Finite Group Databasebeta

← C2×#127×D131 #30679231

Subgroups — C2×#127×D131 #30679231

Order 16768.

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×D131C216yesmaximal, complemented
2C2×#16862243C211yesmaximal, characteristic, complemented
3C2×#16862242C216yesmaximal, complemented
4#127×D131C2116yesmaximal, direct product factor
5C2×C42×D131C211yesmaximal, characteristic, complemented
6C2×#127×C131C211yesmaximal, characteristic, complemented
7C24×D131C2214yes—
8C22×D524C2216yescomplemented
9C22×C22⋊D131C22112yescomplemented
10C2×D4×D131—248no—
11C2×D4×D131C22148yescomplemented
12C22×C4×D131C2213yes—
13#16862243C2218yescomplemented
14#16862242C22148yescomplemented
15C2×C4×Dic131C2213yescomplemented
16C42×D131C2218yescomplemented
17C22×D4×C131C2216yescomplemented
18#127×C131C2218yescomplemented
19C2×C42×C131C2211yescharacteristic, complemented
20C23×D131—248no—
21C23×D131C23116yes—
22C23×D131C2314yes—
23C23×D131C2311yescharacteristic
24C2×D524—224no—
25C2×D524C23124yescomplemented
26C2×C22⋊D131—248no—
27C2×C22⋊D131C23148yescomplemented
28D4×D131—2192no—
29C22×Dic131C2313yes—
30C2×C4×D131D4124yescomplemented
31C2×C4×D131C23112yes—
32C4×Dic131C23112yescomplemented
33C24×C131C2314yes—
34C2×D4×C131—224no—
35C2×D4×C131C23124yescomplemented
36C22×C4×C131C2313yes—
37C42×C131C2314yescomplemented
38C22×D131—224no—
39C22×D131—296no—
40C22×D131—464no—
41C22×D131C2×D4112yes—
42C22×D131C2418yes—
43C22×D131C2412yes—
44D524—248no—
45C22⋊D131—296no—
46C2×Dic131C2×D4112yescomplemented
47C2×Dic131C2416yes—
48C4×D131C2×D4148yescomplemented
49C23×C131—224no—
50C23×C131C2418yes—
51C23×C131C2411yescharacteristic
52D4×C131—248no—
53C2×C4×C131C2×D4112yescomplemented
54C2×C4×C131C2416yes—
55C22×C131—224no—
56C22×C131—416no—
57C22×C131C22×D416yes—
58C22×C131C2511yescharacteristic
59C2×D131—224no—
60C2×D131—416no—
61C2×D131—464no—
62C2×D131#127116yesdirect product factor
63C2×D131C22×D4112yes—
64C4×C131C22×D4112yescomplemented
65Dic131C22×D4112yescomplemented
66C2×C131—48no—
67C2×C131C2×#12714yescomplemented
68C2×C131C23×D413yes—
69D131—48no—
70D131C2×#12718yesdirect product factor
71C131C22×#12711yescharacteristic, complemented, Sylow
72C22×#127—1311nomaximal, Sylow
73C23×D4—1316no—
74C2×#127—1316no—
75C2×#127—13116no—
76C2×#127—1311no—
77C2×#127D13111yescharacteristic, direct product factor
78C22×C42—1311no—
79C25—1314no—
80C22×D4—26248no—
81C22×D4—13112no—
82C22×D4—13148no—
83C22×D4—1316no—
84C22×D4C2×D13116yescomplemented
85C23×C4—1313no—
86#127—13148no—
87#127—1318no—
88#127C2×D13118yesdirect product factor
89C2×C42—1313no—
90C2×C42—1318no—
91C2×C42C2×D13111yescharacteristic, complemented
92C24—26248no—
93C24—13116no—
94C24—1314no—
95C24—1311no—
96C24C22×D13114yes—
97C2×D4—224no—
98C2×D4—262192no—
99C2×D4—26248no—
100C2×D4—26224no—
101C2×D4—13148no—
102C2×D4—13124no—
103C2×D4C22×D131124yescomplemented
104C22×C4—1313no—
105C22×C4—13124no—
106C22×C4—13112no—
107C22×C4C22×D13113yes—
108C42—13112no—
109C42C22×D13114yescomplemented
110C23—224no—
111C23—26296no—
112C23—26224no—
113C23—52464no—
114C23—1318no—
115C23—13112no—
116C23—1312no—
117C23C23×D13118yes—
118C23C23×D13111yescharacteristic
119D4—248no—
120D4—26296no—
121D4—26248no—
122C2×C4—13112no—
123C2×C4—1316no—
124C2×C4—13148no—
125C2×C4D4×D131112yescomplemented
126C2×C4C23×D13116yes—
127C22—224no—
128C22—416no—
129C22—26224no—
130C22—52464no—
131C22—52416no—
132C22—13112no—
133C22—13116no—
134C22C2×D4×D13116yes—
135C22C24×D13111yescharacteristic
136C4—13112no—
137C4C2×D4×D131112yescomplemented
138C2—48no—
139C2—5248no—
140C2—1318no—
141C2#127×D13114yesdirect product factor
142C2C22×D4×D13113yes—
subgroupG/H
centerC23C23×D131
commutator, cosocleC22×C131C25
FrattiniC22C24×D131
FittingC2×#127×C131C2
socleC23×C131C24
hypercenterC2×#127D131
hypocenterC131C22×#127
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×#16862243C2semidirect
3. C2×#16862242C2semidirect
4. #127×D131C2direct (N × H)
5. C2×C42×D131C2semidirect
6. C2×#127×C131C2semidirect
8. C22×D524C22semidirect
9. C22×C22⋊D131C22semidirect
11. C2×D4×D131C22semidirect
13. #16862243C22semidirect
14. #16862242C22semidirect
15. C2×C4×Dic131C22semidirect
16. C42×D131C22semidirect
17. C22×D4×C131C22semidirect
18. #127×C131C22semidirect
19. C2×C42×C131C22semidirect
25. C2×D524C23semidirect
27. C2×C22⋊D131C23semidirect
30. C2×C4×D131D4semidirect
32. C4×Dic131C23semidirect
35. C2×D4×C131C23semidirect
37. C42×C131C23semidirect
46. C2×Dic131C2×D4semidirect
48. C4×D131C2×D4semidirect
53. C2×C4×C131C2×D4semidirect
62. C2×D131#127direct (N × H)
64. C4×C131C22×D4semidirect
65. Dic131C22×D4semidirect
67. C2×C131C2×#127semidirect
70. D131C2×#127direct (N × H)
71. C131C22×#127semidirect
77. C2×#127D131direct (N × H)
84. C22×D4C2×D131semidirect
88. #127C2×D131direct (N × H)
91. C2×C42C2×D131semidirect
103. C2×D4C22×D131semidirect
109. C42C22×D131semidirect
125. C2×C4D4×D131semidirect
137. C4C2×D4×D131semidirect
141. C2#127×D131direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C2×#168622432
C2×#168622422
#127×D1312
C2×C42×D1312
C2×#127×C1312
C22×#127131