Finite Group Databasebeta

← C2×#22550016 #30647919

Subgroups — C2×#22550016 #30647919

Order 16256.

792 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 173 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×Q8×D127C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic, complemented
3C22×#14523235C211yesmaximal, characteristic, complemented
4C2×#16861184C211yesmaximal, characteristic
5#22550016C218yesmaximal, direct product factor
6C2×#16861132C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C2×#103×C127C211yesmaximal, characteristic, complemented
9C22×Dic254—21no—
10C22×Dic254C2211yescharacteristic
11C2×Q8×D127—24no—
12C2×Q8×D127—22no—
13C2×Q8×D127—24no—
14C2×Q8×D127C2214yescomplemented
15C22×D508—21no—
16C22×D508C2211yescharacteristic
17C22×C4×D127—21no—
18C22×C4×D127C2211yescharacteristic
19C2×#14523237—24no—
20C2×#14523237C2214yescomplemented
21C2×#14523235—24no—
22C2×#14523235C2214yescomplemented
23C22×C127⋊C8C2211yescharacteristic
24#16861184—22no—
25#16861184C2214yes—
26C2×(C2×D127)⋊C4—21no—
27#16861132C2214yescomplemented
28(C2×D127)⋊C8C2214yescomplemented
29C22×Q8×C127C2211yescharacteristic
30C22×C8×C127C2211yescharacteristic
31C2×C4⋊C4×C127C2211yescharacteristic
32#103×C127C2214yescomplemented
33C23×D127—21no—
34C23×D127D411yescharacteristic
35C2×Dic254—28no—
36C2×Dic254—24no—
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254D414yescomplemented
40C2×Dic254C2312yes—
41Q8×D127—216no—
42Q8×D127—28no—
43Q8×D127—28no—
44C2×D508—24no—
45C2×D508—22no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—44no—
49C2×D508C2312yes—
50C22×Dic127—21no—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—28no—
53C2×C4×D127—24no—
54C2×C4×D127—22no—
55C2×C4×D127—22no—
56C2×C4×D127D412yes—
57C2×C4×D127C2312yes—
58#14523237—28no—
59#14523235—28no—
60C2×C127⋊C8—22no—
61C2×C127⋊C8C2312yes—
62(C2×D127)⋊C4—24no—
63C2×Q8×C127—22no—
64C2×Q8×C127—21no—
65C2×Q8×C127D414yescomplemented
66C2×Q8×C127C2312yes—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70C2×C8×C127—22no—
71C2×C8×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127C2312yes—
74C22×D127—22no—
75C22×D127—22no—
76C22×D127—22no—
77C22×D127—42no—
78C22×D127—42no—
79C22×D127—42no—
80C22×D127SD1614yescomplemented
81C22×D127C2×D412yes—
82Dic254—216no—
83Dic254—24no—
84Dic254—22no—
85Dic254C2×D414yescomplemented
86D508—22no—
87D508—22no—
88D508—48no—
89D508—42no—
90C2×Dic127—24no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127C2×D412yes—
94C4×D127—216no—
95C4×D127—24no—
96C4×D127—22no—
97C4×D127—22no—
98C127⋊C8—22no—
99C23×C127C2×D411yescharacteristic
100Q8×C127—24no—
101Q8×C127—22no—
102Q8×C127C2×D414yescomplemented
103C2×C4×C127—24no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—42no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2×D412yes—
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127#12012yes—
113C22×C127C2×SD1612yes—
114C22×C127#13712yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—22no—
117C2×D127—22no—
118C2×D127—28no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—42no—
122C2×D127—84no—
123C2×D127C2×SD1614yescomplemented
124C4×C127—24no—
125C4×C127—42no—
126C4×C127#12012yes—
127C4×C127C22×D411yescharacteristic
128C4×C127C22×D411yescharacteristic
129Dic127—24no—
130Dic127—22no—
131C2×C127#44814yescomplemented
132C2×C127C2×#12011yescharacteristic
133C2×C127C22×SD1611yescharacteristic
134C2×C127C2×#13711yescharacteristic
135D127—24no—
136D127—82no—
137C2×#448—1271nomaximal, Sylow
138C127C2×#44811yescharacteristic, complemented, Sylow
139C23×Q8—1271no—
140C22×SD16—1271no—
141C22×SD16—1271no—
142C2×#121—1271no—
143#448—1278no—
144C2×#103—1271no—
145C2×#103D12711yescharacteristic, complemented
146C2×C22⋊C8—1271no—
147C22×Q8—1274no—
148C22×Q8—1271no—
149C22×Q8—2541no—
150C22×Q8—2542no—
151C22×Q8—2544no—
152C22×Q8—2544no—
153C22×Q8C2×D12711yescharacteristic
154C22×D4—1271no—
155C22×D4—2541no—
156C23×C4—1271no—
157C23×C4—2541no—
158C2×SD16—1274no—
159C2×SD16—1274no—
160C2×SD16—2544no—
161C2×SD16—2544no—
162C22×C8—1271no—
163C22×C8C2×D12711yescharacteristic
164#121—1274no—
165#121—2542no—
166C2×C4⋊C4C2×D12711yescharacteristic
167C2×C22⋊C4—2541no—
168#103—1274no—
169#103C2×D12714yescomplemented
170C22⋊C8—1274no—
171C24—1271no—
172C24—2541no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—1274no—
176C2×Q8—1272no—
177C2×Q8—2544no—
178C2×Q8—2548no—
179C2×Q8—2548no—
180C2×Q8—25416no—
181C2×Q8—2548no—
182C2×Q8—2541no—
183C2×Q8—2542no—
184C2×Q8D50814yescomplemented
185C2×Q8C22×D12712yes—
186C2×D4—1272no—
187C2×D4—5084no—
188C2×D4—2544no—
189C2×D4—2541no—
190C2×D4—2542no—
191C2×D4—2542no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—1271no—
195C22×C4—1272no—
196C22×C4—1272no—
197C22×C4—2544no—
198C22×C4—2548no—
199C22×C4—2541no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4C22×D12711yescharacteristic
203SD16—2548no—
204SD16—2548no—
205C2×C8—22no—
206C2×C8—1272no—
207C2×C8—2542no—
208C2×C8C22×D12712yes—
209C4⋊C4—21no—
210C4⋊C4C22×D12712yes—
211C22⋊C4—2544no—
212C23—1274no—
213C23—1272no—
214C23—5082no—
215C23—5082no—
216C23—5082no—
217C23—2542no—
218C23—2542no—
219C23—2542no—
220C23D4×D12711yescharacteristic
221Q8—22no—
222Q8—24no—
223Q8—1274no—
224Q8—25416no—
225Q8—2542no—
226Q8—2544no—
227Q8C2×D50814yescomplemented
228D4—5088no—
229D4—5082no—
230D4—2542no—
231D4—2542no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1272no—
237C2×C4—25416no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2544no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2544no—
244C2×C4D4×D12712yes—
245C2×C4D4×D12711yescharacteristic
246C2×C4C2×D50812yes—
247C2×C4C23×D12711yescharacteristic
248C8—22no—
249C8—2542no—
250C22—1274no—
251C22—10164no—
252C22—5082no—
253C22—5082no—
254C22—5082no—
255C22—2548no—
256C22—2542no—
257C22—2542no—
258C22#1686117912yes—
259C22SD16×D12712yes—
260C22#1686124112yes—
261C22C2×D4×D12711yescharacteristic
262C4—24no—
263C4—42no—
264C4—2542no—
265C4—2544no—
266C4#1686117912yes—
267C4C2×D4×D12711yescharacteristic
268C4C22×D50811yescharacteristic
269C2—10162no—
270C2—2544no—
271C2#2255001614yesdirect product factor
272C2C2×#1686117911yescharacteristic
273C2C2×SD16×D12711yescharacteristic
274C2C2×#1686124111yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861184C2
hypercenterC2×#103D127
hypocenterC127C2×#448
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D127C2semidirect
2. C22×#14523237C2semidirect
3. C22×#14523235C2semidirect
5. #22550016C2direct (N × H)
6. C2×#16861132C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C2×#103×C127C2semidirect
14. C2×Q8×D127C22semidirect
20. C2×#14523237C22semidirect
22. C2×#14523235C22semidirect
27. #16861132C22semidirect
28. (C2×D127)⋊C8C22semidirect
32. #103×C127C22semidirect
39. C2×Dic254D4semidirect
65. C2×Q8×C127D4semidirect
80. C22×D127SD16semidirect
85. Dic254C2×D4semidirect
102. Q8×C127C2×D4semidirect
123. C2×D127C2×SD16semidirect
131. C2×C127#448semidirect
138. C127C2×#448semidirect
145. C2×#103D127semidirect
169. #103C2×D127semidirect
184. C2×Q8D508semidirect
227. Q8C2×D508semidirect
271. C2#22550016direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C22×#145232372
C22×#145232352
C2×#168611842
#225500162
C2×#168611322
C2×(C2×D127)⋊C82
C2×#103×C1272
C2×#448127