Finite Group Databasebeta

← C2×#20265405 #27180691

Subgroups — C2×#20265405 #27180691

Order 12928.

564 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13108062C211yesmaximal, characteristic, complemented
2C22×C8×D101C211yesmaximal, characteristic, complemented
3C2×#15692235C211yesmaximal, characteristic, complemented
4C2×#15692226C211yesmaximal, characteristic
5#20265405C218yesmaximal, direct product factor
6C2×#15692186C211yesmaximal, characteristic, complemented
7C2×#15692180C211yesmaximal, characteristic, complemented
8C2×#102×C101C211yesmaximal, characteristic, complemented
9C22×Dic202C2211yescharacteristic
10C22×C22⋊D101—21no—
11C22×C4×D101C2211yescharacteristic
12C2×#13108062—24no—
13C2×#13108062C2214yescomplemented
14C22×#11717579C2211yescharacteristic
15C2×C8×D101—24no—
16C2×C8×D101C2214yescomplemented
17#15692235—22no—
18#15692235C2214yescomplemented
19#15692226—22no—
20#15692226C2214yes—
21C2×C4⋊Dic101—21no—
22C2×C4⋊Dic101C2211yescharacteristic
23C2×C22⋊Dic101—21no—
24C2×#13108051—21no—
25#15692186C2214yescomplemented
26#15692180C2214yescomplemented
27C22×D4×C101C2211yescharacteristic
28C22×C8×C101C2211yescharacteristic
29C2×C4⋊C4×C101C2211yescharacteristic
30#102×C101C2214yescomplemented
31C23×D101D411yescharacteristic
32C2×Dic202—22no—
33C2×Dic202—22no—
34C2×Dic202—21no—
35C2×Dic202C2312yes—
36C2×C22⋊D101—24no—
37C2×C22⋊D101—44no—
38C22×Dic101—21no—
39C22×Dic101—21no—
40C22×Dic101D411yescharacteristic
41C2×C4×D101—22no—
42C2×C4×D101—22no—
43C2×C4×D101D412yes—
44C2×C4×D101C2312yes—
45#13108062—28no—
46C2×#11717579D414yescomplemented
47C2×#11717579C2312yes—
48C8×D101—28no—
49C4⋊Dic101—21no—
50C4⋊Dic101—24no—
51C4⋊Dic101C2312yes—
52C22⋊Dic101—24no—
53#13108051—24no—
54C24×C101—21no—
55C2×D4×C101—22no—
56C2×D4×C101—21no—
57C2×D4×C101—22no—
58C2×D4×C101C2312yes—
59C22×C4×C101—21no—
60C22×C4×C101C2311yescharacteristic
61C2×C8×C101—22no—
62C2×C8×C101C2312yes—
63C4⋊C4×C101—21no—
64C4⋊C4×C101C2312yes—
65C23×C101—22no—
66C23×C101—42no—
67C23×C101—42no—
68C23×C101—42no—
69C23×C101C2×D411yescharacteristic
70C22×D101—22no—
71C22×D101—22no—
72C22×D101SD1614yescomplemented
73C22×D101C2×D412yes—
74D4×C101—22no—
75D4×C101—22no—
76D4×C101—42no—
77C2×C4×C101—22no—
78C2×C4×C101—42no—
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101C2×D412yes—
81C2×C4×C101#4112yes—
82C2×C4×C101C2411yescharacteristic
83C22⋊D101—48no—
84C2×Dic101—22no—
85C2×Dic101—22no—
86C2×Dic101—22no—
87C2×Dic101—42no—
88C2×Dic101—42no—
89C2×Dic101C2×D412yes—
90C4×D101—24no—
91C4×D101—22no—
92C4×D101—22no—
93Dic202—22no—
94Dic202—22no—
95Dic202—42no—
96C8×C101—22no—
97#11717579C2×D414yescomplemented
98C22×C101—42no—
99C22×C101—42no—
100C22×C101—42no—
101C22×C101—84no—
102C22×C101#12112yes—
103C22×C101C2×SD1612yes—
104C22×C101#13512yes—
105C22×C101C22×D411yescharacteristic
106C2×D101—22no—
107C2×D101—22no—
108C2×D101—28no—
109C2×D101C2×SD1614yescomplemented
110C4×C101—42no—
111C4×C101#12112yes—
112C4×C101C22×D411yescharacteristic
113C4×C101C2×#4111yescharacteristic
114Dic101—22no—
115Dic101—42no—
116Dic101—42no—
117C2×C101—82no—
118C2×C101#46514yescomplemented
119C2×C101C2×#12111yescharacteristic
120C2×C101C22×SD1611yescharacteristic
121C2×C101C2×#13511yescharacteristic
122D101—24no—
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×#106—1011no—
131C2×#103—1011no—
132C2×#102D10111yescharacteristic, complemented
133C22×Q8—1011no—
134C22×D4—2021no—
135C22×D4C2×D10111yescharacteristic
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—1011no—
148C2×C4⋊C4—2021no—
149C2×C4⋊C4C2×D10111yescharacteristic
150C2×C22⋊C4—2021no—
151C2×C22⋊C4—2021no—
152#106—1014no—
153#103—1014no—
154#102C2×D10114yescomplemented
155C24—21no—
156C24—1011no—
157C2×Q8—1012no—
158C2×Q8—2022no—
159C2×Q8—2022no—
160C2×Q8—2021no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—4044no—
165C2×D4—2024no—
166C2×D4C22×D10112yes—
167C22×C4—21no—
168C22×C4—1011no—
169C22×C4—1012no—
170C22×C4—1012no—
171C22×C4—2021no—
172C22×C4—2021no—
173C22×C4—2022no—
174C22×C4—2022no—
175C22×C4C22×D10111yescharacteristic
176SD16—2028no—
177C2×C8—22no—
178C2×C8—1014no—
179C2×C8—1012no—
180C2×C8—2028no—
181C2×C8C22×D10112yes—
182C4⋊C4—21no—
183C4⋊C4—1012no—
184C4⋊C4—2024no—
185C4⋊C4—2021no—
186C4⋊C4C22×D10112yes—
187C22⋊C4—2024no—
188C22⋊C4—2024no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—1012no—
194C23—1014no—
195C23—2022no—
196C23—2022no—
197C23D4×D10111yescharacteristic
198Q8—4042no—
199Q8—2022no—
200Q8—2022no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—4048no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1012no—
208C2×C4—4042no—
209C2×C4—4042no—
210C2×C4—2022no—
211C2×C4—2022no—
212C2×C4—2022no—
213C2×C4—2022no—
214C2×C4—2022no—
215C2×C4—2024no—
216C2×C4D4×D10112yes—
217C2×C4D4×D10111yescharacteristic
218C2×C4#1310807412yes—
219C2×C4C23×D10111yescharacteristic
220C8—22no—
221C8—1014no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—1014no—
227C22—2028no—
228C22—2022no—
229C22—2022no—
230C22#1569222812yes—
231C22SD16×D10112yes—
232C22#1569227412yes—
233C22C2×D4×D10111yescharacteristic
234C4—42no—
235C4—4042no—
236C4—4042no—
237C4—2022no—
238C4#1569222812yes—
239C4C2×D4×D10111yescharacteristic
240C4C2×#1310807411yescharacteristic
241C2—82no—
242C2—2024no—
243C2#2026540514yesdirect product factor
244C2C2×#1569222811yescharacteristic
245C2C2×SD16×D10111yescharacteristic
246C2C2×#1569227411yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#102×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692226C2
hypercenterC2×#102D101
hypocenterC101C2×#465
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#13108062C2semidirect
2. C22×C8×D101C2semidirect
3. C2×#15692235C2semidirect
5. #20265405C2direct (N × H)
6. C2×#15692186C2semidirect
7. C2×#15692180C2semidirect
8. C2×#102×C101C2semidirect
13. C2×#13108062C22semidirect
16. C2×C8×D101C22semidirect
18. #15692235C22semidirect
25. #15692186C22semidirect
26. #15692180C22semidirect
30. #102×C101C22semidirect
46. C2×#11717579D4semidirect
72. C22×D101SD16semidirect
97. #11717579C2×D4semidirect
109. C2×D101C2×SD16semidirect
118. C2×C101#465semidirect
124. C101C2×#465semidirect
132. C2×#102D101semidirect
154. #102C2×D101semidirect
243. C2#20265405direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080622
C22×C8×D1012
C2×#156922352
C2×#156922262
#202654052
C2×#156921862
C2×#156921802
C2×#102×C1012
C2×#465101