Finite Group Databasebeta

← C2×#20265739 #27184435

Subgroups — C2×#20265739 #27184435

Order 12928.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C23×C22⋊D101C212yesmaximal, complemented
2#20265739C2116yesmaximal, direct product factor
3C2×#15692239C214yesmaximal, complemented
4C2×#15692229C214yesmaximal, complemented
5C2×#15692225C214yesmaximal, complemented
6C22×C22⋊C4×C101C211yesmaximal, characteristic, complemented
7C24×D101C2212yescomplemented
8C22×D404C2214yescomplemented
9C22×C22⋊D101—216no—
10C22×C22⋊D101C2214yes—
11C22×C22⋊D101C2214yescomplemented
12C22×C22⋊D101C22116yescomplemented
13C23×Dic101C2212yescomplemented
14#15692239C22132yescomplemented
15#15692229C22132yescomplemented
16#15692225C22132yescomplemented
17C2×C4⋊Dic101C2214yescomplemented
18C2×#13108051C2218yescomplemented
19C25×C101C2211yescharacteristic
20C23×C4×C101C2212yescomplemented
21C2×C22⋊C4×C101C2214yescomplemented
22C2×C22⋊C4×C101C2218yescomplemented
23C23×D101—28no—
24C23×D101—48no—
25C23×D101C2314yes—
26C23×D101C2318yescomplemented
27C2×D404—48no—
28C2×D404C23116yescomplemented
29C2×C22⋊D101—216no—
30C2×C22⋊D101—216no—
31C2×C22⋊D101—264no—
32C2×C22⋊D101C23116yes—
33C2×C22⋊D101C23116yescomplemented
34C22×Dic101—28no—
35C22×Dic101C2314yes—
36C22×Dic101C2318yescomplemented
37C4⋊Dic101C23116yescomplemented
38#13108051C23132yescomplemented
39C24×C101—44no—
40C24×C101D418yescomplemented
41C24×C101C2312yes—
42C24×C101C2311yescharacteristic
43C24×C101C2314yes—
44C22×C4×C101—44no—
45C22×C4×C101C2314yes—
46C22×C4×C101C2318yescomplemented
47C22⋊C4×C101C23116yescomplemented
48C23×C101—24no—
49C23×C101—28no—
50C23×C101—416no—
51C23×C101—48no—
52C23×C101C2×D418yes—
53C23×C101C2×D414yes—
54C23×C101C2×D4116yescomplemented
55C23×C101C2414yes—
56C23×C101C2412yes—
57C23×C101C2411yescharacteristic
58C22×D101—28no—
59C22×D101—216no—
60C22×D101—216no—
61C22×D101—416no—
62C22×D101—432no—
63C22×D101C2418yes—
64C2×C4×C101—28no—
65C2×C4×C101—48no—
66C2×C4×C101C2418yes—
67C22⋊D101—232no—
68C22⋊D101—232no—
69C2×Dic101—28no—
70C2×Dic101—216no—
71C2×Dic101C2418yes—
72D404—416no—
73C22×C101—216no—
74C22×C101—28no—
75C22×C101—24no—
76C22×C101—28no—
77C22×C101—416no—
78C22×C101C22×D418yes—
79C22×C101C22×D414yes—
80C22×C101C22×D412yes—
81C22×C101H(5,𝔽2)14yes—
82C22×C101C2511yescharacteristic
83C2×D101—28no—
84C2×D101—216no—
85C2×D101—432no—
86C2×D101—416no—
87C4×C101—28no—
88Dic101—28no—
89C2×C101—28no—
90C2×C101—24no—
91C2×C101#53414yescomplemented
92C2×C101C23×D411yescharacteristic
93C2×C101C2×H(5,𝔽2)12yes—
94D101—48no—
95C2×#534—1011nomaximal, Sylow
96C101C2×#53411yescharacteristic, complemented, Sylow
97C23×D4—1012no—
98#534—10116no—
99C2×#123—1014no—
100C2×#121—1014no—
101C2×#120—1014no—
102C22×C22⋊C4D10111yescharacteristic, complemented
103C25—1012no—
104C25C2×D10111yescharacteristic
105C22×D4—1014no—
106C22×D4—1014no—
107C22×D4—10116no—
108C22×D4—1014no—
109C22×D4—20216no—
110C23×C4—1012no—
111C23×C4C2×D10112yescomplemented
112#123—10132no—
113#121—10132no—
114#120—10132no—
115C2×C4⋊C4—1014no—
116C2×C22⋊C4—1018no—
117C2×C22⋊C4C2×D10114yescomplemented
118C2×C22⋊C4C2×D10118yescomplemented
119C24—44no—
120C24—1018no—
121C24—1014no—
122C24—4048no—
123C24—2028no—
124C24D40418yescomplemented
125C24C22×D10112yes—
126C24C22×D10111yescharacteristic
127C24C22×D10114yes—
128C2×D4—10116no—
129C2×D4—10116no—
130C2×D4—10116no—
131C2×D4—4048no—
132C2×D4—20216no—
133C2×D4—20216no—
134C2×D4—20264no—
135C22×C4—44no—
136C22×C4—1018no—
137C22×C4—1014no—
138C22×C4—2028no—
139C22×C4C22×D10114yes—
140C22×C4C22×D10118yescomplemented
141C4⋊C4—10116no—
142C22⋊C4—10132no—
143C22⋊C4C22×D101116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—1018no—
149C23—40432no—
150C23—40416no—
151C23—20216no—
152C23—2028no—
153C23—20216no—
154C23C2×D40418yes—
155C23C2×D40414yes—
156C23C2×D404116yescomplemented
157C23C23×D10114yes—
158C23C23×D10112yes—
159C23C23×D10111yescharacteristic
160D4—40416no—
161D4—20232no—
162D4—20232no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—1018no—
166C2×C4—20216no—
167C2×C4—2028no—
168C2×C4C23×D10118yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—40432no—
175C22—40416no—
176C22—2028no—
177C22—20216no—
178C22C22×D40418yes—
179C22C22×D40414yes—
180C22C22×D40412yes—
181C22#1569230014yes—
182C22C24×D10111yescharacteristic
183C4—28no—
184C4—2028no—
185C2—28no—
186C2—24no—
187C2—4048no—
188C2#2026573914yesdirect product factor
189C2C23×D40411yescharacteristic
190C2C2×#1569230012yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC22×C22⋊C4×C101C2
socleC23×C101C24
hypercenterC22×C22⋊C4D101
hypocenterC101C2×#534
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D101C2semidirect
2. #20265739C2direct (N × H)
3. C2×#15692239C2semidirect
4. C2×#15692229C2semidirect
5. C2×#15692225C2semidirect
6. C22×C22⋊C4×C101C2semidirect
7. C24×D101C22semidirect
8. C22×D404C22semidirect
11. C22×C22⋊D101C22semidirect
12. C22×C22⋊D101C22semidirect
13. C23×Dic101C22semidirect
14. #15692239C22semidirect
15. #15692229C22semidirect
16. #15692225C22semidirect
17. C2×C4⋊Dic101C22semidirect
18. C2×#13108051C22semidirect
20. C23×C4×C101C22semidirect
21. C2×C22⋊C4×C101C22semidirect
22. C2×C22⋊C4×C101C22semidirect
26. C23×D101C23semidirect
28. C2×D404C23semidirect
33. C2×C22⋊D101C23semidirect
36. C22×Dic101C23semidirect
37. C4⋊Dic101C23semidirect
38. #13108051C23semidirect
40. C24×C101D4semidirect
46. C22×C4×C101C23semidirect
47. C22⋊C4×C101C23semidirect
54. C23×C101C2×D4semidirect
91. C2×C101#534semidirect
96. C101C2×#534semidirect
102. C22×C22⋊C4D101semidirect
111. C23×C4C2×D101semidirect
117. C2×C22⋊C4C2×D101semidirect
118. C2×C22⋊C4C2×D101semidirect
124. C24D404semidirect
140. C22×C4C22×D101semidirect
143. C22⋊C4C22×D101semidirect
156. C23C2×D404semidirect
188. C2#20265739direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1012
#202657392
C2×#156922392
C2×#156922292
C2×#156922252
C22×C22⋊C4×C1012
C2×#534101