Finite Group Databasebeta

← C2×#22550039 #30647940

Subgroups — C2×#22550039 #30647940

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×#14523247C211yesmaximal, characteristic
2C22×Dic508C211yesmaximal, characteristic, complemented
3C22×#14523236C211yesmaximal, characteristic, complemented
4C2×#16861189C211yesmaximal, characteristic, complemented
5#22550039C218yesmaximal, direct product factor
6C2×#16861132C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C2×#14523247—24no—
10C2×#14523247—22no—
11C2×#14523247—24no—
12C2×#14523247C2214yes—
13C22×Dic254C2211yescharacteristic
14C22×Dic254C2211yescharacteristic
15C22×C22⋊D127—21no—
16C23×Dic127—21no—
17C22×C4×D127C2211yescharacteristic
18C2×Dic508—24no—
19C2×Dic508C2214yescomplemented
20C2×#14523236—24no—
21C2×#14523236C2214yescomplemented
22C22×C127⋊C8C2211yescharacteristic
23#16861189—22no—
24#16861189C2214yescomplemented
25C2×C4⋊Dic127—21no—
26C2×(C2×D127)⋊C4—21no—
27#16861132C2214yescomplemented
28(C2×D127)⋊C8C2214yescomplemented
29C22×D4×C127C2211yescharacteristic
30C22×C8×C127C2211yescharacteristic
31C2×C4⋊C4×C127C2211yescharacteristic
32#102×C127C2214yescomplemented
33C23×D127D411yescharacteristic
34#14523247—216no—
35#14523247—28no—
36#14523247—28no—
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254—21no—
40C2×Dic254—22no—
41C2×Dic254—22no—
42C2×Dic254D414yes—
43C2×Dic254C2312yes—
44C2×Dic254C2312yes—
45C2×C22⋊D127—28no—
46C2×C22⋊D127—24no—
47C22×Dic127—28no—
48C22×Dic127—24no—
49C22×Dic127—21no—
50C22×Dic127—21no—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127D412yes—
55C2×C4×D127C2312yes—
56Dic508—28no—
57#14523236—28no—
58C2×C127⋊C8—22no—
59C2×C127⋊C8C2312yes—
60C4⋊Dic127—24no—
61(C2×D127)⋊C4—24no—
62C24×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—22no—
65C2×D4×C127D414yes—
66C2×D4×C127C2312yes—
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69C2×C8×C127—22no—
70C2×C8×C127C2312yes—
71C4⋊C4×C127—21no—
72C4⋊C4×C127C2312yes—
73C22×D127—22no—
74C22×D127—22no—
75C22×D127—22no—
76C22×D127C2×D412yes—
77Dic254—24no—
78Dic254—22no—
79Dic254—22no—
80Dic254—22no—
81Dic254—42no—
82Dic254C2×D414yes—
83C22⋊D127—216no—
84C2×Dic127—216no—
85C2×Dic127—22no—
86C2×Dic127—24no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—42no—
90C2×Dic127C2×D412yes—
91C4×D127—24no—
92C4×D127—22no—
93C4×D127—22no—
94C127⋊C8—22no—
95C23×C127—24no—
96C23×C127—22no—
97C23×C127—42no—
98C23×C127—42no—
99C23×C127C2×D411yescharacteristic
100D4×C127—24no—
101D4×C127—22no—
102D4×C127C2×D414yes—
103C2×C4×C127—22no—
104C2×C4×C127—42no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2411yescharacteristic
109C8×C127—22no—
110C22×C127—24no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127—48no—
114C22×C127#12012yes—
115C22×C127#13512yes—
116C22×C127#13712yes—
117C22×C127C22×D411yescharacteristic
118C2×D127—22no—
119C2×D127—22no—
120C2×D127—22no—
121C2×D127—44no—
122C4×C127—42no—
123C4×C127#12012yes—
124C4×C127C22×D411yescharacteristic
125C4×C127C22×D411yescharacteristic
126Dic127—22no—
127Dic127—24no—
128Dic127—42no—
129C2×C127—44no—
130C2×C127#45214yescomplemented
131C2×C127C2×#12011yescharacteristic
132C2×C127C2×#13511yescharacteristic
133C2×C127C2×#13711yescharacteristic
134D127—42no—
135C2×#452—1271nomaximal, Sylow
136C127C2×#45211yescharacteristic, complemented, Sylow
137C22×#41—1271no—
138C22×Q16—1271no—
139C22×SD16—1271no—
140C2×C22⋊Q8—1271no—
141#452—1278no—
142C2×#103—1271no—
143C2×#102D12711yescharacteristic, complemented
144C2×C22⋊C8—1271no—
145C2×#41—1274no—
146C2×#41—2542no—
147C2×#41—2544no—
148C2×#41—2544no—
149C22×Q8—1271no—
150C22×Q8—1271no—
151C22×D4—2541no—
152C22×D4C2×D12711yescharacteristic
153C23×C4—1271no—
154C23×C4—2541no—
155C2×Q16—1274no—
156C2×Q16—2544no—
157C2×SD16—1274no—
158C2×SD16—2544no—
159C22×C8—1271no—
160C22×C8C2×D12711yescharacteristic
161C22⋊Q8—1274no—
162C22⋊Q8—2542no—
163C2×C4⋊C4—2541no—
164C2×C4⋊C4C2×D12711yescharacteristic
165C2×C22⋊C4—2541no—
166#103—1274no—
167#102C2×D12714yescomplemented
168C22⋊C8—1274no—
169C24—21no—
170C24—1271no—
171#41—2548no—
172#41—25416no—
173#41—2548no—
174C2×Q8—1274no—
175C2×Q8—1272no—
176C2×Q8—1272no—
177C2×Q8—2542no—
178C2×Q8—2542no—
179C2×Q8—2541no—
180C2×Q8—2541no—
181C2×Q8—2542no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—2544no—
185C2×D4—2548no—
186C2×D4D50814yes—
187C2×D4C22×D12712yes—
188C22×C4—21no—
189C22×C4—1271no—
190C22×C4—1272no—
191C22×C4—1272no—
192C22×C4—2544no—
193C22×C4—2541no—
194C22×C4—2548no—
195C22×C4—2541no—
196C22×C4—2542no—
197C22×C4—2542no—
198C22×C4C22×D12711yescharacteristic
199Q16—2548no—
200SD16—2548no—
201C2×C8—22no—
202C2×C8—1272no—
203C2×C8—2542no—
204C2×C8C22×D12712yes—
205C4⋊C4—21no—
206C4⋊C4—2544no—
207C4⋊C4C22×D12712yes—
208C22⋊C4—2544no—
209C23—22no—
210C23—24no—
211C23—42no—
212C23—42no—
213C23—1272no—
214C23—2542no—
215C23—2542no—
216C23—2542no—
217C23D4×D12711yescharacteristic
218Q8—1274no—
219Q8—5082no—
220Q8—2542no—
221Q8—2542no—
222Q8—2542no—
223Q8—2544no—
224D4—22no—
225D4—24no—
226D4—25416no—
227D4C2×D50814yes—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—1272no—
231C2×C4—5082no—
232C2×C4—2542no—
233C2×C4—25416no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4—2544no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2544no—
240C2×C4D4×D12712yes—
241C2×C4D4×D12711yescharacteristic
242C2×C4C2×D50812yes—
243C2×C4C23×D12711yescharacteristic
244C8—22no—
245C8—2542no—
246C22—24no—
247C22—48no—
248C22—42no—
249C22—42no—
250C22—5084no—
251C22—2542no—
252C22—2542no—
253C22—2542no—
254C22#1686117912yes—
255C22#1686122812yes—
256C22#1686123812yes—
257C22C2×D4×D12711yescharacteristic
258C4—42no—
259C4—5082no—
260C4—2542no—
261C4—2544no—
262C4#1686117912yes—
263C4C2×D4×D12711yescharacteristic
264C4C22×D50811yescharacteristic
265C2—44no—
266C2—5082no—
267C2#2255003914yesdirect product factor
268C2C2×#1686117911yescharacteristic
269C2C2×#1686122811yescharacteristic
270C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523247C2
hypercenterC2×#102D127
hypocenterC127C2×#452
LCSG ▷ C2×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×Dic508C2semidirect
3. C22×#14523236C2semidirect
4. C2×#16861189C2semidirect
5. #22550039C2direct (N × H)
6. C2×#16861132C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C2×#102×C127C2semidirect
19. C2×Dic508C22semidirect
21. C2×#14523236C22semidirect
24. #16861189C22semidirect
27. #16861132C22semidirect
28. (C2×D127)⋊C8C22semidirect
32. #102×C127C22semidirect
130. C2×C127#452semidirect
136. C127C2×#452semidirect
143. C2×#102D127semidirect
167. #102C2×D127semidirect
267. C2#22550039direct (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×Dic5082
C22×#145232362
C2×#168611892
#225500392
C2×#168611322
C2×(C2×D127)⋊C82
C2×#102×C1272
C2×#452127