Finite Group Databasebeta

← C2×#20265407 #27180693

Subgroups — C2×#20265407 #27180693

Order 12928.

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×#13108063C211yesmaximal, characteristic, complemented
2C22×C8×D101C211yesmaximal, characteristic, complemented
3C2×#15692235C211yesmaximal, characteristic, complemented
4C2×#15692230C211yesmaximal, characteristic
5#20265407C218yesmaximal, direct product factor
6C2×#15692178C211yesmaximal, characteristic, complemented
7C2×#15692175C211yesmaximal, characteristic, complemented
8C2×#106×C101C211yesmaximal, characteristic, complemented
9C22×Dic202C2211yescharacteristic
10C22×D404—21no—
11C22×D404C2211yescharacteristic
12C22×C4×D101C2211yescharacteristic
13C2×#13108063—24no—
14C2×#13108063C2214yescomplemented
15C22×#11717579C2211yescharacteristic
16C2×C8×D101—24no—
17C2×C8×D101C2214yescomplemented
18#15692235—22no—
19#15692235C2214yescomplemented
20#15692230—22no—
21#15692230C2214yes—
22C2×C4⋊Dic101—21no—
23C2×#13108051—21no—
24C2×#13108051—21no—
25#15692178C2214yescomplemented
26#15692175C2214yescomplemented
27C22×C8×C101C2211yescharacteristic
28C2×C4⋊C4×C101C2211yescharacteristic
29C2×C4⋊C4×C101C2211yescharacteristic
30#106×C101C2214yescomplemented
31C23×D101—21no—
32C23×D101D411yescharacteristic
33C2×Dic202—22no—
34C2×Dic202—22no—
35C2×Dic202—21no—
36C2×Dic202C2312yes—
37C2×D404—22no—
38C2×D404—22no—
39C2×D404—24no—
40C2×D404—21no—
41C2×D404—44no—
42C2×D404C2312yes—
43C22×Dic101—21no—
44C22×Dic101D411yescharacteristic
45C2×C4×D101—22no—
46C2×C4×D101—22no—
47C2×C4×D101D412yes—
48C2×C4×D101C2312yes—
49#13108063—28no—
50C2×#11717579—22no—
51C2×#11717579C2312yes—
52C8×D101—28no—
53C4⋊Dic101—24no—
54#13108051—24no—
55#13108051—24no—
56C22×C4×C101—21no—
57C22×C4×C101—21no—
58C22×C4×C101C2311yescharacteristic
59C2×C8×C101D414yescomplemented
60C2×C8×C101C2312yes—
61C4⋊C4×C101—21no—
62C4⋊C4×C101—21no—
63C4⋊C4×C101C2312yes—
64C4⋊C4×C101C2312yes—
65C23×C101C2×D411yescharacteristic
66C22×D101—22no—
67C22×D101—22no—
68C22×D101—22no—
69C22×D101—42no—
70C22×D101—42no—
71C22×D101—42no—
72C22×D101SD1614yescomplemented
73C22×D101C2×D412yes—
74C2×C4×C101—22no—
75C2×C4×C101—22no—
76C2×C4×C101—42no—
77C2×C4×C101—42no—
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101C2×D412yes—
80C2×C4×C101#4112yes—
81C2×C4×C101C2411yescharacteristic
82C2×Dic101—22no—
83C2×Dic101—22no—
84C2×Dic101—42no—
85C2×Dic101C2×D412yes—
86D404—22no—
87D404—22no—
88D404—48no—
89D404—42no—
90C4×D101—24no—
91C4×D101—22no—
92C4×D101—22no—
93Dic202—22no—
94Dic202—22no—
95Dic202—42no—
96C8×C101C2×D414yescomplemented
97#11717579—22no—
98C22×C101#12112yes—
99C22×C101C2×SD1612yes—
100C22×C101#13512yes—
101C22×C101C22×D411yescharacteristic
102C2×D101—22no—
103C2×D101—22no—
104C2×D101—28no—
105C2×D101—42no—
106C2×D101—42no—
107C2×D101—42no—
108C2×D101—84no—
109C2×D101C2×SD1614yescomplemented
110C4×C101—42no—
111C4×C101—42no—
112C4×C101#12112yes—
113C4×C101C22×D411yescharacteristic
114C4×C101C2×#4111yescharacteristic
115Dic101—22no—
116Dic101—42no—
117C2×C101#46514yescomplemented
118C2×C101C2×#12111yescharacteristic
119C2×C101C22×SD1611yescharacteristic
120C2×C101C2×#13511yescharacteristic
121D101—24no—
122D101—82no—
123C2×#465—1011nomaximal, Sylow
124C101C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1011no—
126C23×C8—1011no—
127C2×C22⋊Q8—1011no—
128C2×#121—1011no—
129#465—1018no—
130C2×#106D10111yescharacteristic, complemented
131C2×#103—1011no—
132C2×#102—1011no—
133C22×Q8—1011no—
134C22×D4—1011no—
135C22×D4—2021no—
136C23×C4—1011no—
137C2×SD16—1014no—
138C2×SD16—2024no—
139C22×C8—1011no—
140C22×C8—1014no—
141C22×C8—2024no—
142C22×C8C2×D10111yescharacteristic
143C22⋊Q8—1014no—
144C22⋊Q8—2022no—
145#121—1014no—
146#121—2022no—
147C2×C4⋊C4—2021no—
148C2×C4⋊C4C2×D10111yescharacteristic
149C2×C4⋊C4C2×D10111yescharacteristic
150C2×C22⋊C4—2021no—
151C2×C22⋊C4—2021no—
152#106C2×D10114yescomplemented
153#103—1014no—
154#102—1014no—
155C24—1011no—
156C24—2021no—
157C2×Q8—1012no—
158C2×Q8—2022no—
159C2×Q8—2022no—
160C2×Q8—2021no—
161C2×D4—1012no—
162C2×D4—4044no—
163C2×D4—2024no—
164C2×D4—2021no—
165C2×D4—2022no—
166C2×D4—2022no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—1011no—
170C22×C4—1012no—
171C22×C4—1012no—
172C22×C4—2021no—
173C22×C4—2022no—
174C22×C4—2022no—
175C22×C4C22×D10111yescharacteristic
176SD16—2028no—
177C2×C8—1012no—
178C2×C8—2022no—
179C2×C8—2028no—
180C2×C8D40414yescomplemented
181C2×C8C22×D10112yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—2024no—
185C4⋊C4C22×D10112yes—
186C4⋊C4C22×D10112yes—
187C22⋊C4—2024no—
188C22⋊C4—2024no—
189C23—1012no—
190C23—1014no—
191C23—4042no—
192C23—4042no—
193C23—4042no—
194C23—2022no—
195C23—2022no—
196C23—2022no—
197C23D4×D10111yescharacteristic
198Q8—4042no—
199Q8—2022no—
200Q8—2022no—
201D4—4048no—
202D4—4042no—
203D4—2022no—
204D4—2022no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—1012no—
210C2×C4—4042no—
211C2×C4—2022no—
212C2×C4—2022no—
213C2×C4—2022no—
214C2×C4—2022no—
215C2×C4—2024no—
216C2×C4D4×D10111yescharacteristic
217C2×C4C2×D40412yes—
218C2×C4#1310807512yes—
219C2×C4C23×D10111yescharacteristic
220C8—2022no—
221C8C2×D40414yescomplemented
222C22—8084no—
223C22—1014no—
224C22—4042no—
225C22—4042no—
226C22—4042no—
227C22—2028no—
228C22—2022no—
229C22—2022no—
230C22#1569223012yes—
231C22SD16×D10112yes—
232C22#1569227612yes—
233C22C2×D4×D10111yescharacteristic
234C4—42no—
235C4—42no—
236C4—4042no—
237C4—2022no—
238C4#1569223012yes—
239C4C22×D40411yescharacteristic
240C4C2×#1310807511yescharacteristic
241C2—8082no—
242C2—2024no—
243C2#2026540714yesdirect product factor
244C2C2×#1569223011yescharacteristic
245C2C2×SD16×D10111yescharacteristic
246C2C2×#1569227611yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#106×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692230C2
hypercenterC2×#106D101
hypocenterC101C2×#465
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. C22×#13108063C2semidirect
2. C22×C8×D101C2semidirect
3. C2×#15692235C2semidirect
5. #20265407C2direct (N × H)
6. C2×#15692178C2semidirect
7. C2×#15692175C2semidirect
8. C2×#106×C101C2semidirect
14. C2×#13108063C22semidirect
17. C2×C8×D101C22semidirect
19. #15692235C22semidirect
25. #15692178C22semidirect
26. #15692175C22semidirect
30. #106×C101C22semidirect
59. C2×C8×C101D4semidirect
72. C22×D101SD16semidirect
96. C8×C101C2×D4semidirect
109. C2×D101C2×SD16semidirect
117. C2×C101#465semidirect
124. C101C2×#465semidirect
130. C2×#106D101semidirect
152. #106C2×D101semidirect
180. C2×C8D404semidirect
221. C8C2×D404semidirect
243. C2#20265407direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080632
C22×C8×D1012
C2×#156922352
C2×#156922302
#202654072
C2×#156921782
C2×#156921752
C2×#106×C1012
C2×#465101