Finite Group Databasebeta

← C2×#20265501 #27181016

Subgroups — C2×#20265501 #27181016

Order 12928.

468 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#15692229C211yesmaximal, characteristic
2#20265501C218yesmaximal, direct product factor
3C2×#15692188C211yesmaximal, characteristic, complemented
4C2×#15692186C211yesmaximal, characteristic, complemented
5C2×#15692175C211yesmaximal, characteristic, complemented
6C2×#15692175C211yesmaximal, characteristic, complemented
7C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
8C2×#117×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic
10C22×C22⋊D101—21no—
11C22×#11717579C2211yescharacteristic
12C22×#11717579C2211yescharacteristic
13#15692229—22no—
14#15692229C2214yes—
15C2×C4⋊Dic101C2211yescharacteristic
16C2×#13108051—21no—
17#15692188C2214yescomplemented
18#15692186C2214yescomplemented
19#15692175C2214yescomplemented
20#15692175C2214yescomplemented
21C22⋊#11717579C2214yescomplemented
22C23×C4×C101C2211yescharacteristic
23#117×C101—22no—
24#117×C101C2214yescomplemented
25C2×C4⋊C4×C101C2211yescharacteristic
26C2×C4⋊C4×C101C2211yescharacteristic
27C2×C22⋊C4×C101—21no—
28C2×C42×C101—21no—
29C23×D101—21no—
30C2×D404—22no—
31C2×D404—22no—
32C2×D404—21no—
33C2×D404C2312yes—
34C2×C22⋊D101—24no—
35C2×C22⋊D101—44no—
36C22×Dic101—21no—
37C2×#11717579—22no—
38C2×#11717579—22no—
39C2×#11717579C2312yes—
40C2×#11717579C2312yes—
41C4⋊Dic101—21no—
42C4⋊Dic101C2312yes—
43#13108051—24no—
44C24×C101D411yescharacteristic
45C22×C4×C101—22no—
46C22×C4×C101—22no—
47C22×C4×C101—21no—
48C22×C4×C101—21no—
49C22×C4×C101D411yescharacteristic
50C22×C4×C101D412yes—
51C22×C4×C101C2312yes—
52C22×C4×C101C2311yescharacteristic
53C4⋊C4×C101—21no—
54C4⋊C4×C101—21no—
55C4⋊C4×C101C2312yes—
56C4⋊C4×C101C2312yes—
57C22⋊C4×C101—24no—
58C42×C101—24no—
59C23×C101—22no—
60C23×C101—22no—
61C23×C101—22no—
62C23×C101C2×D412yes—
63C23×C101C2×D411yescharacteristic
64C22×D101—22no—
65C22×D101—42no—
66C22×D101—42no—
67C22×D101—42no—
68C2×C4×C101—22no—
69C2×C4×C101—24no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—22no—
74C2×C4×C101—42no—
75C2×C4×C101—42no—
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101#4112yes—
79C2×C4×C101#4112yes—
80C2×C4×C101C2411yescharacteristic
81C22⋊D101—48no—
82C2×Dic101—22no—
83C2×Dic101—42no—
84D404—22no—
85D404—22no—
86D404—42no—
87#11717579—22no—
88#11717579—22no—
89C22×C101—22no—
90C22×C101—22no—
91C22×C101—22no—
92C22×C101—44no—
93C22×C101#12312yes—
94C22×C101#13512yes—
95C22×C101AGL(1,ℤ/8)12yes—
96C22×C101C22×D411yescharacteristic
97C2×D101—42no—
98C2×D101—42no—
99C2×D101—42no—
100C2×D101—84no—
101C4×C101—22no—
102C4×C101—42no—
103C4×C101—42no—
104C4×C101#12312yes—
105C4×C101C2×#4111yescharacteristic
106C4×C101C2×#4111yescharacteristic
107Dic101—42no—
108C2×C101—42no—
109C2×C101#48214yescomplemented
110C2×C101C2×#12311yescharacteristic
111C2×C101C2×#13511yescharacteristic
112C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
113D101—82no—
114C2×#482—1011nomaximal, Sylow
115C101C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—1011no—
117C2×#117D10111yescharacteristic, complemented
118#482—1018no—
119C2×#107—1011no—
120C2×#106—1011no—
121C2×#102—1011no—
122C2×#102—1011no—
123C2×C22⋊C8—1011no—
124C22×D4—1011no—
125C22×D4—2021no—
126C23×C4C2×D10111yescharacteristic
127C22×C8—1011no—
128C22×C8—1011no—
129#121—1014no—
130#121—2022no—
131#117—22no—
132#117C2×D10114yescomplemented
133C2×C4⋊C4—1011no—
134C2×C4⋊C4C2×D10111yescharacteristic
135C2×C4⋊C4C2×D10111yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—2021no—
138C2×C42—21no—
139#107—1014no—
140#106—1014no—
141#102—1014no—
142#102—1014no—
143C22⋊C8—1014no—
144C24—2021no—
145C24C22⋊D10111yescharacteristic
146C2×D4—1012no—
147C2×D4—4044no—
148C2×D4—2024no—
149C2×D4—2021no—
150C2×D4—2022no—
151C2×D4—2022no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—2021no—
157C22×C4C22⋊D10112yes—
158C22×C4C22⋊D10111yescharacteristic
159C22×C4C22×D10112yes—
160C22×C4C22×D10111yescharacteristic
161C2×C8—1012no—
162C2×C8—1012no—
163C2×C8—2022no—
164C2×C8—2022no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1012no—
168C4⋊C4—2021no—
169C4⋊C4C22×D10112yes—
170C4⋊C4C22×D10112yes—
171C22⋊C4—24no—
172C22⋊C4—2024no—
173C42—24no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—4042no—
178C23—4042no—
179C23—4042no—
180C23—2022no—
181C23C2×C22⋊D10112yes—
182C23C2×C22⋊D10111yescharacteristic
183D4—4048no—
184D4—4042no—
185D4—2022no—
186D4—2022no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—4042no—
196C2×C4—2022no—
197C2×C4C2×C22⋊D10112yes—
198C2×C4C2×C22⋊D10111yescharacteristic
199C2×C4#1310807412yes—
200C2×C4#1310807412yes—
201C2×C4C23×D10111yescharacteristic
202C8—2022no—
203C8—2022no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—8084no—
209C22—4042no—
210C22—4042no—
211C22—4042no—
212C22#1569223812yes—
213C22#1569227512yes—
214C22#1569228012yes—
215C22C22×C22⋊D10111yescharacteristic
216C4—22no—
217C4—42no—
218C4—42no—
219C4—4042no—
220C4#1569223812yes—
221C4C2×#1310807411yescharacteristic
222C4C2×#1310807411yescharacteristic
223C2—42no—
224C2—8082no—
225C2#2026550114yesdirect product factor
226C2C2×#1569223811yescharacteristic
227C2C2×#1569227511yescharacteristic
228C2C2×#1569228011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#117×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692229C2
hypercenterC2×#117D101
hypocenterC101C2×#482
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20265501C2direct (N × H)
3. C2×#15692188C2semidirect
4. C2×#15692186C2semidirect
5. C2×#15692175C2semidirect
6. C2×#15692175C2semidirect
7. C2×C22⋊#11717579C2semidirect
8. C2×#117×C101C2semidirect
17. #15692188C22semidirect
18. #15692186C22semidirect
19. #15692175C22semidirect
20. #15692175C22semidirect
21. C22⋊#11717579C22semidirect
24. #117×C101C22semidirect
109. C2×C101#482semidirect
115. C101C2×#482semidirect
117. C2×#117D101semidirect
132. #117C2×D101semidirect
225. C2#20265501direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156922292
#202655012
C2×#156921882
C2×#156921862
C2×#156921752
C2×C22⋊#117175792
C2×#117×C1012
C2×#482101