1278no—
125C2×#102—1272no—
126C2×C22⋊C8—1271no—
127C25—21no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—1271no—
132C22×D4—2541no—
133C22×D4C2×D12712yes—
134C22×D4C2×D12714yescomplemented
135C23×C4C2×D12711yescharacteristic
136C2×D8—1278no—
137C2×D8—2548no—
138C22×C8—1272no—
139#121—1274no—
140#121—2542no—
141C2×C4⋊C4—1271no—
142C2×C22⋊C4—2541no—
143#102—1278no—
144C22⋊C8—1274no—
145C24—28no—
146C24—24no—
147C24—22no—
148C24—44no—
149C24—44no—
150C24—2541no—
151C24C22⋊D12711yescharacteristic
152C2×D4—28no—
153C2×D4—216no—
154C2×D4—216no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—24no—
158C2×D4—1272no—
159C2×D4—5084no—
160C2×D4—2544no—
161C2×D4—2541no—
162C2×D4—2544no—
163C2×D4C22⋊D12718yescomplemented
164C2×D4C22×D12714yes—
165C22×C4—24no—
166C22×C4—2541no—
167C22×C4C22⋊D12712yes—
168C22×C4C22⋊D12711yescharacteristic
169C22×C4C22×D12712yes—
170C22×C4C22×D12711yescharacteristic
171D8—25416no—
172C2×C8—1274no—
173C2×C8—2544no—
174C4⋊C4—1272no—
175C4⋊C4—2541no—
176C22⋊C4—2544no—
177C23—22no—
178C23—22no—
179C23—216no—
180C23—24no—
181C23—28no—
182C23—48no—
183C23—432no—
184C23—44no—
185C23—48no—
186C23—44no—
187C23—5082no—
188C23—5082no—
189C23—5082no—
190C23—2542no—
191C23#1452323314yescomplemented
192C23C2×C22⋊D12712yes—
193C23C2×C22⋊D12711yescharacteristic
194D4—216no—
195D4—24no—
196D4—28no—
197D4—5088no—
198D4—5082no—
199D4—2544no—
200D4C2×C22⋊D12718yescomplemented
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—5082no—
205C2×C4—2542no—
206C2×C4C2×C22⋊D12712yes—
207C2×C4C2×C22⋊D12714yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C23×D12711yescharacteristic
210C8—2544no—
211C22—22no—
212C22—22no—
213C22—28no—
214C22—28no—
215C22—432no—
216C22—44no—
217C22—44no—
218C22—416no—
219C22—10164no—
220C22—5082no—
221C22—5082no—
222C22—5082no—
223C22#1686117812yes—
224C22C2×#1452323312yes—
225C22C2×#1452323314yescomplemented
226C22#1686123512yes—
227C22C22×C22⋊D12711yescharacteristic
228C4—22no—
229C4—5082no—
230C4#1686117812yes—
231C4C22×C22⋊D12712yes—
232C2—24no—
233C2—48no—
234C2—10162no—
235C2#2255001014yesdirect product factor
236C2C2×#1686117811yescharacteristic
237C2C22×#1452323311yescharacteristic
238C2C2×#1686123511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC23×D4×C127C2
socleC23×C127C2×D4
hypercenterC23×D4D127
hypocenterC127C2×#447
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C2×#16861183C2semidirect
3. #22550010C2direct (N × H)
4. C2×#16861130C2semidirect
5. C2×C22⋊(C127⋊C8)C2semidirect
6. C23×D4×C127C2semidirect
7. C22×D508C22semidirect
10. C2×#14523233C22semidirect
11. C22×C127⋊C8C22semidirect
13. #16861183C22semidirect
14. C2×C4⋊Dic127C22semidirect
16. #16861130C22semidirect
17. C22⋊(C127⋊C8)C22semidirect
23. C22×D4×C127C22semidirect
28. C2×D508C23semidirect
34. C2×C127⋊C8C23semidirect
36. C4⋊Dic127C23semidirect
50. C2×D4×C127D4semidirect
77. C23×C127D8semidirect
83. D4×C127C2×D4semidirect
101. C22×C127C2×D8semidirect
114. C2×C127#447semidirect
120. C127C2×#447semidirect
121. C23×D4D127semidirect
134. C22×D4C2×D127semidirect
163. C2×D4C22⋊D127semidirect
191. C23#14523233semidirect
200. D4C2×C22⋊D127semidirect
225. C22C2×#14523233semidirect
235. C2#22550010direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168611832
#225500102
C2×#168611302
C2×C22⋊(C127⋊C8)2
C23×D4×C1272
C2×#447127