Finite Group Databasebeta

← C2×#127×D127 #30652000

Subgroups — C2×#127×D127 #30652000

Order 16256.

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×D127C216yesmaximal, complemented
2C2×#16861206C211yesmaximal, characteristic, complemented
3C2×#16861205C216yesmaximal, complemented
4#127×D127C2116yesmaximal, direct product factor
5C2×C42×D127C211yesmaximal, characteristic, complemented
6C2×#127×C127C211yesmaximal, characteristic, complemented
7C24×D127C2214yes—
8C22×D508C2216yescomplemented
9C22×C22⋊D127C22112yescomplemented
10C2×D4×D127—248no—
11C2×D4×D127C22148yescomplemented
12C22×C4×D127C2213yes—
13#16861206C2218yescomplemented
14#16861205C22148yescomplemented
15C2×C4×Dic127C2213yescomplemented
16C42×D127C2218yescomplemented
17C22×D4×C127C2216yescomplemented
18#127×C127C2218yescomplemented
19C2×C42×C127C2211yescharacteristic, complemented
20C23×D127—248no—
21C23×D127C23116yes—
22C23×D127C2314yes—
23C23×D127C2311yescharacteristic
24C2×D508—224no—
25C2×D508C23124yescomplemented
26C2×C22⋊D127—248no—
27C2×C22⋊D127C23148yescomplemented
28D4×D127—2192no—
29C22×Dic127C2313yes—
30C2×C4×D127D4124yescomplemented
31C2×C4×D127C23112yes—
32C4×Dic127C23112yescomplemented
33C24×C127C2314yes—
34C2×D4×C127—224no—
35C2×D4×C127C23124yescomplemented
36C22×C4×C127C2313yes—
37C42×C127C2314yescomplemented
38C22×D127—224no—
39C22×D127—296no—
40C22×D127—464no—
41C22×D127C2×D4112yes—
42C22×D127C2418yes—
43C22×D127C2412yes—
44D508—248no—
45C22⋊D127—296no—
46C2×Dic127C2×D4112yescomplemented
47C2×Dic127C2416yes—
48C4×D127C2×D4148yescomplemented
49C23×C127—224no—
50C23×C127C2418yes—
51C23×C127C2411yescharacteristic
52D4×C127—248no—
53C2×C4×C127C2×D4112yescomplemented
54C2×C4×C127C2416yes—
55C22×C127—224no—
56C22×C127—416no—
57C22×C127C22×D416yes—
58C22×C127C2511yescharacteristic
59C2×D127—224no—
60C2×D127—416no—
61C2×D127—464no—
62C2×D127#127116yesdirect product factor
63C2×D127C22×D4112yes—
64C4×C127C22×D4112yescomplemented
65Dic127C22×D4112yescomplemented
66C2×C127—48no—
67C2×C127C2×#12714yescomplemented
68C2×C127C23×D413yes—
69D127—48no—
70D127C2×#12718yesdirect product factor
71C22×#127—1271nomaximal, Sylow
72C127C22×#12711yescharacteristic, complemented, Sylow
73C23×D4—1276no—
74C2×#127—1276no—
75C2×#127—12716no—
76C2×#127—1271no—
77C2×#127D12711yescharacteristic, direct product factor
78C22×C42—1271no—
79C25—1274no—
80C22×D4—12712no—
81C22×D4—12748no—
82C22×D4—1276no—
83C22×D4—25448no—
84C22×D4C2×D12716yescomplemented
85C23×C4—1273no—
86#127—12748no—
87#127—1278no—
88#127C2×D12718yesdirect product factor
89C2×C42—1273no—
90C2×C42—1278no—
91C2×C42C2×D12711yescharacteristic, complemented
92C24—1271no—
93C24—12716no—
94C24—1274no—
95C24—25448no—
96C24C22×D12714yes—
97C2×D4—224no—
98C2×D4—12748no—
99C2×D4—12724no—
100C2×D4—254192no—
101C2×D4—25448no—
102C2×D4—25424no—
103C2×D4C22×D127124yescomplemented
104C22×C4—1273no—
105C22×C4—12724no—
106C22×C4—12712no—
107C22×C4C22×D12713yes—
108C42—12712no—
109C42C22×D12714yescomplemented
110C23—224no—
111C23—12712no—
112C23—1272no—
113C23—1278no—
114C23—50864no—
115C23—25496no—
116C23—25424no—
117C23C23×D12718yes—
118C23C23×D12711yescharacteristic
119D4—248no—
120D4—25496no—
121D4—25448no—
122C2×C4—12712no—
123C2×C4—1276no—
124C2×C4—12748no—
125C2×C4D4×D127112yescomplemented
126C2×C4C23×D12716yes—
127C22—224no—
128C22—416no—
129C22—12716no—
130C22—12712no—
131C22—50864no—
132C22—50816no—
133C22—25424no—
134C22C2×D4×D12716yes—
135C22C24×D12711yescharacteristic
136C4—12712no—
137C4C2×D4×D127112yescomplemented
138C2—48no—
139C2—1278no—
140C2—5088no—
141C2#127×D12714yesdirect product factor
142C2C22×D4×D12713yes—
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#127×C127C2
socleC23×C127C24
hypercenterC2×#127D127
hypocenterC127C22×#127
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. C2×#16861206C2semidirect
3. C2×#16861205C2semidirect
4. #127×D127C2direct (N × H)
5. C2×C42×D127C2semidirect
6. C2×#127×C127C2semidirect
8. C22×D508C22semidirect
9. C22×C22⋊D127C22semidirect
11. C2×D4×D127C22semidirect
13. #16861206C22semidirect
14. #16861205C22semidirect
15. C2×C4×Dic127C22semidirect
16. C42×D127C22semidirect
17. C22×D4×C127C22semidirect
18. #127×C127C22semidirect
19. C2×C42×C127C22semidirect
25. C2×D508C23semidirect
27. C2×C22⋊D127C23semidirect
30. C2×C4×D127D4semidirect
32. C4×Dic127C23semidirect
35. C2×D4×C127C23semidirect
37. C42×C127C23semidirect
46. C2×Dic127C2×D4semidirect
48. C4×D127C2×D4semidirect
53. C2×C4×C127C2×D4semidirect
62. C2×D127#127direct (N × H)
64. C4×C127C22×D4semidirect
65. Dic127C22×D4semidirect
67. C2×C127C2×#127semidirect
70. D127C2×#127direct (N × H)
72. C127C22×#127semidirect
77. C2×#127D127direct (N × H)
84. C22×D4C2×D127semidirect
88. #127C2×D127direct (N × H)
91. C2×C42C2×D127semidirect
103. C2×D4C22×D127semidirect
109. C42C22×D127semidirect
125. C2×C4D4×D127semidirect
137. C4C2×D4×D127semidirect
141. C2#127×D127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168612062
C2×#168612052
#127×D1272
C2×C42×D1272
C2×#127×C1272
C22×#127127