Finite Group Databasebeta

← C2×#22550102 #30648287

Subgroups — C2×#22550102 #30648287

Order 16256.

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

normal subgroups only

This group has 564 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×C8×D127C211yesmaximal, characteristic, complemented
2C2×#16861185C211yesmaximal, characteristic, complemented
3C2×#16861180C211yesmaximal, characteristic
4#22550102C218yesmaximal, direct product factor
5C2×#16861141C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×#16861130C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic, complemented
9C22×C22⋊D127—21no—
10C22×C4×D127C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12C2×C8×D127—24no—
13C2×C8×D127C2214yescomplemented
14#16861185—22no—
15#16861185C2214yescomplemented
16#16861180—22no—
17#16861180C2214yes—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4—21no—
21C2×C22⋊Dic127—21no—
22C2×(C2×D127)⋊C4—21no—
23#16861141C2214yescomplemented
24#16861133C2214yescomplemented
25#16861130C2214yescomplemented
26C22×Q8×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×SD16×C127—24no—
29C2×SD16×C127C2214yescomplemented
30C22×C8×C127C2211yescharacteristic
31C23×D127D411yescharacteristic
32C2×C22⋊D127—24no—
33C2×C22⋊D127—44no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37C2×C4×D127—22no—
38C2×C4×D127—22no—
39C2×C4×D127D412yes—
40C2×C4×D127C2312yes—
41C2×C127⋊C8—22no—
42C2×C127⋊C8C2312yes—
43C8×D127—28no—
44C4⋊Dic127—21no—
45C4⋊Dic127—21no—
46C4⋊Dic127C2312yes—
47C4⋊Dic127C2312yes—
48Dic127⋊C4—24no—
49C22⋊Dic127—24no—
50(C2×D127)⋊C4—24no—
51C24×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—22no—
55C2×Q8×C127C2312yes—
56C2×D4×C127—22no—
57C2×D4×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127C2312yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62SD16×C127—28no—
63C2×C8×C127D414yescomplemented
64C2×C8×C127C2312yes—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127SD1614yescomplemented
68C22×D127C2×D412yes—
69C22⋊D127—48no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—42no—
74C2×Dic127—42no—
75C2×Dic127C2×D412yes—
76C4×D127—24no—
77C4×D127—22no—
78C4×D127—22no—
79C127⋊C8—22no—
80C23×C127—22no—
81C23×C127—42no—
82C23×C127—42no—
83C23×C127—42no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—22no—
86Q8×C127—22no—
87Q8×C127—42no—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—42no—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127C2×D414yescomplemented
98C22×C127—42no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127—84no—
102C22×C127#12112yes—
103C22×C127C2×SD1612yes—
104C22×C127#13512yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—22no—
107C2×D127—22no—
108C2×D127—28no—
109C2×D127C2×SD1614yescomplemented
110C4×C127—42no—
111C4×C127#12112yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—22no—
115Dic127—42no—
116Dic127—42no—
117C2×C127—82no—
118C2×C127#46514yescomplemented
119C2×C127C2×#12111yescharacteristic
120C2×C127C22×SD1611yescharacteristic
121C2×C127C2×#13511yescharacteristic
122D127—24no—
123C2×#465—1271nomaximal, Sylow
124C127C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16D12711yescharacteristic, complemented
126C23×C8—1271no—
127C2×C22⋊Q8—1271no—
128C2×#121—1271no—
129#465—1278no—
130C2×#106—1271no—
131C2×#103—1271no—
132C2×#102—1271no—
133C22×Q8C2×D12711yescharacteristic
134C22×D4—2541no—
135C22×D4C2×D12711yescharacteristic
136C23×C4—1271no—
137C2×SD16—24no—
138C2×SD16C2×D12714yescomplemented
139C22×C8—1271no—
140C22×C8—1274no—
141C22×C8—2544no—
142C22×C8C2×D12711yescharacteristic
143C22⋊Q8—1274no—
144C22⋊Q8—2542no—
145#121—1274no—
146#121—2542no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4—2541no—
150C2×C22⋊C4—2541no—
151C2×C22⋊C4—2541no—
152#106—1274no—
153#103—1274no—
154#102—1274no—
155C24—21no—
156C24—1271no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D12712yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—5084no—
165C2×D4—2544no—
166C2×D4C22×D12712yes—
167C22×C4—21no—
168C22×C4—1271no—
169C22×C4—1272no—
170C22×C4—1272no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4—2542no—
174C22×C4—2542no—
175C22×C4C22×D12711yescharacteristic
176SD16—28no—
177C2×C8—1272no—
178C2×C8—2542no—
179C2×C8—2548no—
180C2×C8C22⋊D12714yescomplemented
181C2×C8C22×D12712yes—
182C4⋊C4—1272no—
183C4⋊C4—1272no—
184C4⋊C4—2544no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C22⋊C4—2544no—
188C22⋊C4—2544no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—1274no—
194C23—1272no—
195C23—2542no—
196C23—2542no—
197C23D4×D12711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—5088no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1272no—
208C2×C4—5082no—
209C2×C4—5082no—
210C2×C4—2542no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2544no—
216C2×C4D4×D12711yescharacteristic
217C2×C4C2×C22⋊D12712yes—
218C2×C4#1452324712yes—
219C2×C4C23×D12711yescharacteristic
220C8—2542no—
221C8C2×C22⋊D12714yescomplemented
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—1274no—
227C22—2548no—
228C22—2542no—
229C22—2542no—
230C22#1686118012yes—
231C22SD16×D12712yes—
232C22#1686123012yes—
233C22C2×D4×D12711yescharacteristic
234C4—42no—
235C4—5082no—
236C4—5082no—
237C4—2542no—
238C4#1686118012yes—
239C4C22×C22⋊D12711yescharacteristic
240C4C2×#1452324711yescharacteristic
241C2—82no—
242C2—2544no—
243C2#2255010214yesdirect product factor
244C2C2×#1686118011yescharacteristic
245C2C2×SD16×D12711yescharacteristic
246C2C2×#1686123011yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
hypercenterC22×SD16D127
hypocenterC127C2×#465
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C8×D127C2semidirect
2. C2×#16861185C2semidirect
4. #22550102C2direct (N × H)
5. C2×#16861141C2semidirect
6. C2×#16861133C2semidirect
7. C2×#16861130C2semidirect
8. C22×SD16×C127C2semidirect
13. C2×C8×D127C22semidirect
15. #16861185C22semidirect
23. #16861141C22semidirect
24. #16861133C22semidirect
25. #16861130C22semidirect
29. C2×SD16×C127C22semidirect
63. C2×C8×C127D4semidirect
67. C22×D127SD16semidirect
97. C8×C127C2×D4semidirect
109. C2×D127C2×SD16semidirect
118. C2×C127#465semidirect
124. C127C2×#465semidirect
125. C22×SD16D127semidirect
138. C2×SD16C2×D127semidirect
180. C2×C8C22⋊D127semidirect
221. C8C2×C22⋊D127semidirect
243. C2#22550102direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C8×D1272
C2×#168611852
C2×#168611802
#225501022
C2×#168611412
C2×#168611332
C2×#168611302
C22×SD16×C1272
C2×#465127