Finite Group Databasebeta

← C2×#22549874 #30647239

Subgroups — C2×#22549874 #30647239

Order 16256.

816 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 229 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×#14523247C211yesmaximal, characteristic
2C22×#14523231C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549874C218yesmaximal, direct product factor
5C2×#16861134C211yesmaximal, characteristic, complemented
6C2×#16861132C211yesmaximal, characteristic, complemented
7C2×#16861130C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C2×#14523247—22no—
10C2×#14523247—24no—
11C2×#14523247C418yescomplemented
12C2×#14523247C2214yes—
13C22×Dic254C2211yescharacteristic
14C22×C22⋊D127—21no—
15C23×Dic127—21no—
16C22×C4×D127—21no—
17C22×C4×D127C2211yescharacteristic
18C2×#14523231—24no—
19C2×#14523231C2214yescomplemented
20C22×C127⋊C8C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4—21no—
23C4⋊C4×D127—22no—
24C4⋊C4×D127C2214yescomplemented
25#16861134C2214yescomplemented
26#16861132C2214yescomplemented
27#16861130C2214yescomplemented
28C22×D4×C127C2211yescharacteristic
29C22×C8×C127C2211yescharacteristic
30C2×C4⋊C4×C127C2211yescharacteristic
31#102×C127C2214yescomplemented
32C23×D127D411yescharacteristic
33#14523247—216no—
34#14523247—28no—
35#14523247C2×C4116yescomplemented
36C2×Dic254—21no—
37C2×Dic254—22no—
38C2×Dic254C2×C414yescomplemented
39C2×Dic254C2312yes—
40C2×C22⋊D127—28no—
41C2×C22⋊D127—24no—
42C22×Dic127—28no—
43C22×Dic127—24no—
44C22×Dic127—21no—
45C22×Dic127—21no—
46C22×Dic127D411yescharacteristic
47C2×C4×D127—24no—
48C2×C4×D127—44no—
49C2×C4×D127C2×C414yes—
50C2×C4×D127D412yes—
51C2×C4×D127D414yes—
52C2×C4×D127C2312yes—
53#14523231—28no—
54C2×C127⋊C8—22no—
55C2×C127⋊C8C2312yes—
56C4⋊Dic127—21no—
57C4⋊Dic127C2312yes—
58Dic127⋊C4—24no—
59C24×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127C2×C414yescomplemented
63C2×D4×C127C2312yes—
64C22×C4×C127—21no—
65C22×C4×C127C2311yescharacteristic
66C2×C8×C127—22no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127—21no—
69C4⋊C4×C127C2312yes—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127C22⋊C414yes—
73C22×D127C2×D412yes—
74Dic254—24no—
75Dic254—22no—
76Dic254C22×C414yescomplemented
77C22⋊D127—216no—
78C2×Dic127—24no—
79C2×Dic127—22no—
80C2×Dic127—216no—
81C2×Dic127—22no—
82C2×Dic127—42no—
83C2×Dic127C22⋊C414yes—
84C2×Dic127C2×D412yes—
85C4×D127—48no—
86C4×D127C22⋊C418yes—
87C4×D127C22×C414yes—
88C4×D127C2×D414yes—
89C127⋊C8—22no—
90C23×C127—24no—
91C23×C127—22no—
92C23×C127—42no—
93C23×C127—42no—
94C23×C127C2×D411yescharacteristic
95D4×C127—24no—
96D4×C127—22no—
97D4×C127C22×C414yescomplemented
98C2×C4×C127—22no—
99C2×C4×C127—42no—
100C2×C4×C127C22×C412yes—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2411yescharacteristic
104C8×C127—22no—
105C22×C127—24no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—48no—
109C22×C127C2×C22⋊C412yes—
110C22×C127AGL(1,ℤ/8)12yes—
111C22×C127#13712yes—
112C22×C127C22×D411yescharacteristic
113C2×D127—22no—
114C2×D127—22no—
115C2×D127—28no—
116C2×D127C2×C22⋊C414yes—
117C4×C127—42no—
118C4×C127C2×C22⋊C412yes—
119C4×C127C23×C411yescharacteristic
120C4×C127C22×D411yescharacteristic
121Dic127—24no—
122Dic127—42no—
123Dic127C2×C22⋊C414yes—
124C2×C127—44no—
125C2×C127#41714yescomplemented
126C2×C127C22×C22⋊C411yescharacteristic
127C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C127C2×#13711yescharacteristic
129D127—24no—
130C2×#417—1271nomaximal, Sylow
131C127C2×#41711yescharacteristic, complemented, Sylow
132C22×#41—1271no—
133C22×M16—1271no—
134C22×C4⋊C4—1271no—
135#417—1278no—
136C2×#103—1271no—
137C2×#103—1271no—
138C2×#102—1271no—
139C2×#102D12711yescharacteristic, complemented
140C2×#41—1278no—
141C2×#41—1274no—
142C2×#41—2542no—
143C2×#41—2544no—
144C22×Q8—1271no—
145C22×D4—2541no—
146C22×D4C2×D12711yescharacteristic
147C23×C4—1271no—
148C23×C4—2541no—
149C23×C4—2541no—
150C2×M16—1274no—
151C2×M16—2544no—
152C22×C8—1271no—
153C22×C8C2×D12711yescharacteristic
154C2×C4⋊C4—1271no—
155C2×C4⋊C4—1274no—
156C2×C4⋊C4—2542no—
157C2×C4⋊C4—2541no—
158C2×C4⋊C4C2×D12711yescharacteristic
159#103—1274no—
160#103—1274no—
161#102—1274no—
162#102C2×D12714yescomplemented
163C24—21no—
164C24—1271no—
165#41—12716no—
166#41—2548no—
167#41—25416no—
168C2×Q8—1274no—
169C2×Q8—1272no—
170C2×Q8—2541no—
171C2×Q8—2542no—
172C2×D4—22no—
173C2×D4—21no—
174C2×D4—2544no—
175C2×D4—2548no—
176C2×D4C4×D12714yescomplemented
177C2×D4C22×D12712yes—
178C22×C4—21no—
179C22×C4—1271no—
180C22×C4—1274no—
181C22×C4—1272no—
182C22×C4—1274no—
183C22×C4—1272no—
184C22×C4—5084no—
185C22×C4—2544no—
186C22×C4—2544no—
187C22×C4—2548no—
188C22×C4—2541no—
189C22×C4—2541no—
190C22×C4C22×D12711yescharacteristic
191M16—2548no—
192C2×C8—22no—
193C2×C8—1272no—
194C2×C8—2542no—
195C2×C8C22×D12712yes—
196C4⋊C4—21no—
197C4⋊C4—1272no—
198C4⋊C4—2544no—
199C4⋊C4—2541no—
200C4⋊C4C22×D12712yes—
201C23—22no—
202C23—24no—
203C23—42no—
204C23—42no—
205C23—1272no—
206C23—1274no—
207C23—2542no—
208C23—2542no—
209C23D4×D12711yescharacteristic
210Q8—1274no—
211Q8—2542no—
212Q8—2544no—
213D4—22no—
214D4—24no—
215D4—25416no—
216D4C2×C4×D12714yescomplemented
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—1272no—
220C2×C4—1274no—
221C2×C4—1274no—
222C2×C4—1274no—
223C2×C4—1278no—
224C2×C4—5088no—
225C2×C4—5082no—
226C2×C4—25416no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4—2544no—
230C2×C4C2×C4×D12712yes—
231C2×C4D4×D12712yes—
232C2×C4D4×D12711yescharacteristic
233C2×C4C23×D12711yescharacteristic
234C8—22no—
235C8—2542no—
236C22—24no—
237C22—48no—
238C22—42no—
239C22—42no—
240C22—1274no—
241C22—2548no—
242C22—2542no—
243C22—2542no—
244C22C22⋊C4×D12712yes—
245C22#1686123312yes—
246C22#1686123812yes—
247C22C2×D4×D12711yescharacteristic
248C4—42no—
249C4—1274no—
250C4—5082no—
251C4—2544no—
252C4C22⋊C4×D12712yes—
253C4C22×C4×D12711yescharacteristic
254C4C2×D4×D12711yescharacteristic
255C2—44no—
256C2—2544no—
257C2#2254987414yesdirect product factor
258C2C2×C22⋊C4×D12711yescharacteristic
259C2C2×#1686123311yescharacteristic
260C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523247C2
cosocleC2×C4×C127C24
hypercenterC2×#102D127
hypocenterC127C2×#417
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C22×#14523231C2semidirect
3. C2×C4⋊C4×D127C2semidirect
4. #22549874C2direct (N × H)
5. C2×#16861134C2semidirect
6. C2×#16861132C2semidirect
7. C2×#16861130C2semidirect
8. C2×#102×C127C2semidirect
11. C2×#14523247C4semidirect
19. C2×#14523231C22semidirect
24. C4⋊C4×D127C22semidirect
25. #16861134C22semidirect
26. #16861132C22semidirect
27. #16861130C22semidirect
31. #102×C127C22semidirect
35. #14523247C2×C4semidirect
38. C2×Dic254C2×C4semidirect
62. C2×D4×C127C2×C4semidirect
76. Dic254C22×C4semidirect
97. D4×C127C22×C4semidirect
125. C2×C127#417semidirect
131. C127C2×#417semidirect
139. C2×#102D127semidirect
162. #102C2×D127semidirect
176. C2×D4C4×D127semidirect
216. D4C2×C4×D127semidirect
257. C2#22549874direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232472
C22×#145232312
C2×C4⋊C4×D1272
#225498742
C2×#168611342
C2×#168611322
C2×#168611302
C2×#102×C1272
C2×#417127