Finite Group Databasebeta

← C2×#127×D101 #27184402

Subgroups — C2×#127×D101 #27184402

Order 12928.

3000 conjugacy classes of subgroups; only the 661 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 661 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×D101C216yesmaximal, complemented
2C2×#15692252C211yesmaximal, characteristic, complemented
3C2×#15692251C216yesmaximal, complemented
4#127×D101C2116yesmaximal, direct product factor
5C2×C42×D101C211yesmaximal, characteristic, complemented
6C2×#127×C101C211yesmaximal, characteristic, complemented
7C24×D101C2214yes—
8C22×D404C2216yescomplemented
9C22×C22⋊D101C22112yescomplemented
10C2×D4×D101—248no—
11C2×D4×D101C22148yescomplemented
12C22×C4×D101C2213yes—
13#15692252C2218yescomplemented
14#15692251C22148yescomplemented
15C2×C4×Dic101C2213yescomplemented
16C42×D101C2218yescomplemented
17C22×D4×C101C2216yescomplemented
18#127×C101C2218yescomplemented
19C2×C42×C101C2211yescharacteristic, complemented
20C23×D101—248no—
21C23×D101C23116yes—
22C23×D101C2314yes—
23C23×D101C2311yescharacteristic
24C2×D404—224no—
25C2×D404C23124yescomplemented
26C2×C22⋊D101—248no—
27C2×C22⋊D101C23148yescomplemented
28D4×D101—2192no—
29C22×Dic101C2313yes—
30C2×C4×D101D4124yescomplemented
31C2×C4×D101C23112yes—
32C4×Dic101C23112yescomplemented
33C24×C101C2314yes—
34C2×D4×C101—224no—
35C2×D4×C101C23124yescomplemented
36C22×C4×C101C2313yes—
37C42×C101C2314yescomplemented
38C23×C101—224no—
39C23×C101C2418yes—
40C23×C101C2411yescharacteristic
41C22×D101—224no—
42C22×D101—296no—
43C22×D101—464no—
44C22×D101C2×D4112yes—
45C22×D101C2418yes—
46C22×D101C2412yes—
47D4×C101—248no—
48C2×C4×C101C2×D4112yescomplemented
49C2×C4×C101C2416yes—
50C22⋊D101—296no—
51C2×Dic101C2×D4112yescomplemented
52C2×Dic101C2416yes—
53D404—248no—
54C4×D101C2×D4148yescomplemented
55C22×C101—224no—
56C22×C101—416no—
57C22×C101C22×D416yes—
58C22×C101C2511yescharacteristic
59C2×D101—224no—
60C2×D101—416no—
61C2×D101—464no—
62C2×D101#127116yesdirect product factor
63C2×D101C22×D4112yes—
64C4×C101C22×D4112yescomplemented
65Dic101C22×D4112yescomplemented
66C2×C101—48no—
67C2×C101C2×#12714yescomplemented
68C2×C101C23×D413yes—
69D101—48no—
70D101C2×#12718yesdirect product factor
71C22×#127—1011nomaximal, Sylow
72C101C22×#12711yescharacteristic, complemented, Sylow
73C23×D4—1016no—
74C2×#127—1016no—
75C2×#127—10116no—
76C2×#127—1011no—
77C2×#127D10111yescharacteristic, direct product factor
78C22×C42—1011no—
79C25—1014no—
80C22×D4—10112no—
81C22×D4—10148no—
82C22×D4—1016no—
83C22×D4—20248no—
84C22×D4C2×D10116yescomplemented
85C23×C4—1013no—
86#127—10148no—
87#127—1018no—
88#127C2×D10118yesdirect product factor
89C2×C42—1013no—
90C2×C42—1018no—
91C2×C42C2×D10111yescharacteristic, complemented
92C24—1011no—
93C24—10116no—
94C24—1014no—
95C24—20248no—
96C24C22×D10114yes—
97C2×D4—224no—
98C2×D4—10148no—
99C2×D4—10124no—
100C2×D4—202192no—
101C2×D4—20248no—
102C2×D4—20224no—
103C2×D4C22×D101124yescomplemented
104C22×C4—1013no—
105C22×C4—10124no—
106C22×C4—10112no—
107C22×C4C22×D10113yes—
108C42—10112no—
109C42C22×D10114yescomplemented
110C23—224no—
111C23—10112no—
112C23—1012no—
113C23—1018no—
114C23—40464no—
115C23—20296no—
116C23—20224no—
117C23C23×D10118yes—
118C23C23×D10111yescharacteristic
119D4—248no—
120D4—20296no—
121D4—20248no—
122C2×C4—10112no—
123C2×C4—1016no—
124C2×C4—10148no—
125C2×C4D4×D101112yescomplemented
126C2×C4C23×D10116yes—
127C22—224no—
128C22—416no—
129C22—10112no—
130C22—10116no—
131C22—40464no—
132C22—40416no—
133C22—20224no—
134C22C2×D4×D10116yes—
135C22C24×D10111yescharacteristic
136C4—10112no—
137C4C2×D4×D101112yescomplemented
138C2—48no—
139C2—1018no—
140C2—4048no—
141C2#127×D10114yesdirect product factor
142C2C22×D4×D10113yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#127×C101C2
socleC23×C101C24
hypercenterC2×#127D101
hypocenterC101C22×#127
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. C2×#15692252C2semidirect
3. C2×#15692251C2semidirect
4. #127×D101C2direct (N × H)
5. C2×C42×D101C2semidirect
6. C2×#127×C101C2semidirect
8. C22×D404C22semidirect
9. C22×C22⋊D101C22semidirect
11. C2×D4×D101C22semidirect
13. #15692252C22semidirect
14. #15692251C22semidirect
15. C2×C4×Dic101C22semidirect
16. C42×D101C22semidirect
17. C22×D4×C101C22semidirect
18. #127×C101C22semidirect
19. C2×C42×C101C22semidirect
25. C2×D404C23semidirect
27. C2×C22⋊D101C23semidirect
30. C2×C4×D101D4semidirect
32. C4×Dic101C23semidirect
35. C2×D4×C101C23semidirect
37. C42×C101C23semidirect
48. C2×C4×C101C2×D4semidirect
51. C2×Dic101C2×D4semidirect
54. C4×D101C2×D4semidirect
62. C2×D101#127direct (N × H)
64. C4×C101C22×D4semidirect
65. Dic101C22×D4semidirect
67. C2×C101C2×#127semidirect
70. D101C2×#127direct (N × H)
72. C101C22×#127semidirect
77. C2×#127D101direct (N × H)
84. C22×D4C2×D101semidirect
88. #127C2×D101direct (N × H)
91. C2×C42C2×D101semidirect
103. C2×D4C22×D101semidirect
109. C42C22×D101semidirect
125. C2×C4D4×D101semidirect
137. C4C2×D4×D101semidirect
141. C2#127×D101direct (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
C2×#156922522
C2×#156922512
#127×D1012
C2×C42×D1012
C2×#127×C1012
C22×#127101