Finite Group Databasebeta

← C2×#22550127 #30648324

Subgroups — C2×#22550127 #30648324

Order 16256.

596 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14523233C211yesmaximal, characteristic, complemented
2C22×#14523232C211yesmaximal, characteristic, complemented
3C2×#16861183C211yesmaximal, characteristic, complemented
4#22550127C218yesmaximal, direct product factor
5C2×#16861140C211yesmaximal, characteristic, complemented
6C2×#16861130C211yesmaximal, characteristic, complemented
7C2×#16861129C211yesmaximal, characteristic, complemented
8C2×#121×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C2×#14523233—24no—
12C2×#14523233C2214yescomplemented
13C2×#14523232—24no—
14C2×#14523232C2214yescomplemented
15C22×C127⋊C8C2211yescharacteristic
16C22×C127⋊C8C2211yescharacteristic
17#16861183—22no—
18#16861183C2214yescomplemented
19C2×C4⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4—21no—
21#16861140C2214yescomplemented
22#16861130C2214yescomplemented
23#16861129C2214yescomplemented
24C22×D4×C127—21no—
25C22×D4×C127C2211yescharacteristic
26C23×C4×C127C2211yescharacteristic
27#121×C127—22no—
28#121×C127C2214yescomplemented
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127—21no—
31C23×D127—21no—
32C2×D508—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508C2312yes—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—44no—
38C22×Dic127—21no—
39#14523233—28no—
40#14523232—28no—
41C2×C127⋊C8—22no—
42C2×C127⋊C8D414yescomplemented
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—21no—
46C4⋊Dic127C2312yes—
47(C2×D127)⋊C4—24no—
48C24×C127—21no—
49C24×C127D411yescharacteristic
50C2×D4×C127—24no—
51C2×D4×C127—22no—
52C2×D4×C127—21no—
53C2×D4×C127—22no—
54C2×D4×C127—44no—
55C2×D4×C127C2312yes—
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127—21no—
59C22×C4×C127D411yescharacteristic
60C22×C4×C127D412yes—
61C22×C4×C127C2312yes—
62C22×C4×C127C2311yescharacteristic
63C4⋊C4×C127—21no—
64C4⋊C4×C127C2312yes—
65C22⋊C4×C127—24no—
66C22×D127—22no—
67C22×D127—42no—
68C22×D127—42no—
69C22×D127—42no—
70D508—22no—
71D508—22no—
72D508—42no—
73C22⋊D127—48no—
74C2×Dic127—22no—
75C2×Dic127—42no—
76C127⋊C8—22no—
77C127⋊C8C2×D414yescomplemented
78C23×C127—22no—
79C23×C127—22no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—42no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127C2×D412yes—
86C23×C127C2×D411yescharacteristic
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—48no—
90D4×C127—42no—
91C2×C4×C127—22no—
92C2×C4×C127—24no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—42no—
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4112yes—
101C2×C4×C127C2411yescharacteristic
102C22×C127—22no—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—44no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—84no—
110C22×C127#12112yes—
111C22×C127AGL(1,ℤ/8)12yes—
112C22×C127AGL(1,ℤ/8)12yes—
113C22×C127C22×D411yescharacteristic
114C2×D127—42no—
115C2×D127—42no—
116C2×D127—42no—
117C2×D127—84no—
118C4×C127—22no—
119C4×C127—42no—
120C4×C127#12112yes—
121C4×C127C22×D411yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—42no—
124C2×C127—42no—
125C2×C127—82no—
126C2×C127#46914yescomplemented
127C2×C127C2×#12111yescharacteristic
128C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
130D127—82no—
131C2×#469—1271nomaximal, Sylow
132C127C2×#46911yescharacteristic, complemented, Sylow
133C22×D8—1271no—
134C22×M16—1271no—
135C2×#121—1271no—
136C2×#121D12711yescharacteristic, complemented
137#469—1278no—
138C2×#106—1271no—
139C2×#102—1271no—
140C2×#102—1271no—
141C22×D4—21no—
142C22×D4—1271no—
143C22×D4—2541no—
144C22×D4C2×D12711yescharacteristic
145C23×C4C2×D12711yescharacteristic
146C2×D8—1274no—
147C2×D8—2544no—
148C2×M16—1274no—
149C2×M16—2544no—
150C22×C8—1271no—
151C22×C8—1271no—
152#121—22no—
153#121—1274no—
154#121—2542no—
155#121C2×D12714yescomplemented
156C2×C4⋊C4—1271no—
157C2×C4⋊C4C2×D12711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—2541no—
160#106—1274no—
161#102—1274no—
162#102—1274no—
163C24—21no—
164C24—2541no—
165C24C22⋊D12711yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—1272no—
172C2×D4—5084no—
173C2×D4—2544no—
174C2×D4—2542no—
175C2×D4—2541no—
176C2×D4—2542no—
177C2×D4C22×D12712yes—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—2541no—
182C22×C4C22⋊D12712yes—
183C22×C4C22⋊D12711yescharacteristic
184C22×C4C22×D12712yes—
185C22×C4C22×D12711yescharacteristic
186D8—2548no—
187M16—2548no—
188C2×C8—1274no—
189C2×C8—1272no—
190C2×C8—1272no—
191C2×C8—2542no—
192C4⋊C4—21no—
193C4⋊C4—1272no—
194C4⋊C4—2541no—
195C4⋊C4C22×D12712yes—
196C22⋊C4—24no—
197C22⋊C4—2544no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—5082no—
206C23—5082no—
207C23—5082no—
208C23—2542no—
209C23C2×C22⋊D12712yes—
210C23C2×C22⋊D12711yescharacteristic
211D4—22no—
212D4—22no—
213D4—48no—
214D4—42no—
215D4—5088no—
216D4—5082no—
217D4—2542no—
218D4—2542no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—5082no—
226C2×C4—2542no—
227C2×C4D4×D12712yes—
228C2×C4C2×C22⋊D12712yes—
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4#1452324712yes—
231C2×C4C23×D12711yescharacteristic
232C8—1274no—
233C8—2542no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—10164no—
243C22—5082no—
244C22—5082no—
245C22—5082no—
246C22#1686118112yes—
247C22#1686123412yes—
248C22#1686123512yes—
249C22C22×C22⋊D12711yescharacteristic
250C4—22no—
251C4—42no—
252C4—5082no—
253C4#1686118112yes—
254C4C2×D4×D12711yescharacteristic
255C4C2×#1452324711yescharacteristic
256C2—42no—
257C2—82no—
258C2—10162no—
259C2#2255012714yesdirect product factor
260C2C2×#1686118111yescharacteristic
261C2C2×#1686123411yescharacteristic
262C2C2×#1686123511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#121×C127C2
socleC23×C127C2×D4
hypercenterC2×#121D127
hypocenterC127C2×#469
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C22×#14523232C2semidirect
3. C2×#16861183C2semidirect
4. #22550127C2direct (N × H)
5. C2×#16861140C2semidirect
6. C2×#16861130C2semidirect
7. C2×#16861129C2semidirect
8. C2×#121×C127C2semidirect
12. C2×#14523233C22semidirect
14. C2×#14523232C22semidirect
18. #16861183C22semidirect
21. #16861140C22semidirect
22. #16861130C22semidirect
23. #16861129C22semidirect
28. #121×C127C22semidirect
42. C2×C127⋊C8D4semidirect
77. C127⋊C8C2×D4semidirect
126. C2×C127#469semidirect
132. C127C2×#469semidirect
136. C2×#121D127semidirect
155. #121C2×D127semidirect
259. C2#22550127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232332
C22×#145232322
C2×#168611832
#225501272
C2×#168611402
C2×#168611302
C2×#168611292
C2×#121×C1272
C2×#469127