Finite Group Databasebeta

← C2×#20268020 #27204704

Subgroups — C2×#20268020 #27204704

Order 13184.

564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13110094C211yesmaximal, characteristic, complemented
2C2×#15693000C211yesmaximal, characteristic, complemented
3C2×#15692995C211yesmaximal, characteristic
4#20268020C218yesmaximal, direct product factor
5C2×#15692953C211yesmaximal, characteristic, complemented
6C2×#15692946C211yesmaximal, characteristic, complemented
7C2×#15692943C211yesmaximal, characteristic, complemented
8C23×C8×C103C211yesmaximal, characteristic, complemented
9C22×Dic206C2211yescharacteristic
10C22×D412C2211yescharacteristic
11C22×C22⋊D103—21no—
12C2×#13110094—24no—
13C2×#13110094C2214yescomplemented
14#15693000—22no—
15#15693000C2214yescomplemented
16#15692995—22no—
17#15692995C2214yes—
18C2×C4⋊Dic103C2211yescharacteristic
19C2×C4⋊Dic103C2211yescharacteristic
20C2×Dic103⋊C4—21no—
21C2×C22⋊Dic103—21no—
22C2×(C2×D103)⋊C4—21no—
23#15692953C2214yescomplemented
24#15692946C2214yescomplemented
25#15692943C2214yescomplemented
26C23×C4×C103C2211yescharacteristic
27C22×C8×C103—24no—
28C22×C8×C103C2211yescharacteristic
29C22×C8×C103C2211yescharacteristic
30C22×C8×C103C2214yescomplemented
31C23×D103—21no—
32C2×Dic206—22no—
33C2×Dic206—22no—
34C2×Dic206—21no—
35C2×Dic206C2312yes—
36C2×D412—22no—
37C2×D412—22no—
38C2×D412—21no—
39C2×D412C2312yes—
40C2×C22⋊D103—24no—
41C2×C22⋊D103—44no—
42C22×Dic103—21no—
43C22×Dic103—21no—
44C22×Dic103—21no—
45#13110094—28no—
46C4⋊Dic103—21no—
47C4⋊Dic103—21no—
48C4⋊Dic103C2312yes—
49C4⋊Dic103C2312yes—
50Dic103⋊C4—24no—
51C22⋊Dic103—24no—
52(C2×D103)⋊C4—24no—
53C24×C103D411yescharacteristic
54C22×C4×C103—22no—
55C22×C4×C103—22no—
56C22×C4×C103D411yescharacteristic
57C22×C4×C103D412yes—
58C22×C4×C103C2312yes—
59C22×C4×C103C2311yescharacteristic
60C2×C8×C103—22no—
61C2×C8×C103—28no—
62C2×C8×C103D414yescomplemented
63C2×C8×C103C2312yes—
64C2×C8×C103C2312yes—
65C23×C103—22no—
66C23×C103—22no—
67C23×C103SD1614yescomplemented
68C23×C103C2×D412yes—
69C23×C103C2×D411yescharacteristic
70C22×D103—22no—
71C22×D103—42no—
72C22×D103—42no—
73C22×D103—42no—
74C2×C4×C103—22no—
75C2×C4×C103—24no—
76C2×C4×C103—22no—
77C2×C4×C103—22no—
78C2×C4×C103C2×D412yes—
79C2×C4×C103C2×D412yes—
80C2×C4×C103C2×D411yescharacteristic
81C2×C4×C103#4112yes—
82C2×C4×C103C2411yescharacteristic
83C22⋊D103—48no—
84C2×Dic103—22no—
85C2×Dic103—22no—
86C2×Dic103—22no—
87C2×Dic103—42no—
88C2×Dic103—42no—
89C2×Dic103—42no—
90D412—22no—
91D412—22no—
92D412—42no—
93Dic206—22no—
94Dic206—22no—
95Dic206—42no—
96C8×C103—22no—
97C8×C103C2×D414yescomplemented
98C22×C103—22no—
99C22×C103—22no—
100C22×C103—28no—
101C22×C103#12112yes—
102C22×C103C2×SD1612yes—
103C22×C103C2×SD1614yescomplemented
104C22×C103#13512yes—
105C22×C103C22×D411yescharacteristic
106C2×D103—42no—
107C2×D103—42no—
108C2×D103—42no—
109C2×D103—84no—
110C4×C103—22no—
111C4×C103#12112yes—
112C4×C103C22×D411yescharacteristic
113C4×C103C2×#4111yescharacteristic
114Dic103—42no—
115Dic103—42no—
116Dic103—42no—
117C2×C103—24no—
118C2×C103#46514yescomplemented
119C2×C103C2×#12111yescharacteristic
120C2×C103C22×SD1611yescharacteristic
121C2×C103C2×#13511yescharacteristic
122D103—82no—
123C2×#465—1031nomaximal, Sylow
124C103C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1031no—
126C23×C8D10311yescharacteristic, complemented
127C2×C22⋊Q8—1031no—
128C2×#121—1031no—
129#465—1038no—
130C2×#106—1031no—
131C2×#103—1031no—
132C2×#102—1031no—
133C22×Q8—1031no—
134C22×D4—1031no—
135C22×D4—2061no—
136C23×C4C2×D10311yescharacteristic
137C2×SD16—1034no—
138C2×SD16—2064no—
139C22×C8—24no—
140C22×C8C2×D10311yescharacteristic
141C22×C8C2×D10311yescharacteristic
142C22×C8C2×D10314yescomplemented
143C22⋊Q8—1034no—
144C22⋊Q8—2062no—
145#121—1034no—
146#121—2062no—
147C2×C4⋊C4—1031no—
148C2×C4⋊C4—1031no—
149C2×C4⋊C4—2061no—
150C2×C22⋊C4—2061no—
151C2×C22⋊C4—2061no—
152#106—1034no—
153#103—1034no—
154#102—1034no—
155C24—2061no—
156C24D41211yescharacteristic
157C2×Q8—1032no—
158C2×Q8—2062no—
159C2×Q8—2062no—
160C2×Q8—2061no—
161C2×D4—1032no—
162C2×D4—4124no—
163C2×D4—2064no—
164C2×D4—2062no—
165C2×D4—2061no—
166C2×D4—2062no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—2061no—
170C22×C4—2061no—
171C22×C4—2061no—
172C22×C4D41212yes—
173C22×C4D41211yescharacteristic
174C22×C4C22×D10312yes—
175C22×C4C22×D10311yescharacteristic
176SD16—2068no—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8C22⋊D10314yescomplemented
180C2×C8C22×D10312yes—
181C2×C8C22×D10312yes—
182C4⋊C4—1032no—
183C4⋊C4—1032no—
184C4⋊C4—2064no—
185C4⋊C4—2061no—
186C4⋊C4—2061no—
187C22⋊C4—2064no—
188C22⋊C4—2064no—
189C23—22no—
190C23—22no—
191C23—4122no—
192C23—4122no—
193C23—4122no—
194C23—2062no—
195C23#1311009414yescomplemented
196C23C2×D41212yes—
197C23C2×D41211yescharacteristic
198Q8—4122no—
199Q8—2062no—
200Q8—2062no—
201D4—4128no—
202D4—4122no—
203D4—2062no—
204D4—2062no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—4122no—
210C2×C4—4122no—
211C2×C4—4122no—
212C2×C4—2062no—
213C2×C4—2062no—
214C2×C4—2062no—
215C2×C4C2×C22⋊D10312yes—
216C2×C4C2×D41212yes—
217C2×C4C2×D41211yescharacteristic
218C2×C4#1311010512yes—
219C2×C4C23×D10311yescharacteristic
220C8—22no—
221C8C2×C22⋊D10314yescomplemented
222C22—22no—
223C22—28no—
224C22—22no—
225C22—8244no—
226C22—4122no—
227C22—4122no—
228C22—4122no—
229C22#1569299512yes—
230C22C2×#1311009412yes—
231C22C2×#1311009414yescomplemented
232C22#1569304312yes—
233C22C22×D41211yescharacteristic
234C4—22no—
235C4—4122no—
236C4—4122no—
237C4—4122no—
238C4#1569299512yes—
239C4C22×C22⋊D10311yescharacteristic
240C4C2×#1311010511yescharacteristic
241C2—24no—
242C2—8242no—
243C2#2026802014yesdirect product factor
244C2C2×#1569299511yescharacteristic
245C2C22×#1311009411yescharacteristic
246C2C2×#1569304311yescharacteristic
subgroupG/H
centerC23C2×D412
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
FittingC23×C8×C103C2
socleC23×C103C2×D4
prime radicalC2×#15692995C2
hypercenterC23×C8D103
hypocenterC103C2×#465
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C22×#13110094C2semidirect
2. C2×#15693000C2semidirect
4. #20268020C2direct (N × H)
5. C2×#15692953C2semidirect
6. C2×#15692946C2semidirect
7. C2×#15692943C2semidirect
8. C23×C8×C103C2semidirect
13. C2×#13110094C22semidirect
15. #15693000C22semidirect
23. #15692953C22semidirect
24. #15692946C22semidirect
25. #15692943C22semidirect
30. C22×C8×C103C22semidirect
62. C2×C8×C103D4semidirect
67. C23×C103SD16semidirect
97. C8×C103C2×D4semidirect
103. C22×C103C2×SD16semidirect
118. C2×C103#465semidirect
124. C103C2×#465semidirect
126. C23×C8D103semidirect
142. C22×C8C2×D103semidirect
179. C2×C8C22⋊D103semidirect
195. C23#13110094semidirect
221. C8C2×C22⋊D103semidirect
231. C22C2×#13110094semidirect
243. C2#20268020direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131100942
C2×#156930002
C2×#156929952
#202680202
C2×#156929532
C2×#156929462
C2×#156929432
C23×C8×C1032
C2×#465103