Finite Group Databasebeta

← C2×#22550019 #30647929

Subgroups — C2×#22550019 #30647929

Order 16256.

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

normal subgroups only

This group has 812 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
1C22×#14523248C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C22×#14523233C211yesmaximal, characteristic, complemented
4#22550019C218yesmaximal, direct product factor
5C2×#16861132C211yesmaximal, characteristic, complemented
6C2×#16861129C211yesmaximal, characteristic, complemented
7C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
8C2×#121×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—24no—
10C2×#14523248—24no—
11C2×#14523248—22no—
12C2×#14523248C2214yescomplemented
13C22×Dic254C2211yescharacteristic
14C22×D508C2211yescharacteristic
15C22×C22⋊D127—21no—
16C22×C4×D127—21no—
17C2×#14523236—24no—
18C2×#14523236C2214yescomplemented
19C2×#14523233—24no—
20C2×#14523233C2214yescomplemented
21C22×C127⋊C8C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861132C2214yescomplemented
24#16861129C2214yescomplemented
25C22⋊(C127⋊C8)C2214yescomplemented
26C22×D4×C127—21no—
27C22×D4×C127C2211yescharacteristic
28C23×C4×C127C2211yescharacteristic
29#121×C127—22no—
30#121×C127C2214yescomplemented
31C2×C4⋊C4×C127C2211yescharacteristic
32C2×C22⋊C4×C127—21no—
33C23×D127—21no—
34#14523248—216no—
35#14523248—28no—
36#14523248—28no—
37C2×Dic254—22no—
38C2×Dic254—21no—
39C2×Dic254D414yescomplemented
40C2×Dic254C2312yes—
41C2×D508—22no—
42C2×D508—21no—
43C2×D508D414yescomplemented
44C2×D508C2312yes—
45C2×C22⋊D127—28no—
46C2×C22⋊D127—24no—
47C22×Dic127—21no—
48C2×C4×D127—28no—
49C2×C4×D127—24no—
50#14523236—28no—
51#14523233—28no—
52C2×C127⋊C8—22no—
53C2×C127⋊C8—22no—
54C2×C127⋊C8C2312yes—
55C2×C127⋊C8C2312yes—
56C24×C127—21no—
57C24×C127D411yescharacteristic
58C2×D4×C127—24no—
59C2×D4×C127—22no—
60C2×D4×C127—22no—
61C2×D4×C127—21no—
62C2×D4×C127—44no—
63C2×D4×C127C2312yes—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—21no—
67C22×C4×C127D411yescharacteristic
68C22×C4×C127D412yes—
69C22×C4×C127C2312yes—
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127—21no—
72C4⋊C4×C127C2312yes—
73C22⋊C4×C127—24no—
74C22×D127—24no—
75C22×D127—22no—
76C22×D127—42no—
77C22×D127—42no—
78Dic254—24no—
79Dic254—22no—
80Dic254C2×D414yescomplemented
81D508—24no—
82D508—22no—
83D508C2×D414yescomplemented
84C22⋊D127—216no—
85C2×Dic127—24no—
86C2×Dic127—22no—
87C4×D127—216no—
88C127⋊C8—22no—
89C127⋊C8—22no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—42no—
95C23×C127—42no—
96C23×C127—42no—
97C23×C127C2×D412yes—
98C23×C127C2×D411yescharacteristic
99D4×C127—22no—
100D4×C127—22no—
101D4×C127—48no—
102D4×C127—42no—
103C2×C4×C127—22no—
104C2×C4×C127—24no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D412yes—
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D412yes—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2411yescharacteristic
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—22no—
117C22×C127—44no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—84no—
122C22×C127#12012yes—
123C22×C127#13512yes—
124C22×C127AGL(1,ℤ/8)12yes—
125C22×C127C22×D411yescharacteristic
126C2×D127—24no—
127C2×D127—42no—
128C2×D127—42no—
129C2×D127—48no—
130C4×C127—22no—
131C4×C127—42no—
132C4×C127#12012yes—
133C4×C127C22×D411yescharacteristic
134C4×C127C22×D411yescharacteristic
135Dic127—24no—
136C2×C127—42no—
137C2×C127—82no—
138C2×C127#44914yescomplemented
139C2×C127C2×#12011yescharacteristic
140C2×C127C2×#13511yescharacteristic
141C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
142D127—44no—
143C2×#449—1271nomaximal, Sylow
144C127C2×#44911yescharacteristic, complemented, Sylow
145C22×#41—1271no—
146C22×SD16—1271no—
147C22×D8—1271no—
148C2×#121D12711yescharacteristic, complemented
149#449—1278no—
150C2×#103—1271no—
151C2×#102—1271no—
152C2×C22⋊C8—1271no—
153C2×#41—1274no—
154C2×#41—2544no—
155C2×#41—2544no—
156C2×#41—2542no—
157C22×Q8—1271no—
158C22×D4—21no—
159C22×D4—1271no—
160C22×D4—2541no—
161C22×D4C2×D12711yescharacteristic
162C23×C4—2541no—
163C23×C4C2×D12711yescharacteristic
164C2×SD16—1274no—
165C2×SD16—2544no—
166C2×D8—1274no—
167C2×D8—2544no—
168C22×C8—1271no—
169C22×C8—1271no—
170#121—22no—
171#121C2×D12714yescomplemented
172C2×C4⋊C4C2×D12711yescharacteristic
173C2×C22⋊C4—21no—
174#103—1274no—
175#102—1274no—
176C22⋊C8—1274no—
177C24—21no—
178C24—2541no—
179C24C22⋊D12711yescharacteristic
180#41—2548no—
181#41—2548no—
182#41—25416no—
183C2×Q8—1274no—
184C2×Q8—1272no—
185C2×Q8—2542no—
186C2×Q8—2541no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—44no—
192C2×D4—1274no—
193C2×D4—1272no—
194C2×D4—2548no—
195C2×D4—2544no—
196C2×D4—2542no—
197C2×D4—2541no—
198C2×D4C22×D12712yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—2548no—
203C22×C4—2541no—
204C22×C4—2544no—
205C22×C4C22⋊D12712yes—
206C22×C4C22⋊D12711yescharacteristic
207C22×C4C22×D12712yes—
208C22×C4C22×D12711yescharacteristic
209SD16—2548no—
210D8—2548no—
211C2×C8—1272no—
212C2×C8—1272no—
213C2×C8—2542no—
214C2×C8—2542no—
215C4⋊C4—21no—
216C4⋊C4C22×D12712yes—
217C22⋊C4—24no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—5082no—
226C23—5082no—
227C23—2542no—
228C23—2544no—
229C23C2×C22⋊D12712yes—
230C23C2×C22⋊D12711yescharacteristic
231Q8—1274no—
232Q8—2542no—
233Q8—2544no—
234D4—22no—
235D4—22no—
236D4—48no—
237D4—42no—
238D4—1274no—
239D4—25416no—
240D4—2542no—
241D4—2544no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—25416no—
249C2×C4—2542no—
250C2×C4—2544no—
251C2×C4D4×D12712yes—
252C2×C4D4×D12712yes—
253C2×C4C2×C22⋊D12712yes—
254C2×C4C2×C22⋊D12711yescharacteristic
255C2×C4C23×D12711yescharacteristic
256C8—2542no—
257C8—2542no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—5088no—
267C22—5082no—
268C22—5082no—
269C22—2544no—
270C22#1686117712yes—
271C22#1686122912yes—
272C22#1686123512yes—
273C22C22×C22⋊D12711yescharacteristic
274C4—22no—
275C4—42no—
276C4—2544no—
277C4#1686117712yes—
278C4C2×D4×D12711yescharacteristic
279C4C2×D4×D12711yescharacteristic
280C2—42no—
281C2—82no—
282C2—5084no—
283C2#2255001914yesdirect product factor
284C2C2×#1686117711yescharacteristic
285C2C2×#1686122911yescharacteristic
286C2C2×#1686123511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#121×C127C2
socleC23×C127C2×D4
hypercenterC2×#121D127
hypocenterC127C2×#449
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×#14523248C2semidirect
2. C22×#14523236C2semidirect
3. C22×#14523233C2semidirect
4. #22550019C2direct (N × H)
5. C2×#16861132C2semidirect
6. C2×#16861129C2semidirect
7. C2×C22⋊(C127⋊C8)C2semidirect
8. C2×#121×C127C2semidirect
12. C2×#14523248C22semidirect
18. C2×#14523236C22semidirect
20. C2×#14523233C22semidirect
23. #16861132C22semidirect
24. #16861129C22semidirect
25. C22⋊(C127⋊C8)C22semidirect
30. #121×C127C22semidirect
39. C2×Dic254D4semidirect
43. C2×D508D4semidirect
80. Dic254C2×D4semidirect
83. D508C2×D4semidirect
138. C2×C127#449semidirect
144. C127C2×#449semidirect
148. C2×#121D127semidirect
171. #121C2×D127semidirect
283. C2#22550019direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232482
C22×#145232362
C22×#145232332
#225500192
C2×#168611322
C2×#168611292
C2×C22⋊(C127⋊C8)2
C2×#121×C1272
C2×#449127