Finite Group Databasebeta

← C2×#20265812 #27184571

Subgroups — C2×#20265812 #27184571

Order 12928.

2536 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 2536 conjugacy classes of subgroups — too many to draw. Only its 581 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×D4×D101C214yesmaximal, complemented
2#20265812C2116yesmaximal, direct product factor
3C2×#15692226C214yesmaximal, complemented
4C2×#15692223C214yesmaximal, complemented
5C2×C4×D404C212yesmaximal, complemented
6C2×#127×C101C211yesmaximal, characteristic, complemented
7C24×D101C2214yescomplemented
8C22×D404C2212yes—
9C22×C22⋊D101C2218yescomplemented
10C2×D4×D101—232no—
11C2×D4×D101C22132yescomplemented
12C22×C4×D101C2214yescomplemented
13#15692226C22132yescomplemented
14#15692223C22132yescomplemented
15C4×D404C22116yescomplemented
16C2×C4⋊Dic101C2212yescomplemented
17C2×C22⋊Dic101C2214yescomplemented
18C2×#13108051C2214yescomplemented
19C22×D4×C101C2214yescomplemented
20C22×D4×C101C2212yescomplemented
21#127×C101C2218yescomplemented
22C2×C42×C101C2211yescharacteristic, complemented
23C23×D101—232no—
24C23×D101—48no—
25C23×D101C2314yes—
26C23×D101C23116yescomplemented
27C2×D404D4116yescomplemented
28C2×D404C2318yes—
29C2×C22⋊D101—232no—
30C2×C22⋊D101C23132yescomplemented
31D4×D101—2128no—
32C22×Dic101C2314yes—
33C2×C4×D101—216no—
34C2×C4×D101C23116yescomplemented
35C4⋊Dic101C2318yescomplemented
36C22⋊Dic101C23116yescomplemented
37#13108051C23116yescomplemented
38C24×C101C2314yes—
39C2×D4×C101—216no—
40C2×D4×C101—44no—
41C2×D4×C101C23116yescomplemented
42C2×D4×C101C2318yescomplemented
43C22×C4×C101C2311yescharacteristic
44C22×C4×C101C2312yes—
45C42×C101C2314yescomplemented
46C23×C101—216no—
47C23×C101—28no—
48C23×C101C2418yes—
49C23×C101C2411yescharacteristic
50C22×D101—28no—
51C22×D101—28no—
52C22×D101—264no—
53C22×D101—416no—
54C22×D101—464no—
55C22×D101C2×D4116yescomplemented
56C22×D101C2418yes—
57D4×C101—232no—
58D4×C101—48no—
59C2×C4×C101—22no—
60C2×C4×C101C2×D418yescomplemented
61C2×C4×C101C2412yes—
62C2×C4×C101C2414yes—
63C22⋊D101—264no—
64C2×Dic101—28no—
65C2×Dic101C2418yes—
66D404C2×D4132yescomplemented
67C4×D101—232no—
68C22×C101—216no—
69C22×C101—28no—
70C22×C101—416no—
71C22×C101C22×D414yes—
72C22×C101H(5,𝔽2)12yes—
73C22×C101C2511yescharacteristic
74C2×D101—232no—
75C2×D101—28no—
76C2×D101—28no—
77C2×D101—464no—
78C2×D101C22×D4116yescomplemented
79C4×C101—22no—
80C4×C101C22×D418yescomplemented
81Dic101—28no—
82C2×C101—48no—
83C2×C101D4214yescomplemented
84C2×C101C23×D412yes—
85C2×C101C2×H(5,𝔽2)11yescharacteristic
86D101—216no—
87C2×D42—1011nomaximal, Sylow
88C101C2×D4211yescharacteristic, complemented, Sylow
89C23×D4—1014no—
90D42—10116no—
91C2×#127D10111yescharacteristic, complemented
92C2×#121—1014no—
93C2×#120—1014no—
94C2×C4×D4—1012no—
95C25—1014no—
96C22×D4—1018no—
97C22×D4—10132no—
98C22×D4—1012no—
99C22×D4—20232no—
100C22×D4C2×D10114yescomplemented
101C22×D4C2×D10112yescomplemented
102C23×C4—1014no—
103#127C2×D10118yescomplemented
104#121—10132no—
105#120—10132no—
106C4×D4—10116no—
107C2×C4⋊C4—1012no—
108C2×C22⋊C4—1014no—
109C2×C22⋊C4—1014no—
110C2×C42C2×D10111yescharacteristic, complemented
111C24—1014no—
112C24—10116no—
113C24—4048no—
114C24—20232no—
115C24C22×D10114yes—
116C2×D4—216no—
117C2×D4—44no—
118C2×D4—10132no—
119C2×D4—10116no—
120C2×D4—1018no—
121C2×D4—202128no—
122C2×D4—20232no—
123C2×D4C22×D101116yescomplemented
124C2×D4C22×D10118yescomplemented
125C22×C4—1014no—
126C22×C4—10116no—
127C22×C4—20216no—
128C22×C4C22×D10111yescharacteristic
129C22×C4C22×D10112yes—
130C4⋊C4—1018no—
131C22⋊C4—10116no—
132C22⋊C4—10116no—
133C42C22×D10114yescomplemented
134C23—216no—
135C23—28no—
136C23—10116no—
137C23—1018no—
138C23—40416no—
139C23—40464no—
140C23—20264no—
141C23—2028no—
142C23—2028no—
143C23C23×D10118yes—
144C23C23×D10111yescharacteristic
145D4—232no—
146D4—48no—
147D4—10132no—
148D4—20264no—
149C2×C4—22no—
150C2×C4—1018no—
151C2×C4—2028no—
152C2×C4—20232no—
153C2×C4D4×D10118yescomplemented
154C2×C4C23×D10112yes—
155C2×C4C23×D10114yes—
156C22—216no—
157C22—28no—
158C22—416no—
159C22—10116no—
160C22—40464no—
161C22—2028no—
162C22—2028no—
163C22—20232no—
164C22C2×D4×D10114yes—
165C22#1569230012yes—
166C22C24×D10111yescharacteristic
167C4—22no—
168C4—2028no—
169C4C2×D4×D10118yescomplemented
170C2—48no—
171C2—20216no—
172C2#2026581214yesdirect product factor
173C2C22×D4×D10112yes—
174C2C2×#1569230011yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#127×C101C2
socleC23×C101C24
hypercenterC2×#127D101
hypocenterC101C2×D42
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D101C2semidirect
2. #20265812C2direct (N × H)
3. C2×#15692226C2semidirect
4. C2×#15692223C2semidirect
5. C2×C4×D404C2semidirect
6. C2×#127×C101C2semidirect
7. C24×D101C22semidirect
9. C22×C22⋊D101C22semidirect
11. C2×D4×D101C22semidirect
12. C22×C4×D101C22semidirect
13. #15692226C22semidirect
14. #15692223C22semidirect
15. C4×D404C22semidirect
16. C2×C4⋊Dic101C22semidirect
17. C2×C22⋊Dic101C22semidirect
18. C2×#13108051C22semidirect
19. C22×D4×C101C22semidirect
20. C22×D4×C101C22semidirect
21. #127×C101C22semidirect
22. C2×C42×C101C22semidirect
26. C23×D101C23semidirect
27. C2×D404D4semidirect
30. C2×C22⋊D101C23semidirect
34. C2×C4×D101C23semidirect
35. C4⋊Dic101C23semidirect
36. C22⋊Dic101C23semidirect
37. #13108051C23semidirect
41. C2×D4×C101C23semidirect
42. C2×D4×C101C23semidirect
45. C42×C101C23semidirect
55. C22×D101C2×D4semidirect
60. C2×C4×C101C2×D4semidirect
66. D404C2×D4semidirect
78. C2×D101C22×D4semidirect
80. C4×C101C22×D4semidirect
83. C2×C101D42semidirect
88. C101C2×D42semidirect
91. C2×#127D101semidirect
100. C22×D4C2×D101semidirect
101. C22×D4C2×D101semidirect
103. #127C2×D101semidirect
110. C2×C42C2×D101semidirect
123. C2×D4C22×D101semidirect
124. C2×D4C22×D101semidirect
133. C42C22×D101semidirect
153. C2×C4D4×D101semidirect
169. C4C2×D4×D101semidirect
172. C2#20265812direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1012
#202658122
C2×#156922262
C2×#156922232
C2×C4×D4042
C2×#127×C1012
C2×D42101