Finite Group Databasebeta

← C2×#22550036 #30647937

Subgroups — C2×#22550036 #30647937

Order 16256.

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

normal subgroups only

This group has 760 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
2C22×#14523238C211yesmaximal, characteristic, complemented
3C22×#14523235C211yesmaximal, characteristic, complemented
4#22550036C218yesmaximal, direct product factor
5C2×#16861132C211yesmaximal, characteristic, complemented
6C2×#16861129C211yesmaximal, characteristic, complemented
7C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—24no—
10C2×#14523248—24no—
11C2×#14523248—22no—
12C2×#14523248C2214yes—
13C22×Dic254C2211yescharacteristic
14C22×D508C2211yescharacteristic
15C22×C22⋊D127—21no—
16C22×C4×D127—21no—
17C2×#14523238—24no—
18C2×#14523238C2214yescomplemented
19C2×#14523235—24no—
20C2×#14523235C2214yescomplemented
21C22×C127⋊C8C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861132C2214yescomplemented
24#16861129C2214yescomplemented
25C22⋊(C127⋊C8)C2214yescomplemented
26C22×Q8×C127C2211yescharacteristic
27C23×C4×C127C2211yescharacteristic
28C22⋊Q8×C127—22no—
29C22⋊Q8×C127C2214yescomplemented
30C2×C4⋊C4×C127—21no—
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×Dic254D414yes—
40C2×Dic254C2312yes—
41C2×D508—22no—
42C2×D508—21no—
43C2×D508D414yes—
44C2×D508C2312yes—
45C2×C22⋊D127—28no—
46C2×C22⋊D127—24no—
47C22×Dic127—21no—
48C2×C4×D127—28no—
49C2×C4×D127—24no—
50#14523238—28no—
51#14523235—28no—
52C2×C127⋊C8—22no—
53C2×C127⋊C8—22no—
54C2×C127⋊C8C2312yes—
55C2×C127⋊C8C2312yes—
56C24×C127D411yescharacteristic
57C2×Q8×C127—21no—
58C2×Q8×C127—22no—
59C2×Q8×C127—22no—
60C2×Q8×C127C2312yes—
61C22×C4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127D411yescharacteristic
66C22×C4×C127D412yes—
67C22×C4×C127C2312yes—
68C22×C4×C127C2311yescharacteristic
69C4⋊C4×C127—24no—
70C4⋊C4×C127—21no—
71C4⋊C4×C127C2312yes—
72C22⋊C4×C127—24no—
73C22×D127—24no—
74C22×D127—22no—
75C22×D127—42no—
76C22×D127—42no—
77Dic254—24no—
78Dic254—22no—
79Dic254C2×D414yes—
80D508—24no—
81D508—22no—
82D508C2×D414yes—
83C22⋊D127—216no—
84C2×Dic127—24no—
85C2×Dic127—22no—
86C4×D127—216no—
87C127⋊C8—22no—
88C127⋊C8—22no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127C2×D412yes—
93C23×C127C2×D411yescharacteristic
94Q8×C127—22no—
95Q8×C127—22no—
96Q8×C127—42no—
97C2×C4×C127—22no—
98C2×C4×C127—24no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—42no—
104C2×C4×C127—42no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—44no—
114C22×C127#12012yes—
115C22×C127#13512yes—
116C22×C127#13712yes—
117C22×C127C22×D411yescharacteristic
118C2×D127—24no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—48no—
122C4×C127—22no—
123C4×C127—42no—
124C4×C127—42no—
125C4×C127#12012yes—
126C4×C127C22×D411yescharacteristic
127C4×C127C22×D411yescharacteristic
128Dic127—24no—
129C2×C127—42no—
130C2×C127#45214yescomplemented
131C2×C127C2×#12011yescharacteristic
132C2×C127C2×#13511yescharacteristic
133C2×C127C2×#13711yescharacteristic
134D127—44no—
135C2×#452—1271nomaximal, Sylow
136C127C2×#45211yescharacteristic, complemented, Sylow
137C22×#41—1271no—
138C22×Q16—1271no—
139C22×SD16—1271no—
140C2×C22⋊Q8D12711yescharacteristic, complemented
141#452—1278no—
142C2×#103—1271no—
143C2×#102—1271no—
144C2×C22⋊C8—1271no—
145C2×#41—1274no—
146C2×#41—2544no—
147C2×#41—2544no—
148C2×#41—2542no—
149C22×Q8—1271no—
150C22×Q8C2×D12711yescharacteristic
151C22×D4—1271no—
152C22×D4—2541no—
153C23×C4—2541no—
154C23×C4C2×D12711yescharacteristic
155C2×Q16—1274no—
156C2×Q16—2544no—
157C2×SD16—1274no—
158C2×SD16—2544no—
159C22×C8—1271no—
160C22×C8—1271no—
161C22⋊Q8—22no—
162C22⋊Q8C2×D12714yescomplemented
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×D12711yescharacteristic
165C2×C22⋊C4—21no—
166#103—1274no—
167#102—1274no—
168C22⋊C8—1274no—
169C24—2541no—
170C24C22⋊D12711yescharacteristic
171#41—2548no—
172#41—2548no—
173#41—25416no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—1274no—
178C2×Q8—1272no—
179C2×Q8—2542no—
180C2×Q8—2541no—
181C2×Q8C22×D12712yes—
182C2×D4—1274no—
183C2×D4—1272no—
184C2×D4—2548no—
185C2×D4—2544no—
186C2×D4—2542no—
187C2×D4—2541no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—2548no—
193C22×C4—2541no—
194C22×C4—2544no—
195C22×C4C22⋊D12712yes—
196C22×C4C22⋊D12711yescharacteristic
197C22×C4C22×D12712yes—
198C22×C4C22×D12711yescharacteristic
199Q16—2548no—
200SD16—2548no—
201C2×C8—1272no—
202C2×C8—1272no—
203C2×C8—2542no—
204C2×C8—2542no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D12712yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—5082no—
213C23—5082no—
214C23—2542no—
215C23—2544no—
216C23C2×C22⋊D12712yes—
217C23C2×C22⋊D12711yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—1274no—
222Q8—2542no—
223Q8—2544no—
224D4—1274no—
225D4—25416no—
226D4—2542no—
227D4—2544no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—25416no—
237C2×C4—2542no—
238C2×C4—2544no—
239C2×C4D4×D12712yes—
240C2×C4D4×D12712yes—
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—2542no—
245C8—2542no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—5088no—
251C22—5082no—
252C22—5082no—
253C22—2544no—
254C22#1686117712yes—
255C22#1686122912yes—
256C22#1686124012yes—
257C22C22×C22⋊D12711yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—2544no—
262C4#1686117712yes—
263C4C2×D4×D12711yescharacteristic
264C4C2×D4×D12711yescharacteristic
265C2—42no—
266C2—5084no—
267C2#2255003614yesdirect product factor
268C2C2×#1686117711yescharacteristic
269C2C2×#1686122911yescharacteristic
270C2C2×#1686124011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523248C2
hypercenterC2×C22⋊Q8D127
hypocenterC127C2×#452
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#14523238C2semidirect
3. C22×#14523235C2semidirect
4. #22550036C2direct (N × H)
5. C2×#16861132C2semidirect
6. C2×#16861129C2semidirect
7. C2×C22⋊(C127⋊C8)C2semidirect
8. C2×C22⋊Q8×C127C2semidirect
18. C2×#14523238C22semidirect
20. C2×#14523235C22semidirect
23. #16861132C22semidirect
24. #16861129C22semidirect
25. C22⋊(C127⋊C8)C22semidirect
29. C22⋊Q8×C127C22semidirect
130. C2×C127#452semidirect
136. C127C2×#452semidirect
140. C2×C22⋊Q8D127semidirect
162. C22⋊Q8C2×D127semidirect
267. C2#22550036direct (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×#145232382
C22×#145232352
#225500362
C2×#168611322
C2×#168611292
C2×C22⋊(C127⋊C8)2
C2×C22⋊Q8×C1272
C2×#452127