Finite Group Databasebeta

← C2×#127×D31 #16858981

Subgroups — C2×#127×D31 #16858981

Order 3968.

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×D31C216yesmaximal, complemented
2C2×#11758451C211yesmaximal, characteristic, complemented
3C2×#11758517C216yesmaximal, complemented
4#127×D31C2116yesmaximal, direct product factor
5C2×C42×D31C211yesmaximal, characteristic, complemented
6C2×C31×#127C211yesmaximal, characteristic, complemented
7C24×D31C2214yes—
8C22×D4×C31C2216yescomplemented
9C22×C22⋊D31C22112yescomplemented
10C2×D4×D31—248no—
11C2×D4×D31C22148yescomplemented
12C22×D124C2216yescomplemented
13C22×C4×D31C2213yes—
14C31×#127C2218yescomplemented
15C2×C42×C31C2211yescharacteristic, complemented
16#11758517C22148yescomplemented
17C2×C4×Dic31C2213yescomplemented
18#11758451C2218yescomplemented
19C42×D31C2218yescomplemented
20C24×C31C2314yes—
21C23×D31—248no—
22C23×D31C2314yes—
23C23×D31C23116yes—
24C23×D31C2311yescharacteristic
25C2×D4×C31—224no—
26C2×D4×C31C23124yescomplemented
27C22×C4×C31C2313yes—
28C2×C22⋊D31—248no—
29C2×C22⋊D31C23148yescomplemented
30C22×Dic31C2313yes—
31D4×D31—2192no—
32C2×D124—224no—
33C2×D124C23124yescomplemented
34C2×C4×D31D4124yescomplemented
35C2×C4×D31C23112yes—
36C42×C31C2314yescomplemented
37C4×Dic31C23112yescomplemented
38C23×C31—224no—
39C23×C31C2418yes—
40C23×C31C2411yescharacteristic
41C22×D31—224no—
42C22×D31—296no—
43C22×D31—464no—
44C22×D31C2×D4112yes—
45C22×D31C2418yes—
46C22×D31C2412yes—
47D4×C31—248no—
48C2×C4×C31C2×D4112yescomplemented
49C2×C4×C31C2416yes—
50C22⋊D31—296no—
51C2×Dic31C2×D4112yescomplemented
52C2×Dic31C2416yes—
53D124—248no—
54C4×D31C2×D4148yescomplemented
55C22×#127—311nomaximal, Sylow
56C22×C31—224no—
57C22×C31—416no—
58C22×C31C22×D416yes—
59C22×C31C2511yescharacteristic
60C2×D31—224no—
61C2×D31—416no—
62C2×D31—464no—
63C2×D31#127116yesdirect product factor
64C2×D31C22×D4112yes—
65C4×C31C22×D4112yescomplemented
66Dic31C22×D4112yescomplemented
67C23×D4—316no—
68C2×#127—316no—
69C2×#127—3116no—
70C2×#127—311no—
71C2×#127D3111yescharacteristic, direct product factor
72C22×C42—311no—
73C2×C31—48no—
74C2×C31C2×#12714yescomplemented
75C2×C31C23×D413yes—
76D31—48no—
77D31C2×#12718yesdirect product factor
78C25—314no—
79C22×D4—3112no—
80C22×D4—3148no—
81C22×D4—316no—
82C22×D4—6248no—
83C22×D4C2×D3116yescomplemented
84C23×C4—313no—
85#127—3148no—
86#127—318no—
87#127C2×D3118yesdirect product factor
88C2×C42—313no—
89C2×C42—318no—
90C2×C42C2×D3111yescharacteristic, complemented
91C31C22×#12711yescharacteristic, complemented, Sylow
92C24—311no—
93C24—3116no—
94C24—314no—
95C24—6248no—
96C24C22×D3114yes—
97C2×D4—224no—
98C2×D4—3148no—
99C2×D4—3124no—
100C2×D4—62192no—
101C2×D4—6248no—
102C2×D4—6224no—
103C2×D4C22×D31124yescomplemented
104C22×C4—313no—
105C22×C4—3124no—
106C22×C4—3112no—
107C22×C4C22×D3113yes—
108C42—3112no—
109C42C22×D3114yescomplemented
110C23—224no—
111C23—3112no—
112C23—312no—
113C23—318no—
114C23—6296no—
115C23—6224no—
116C23—12464no—
117C23C23×D3118yes—
118C23C23×D3111yescharacteristic
119D4—248no—
120D4—6296no—
121D4—6248no—
122C2×C4—3112no—
123C2×C4—316no—
124C2×C4—3148no—
125C2×C4D4×D31112yescomplemented
126C2×C4C23×D3116yes—
127C22—224no—
128C22—416no—
129C22—3112no—
130C22—3116no—
131C22—6224no—
132C22—12464no—
133C22—12416no—
134C22C2×D4×D3116yes—
135C22C24×D3111yescharacteristic
136C4—3112no—
137C4C2×D4×D31112yescomplemented
138C2—48no—
139C2—318no—
140C2—1248no—
141C2#127×D3114yesdirect product factor
142C2C22×D4×D3113yes—
subgroupG/H
centerC23C23×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC2×C31×#127C2
socleC23×C31C24
hypercenterC2×#127D31
hypocenterC31C22×#127
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D31C2semidirect
2. C2×#11758451C2semidirect
3. C2×#11758517C2semidirect
4. #127×D31C2direct (N × H)
5. C2×C42×D31C2semidirect
6. C2×C31×#127C2semidirect
8. C22×D4×C31C22semidirect
9. C22×C22⋊D31C22semidirect
11. C2×D4×D31C22semidirect
12. C22×D124C22semidirect
14. C31×#127C22semidirect
15. C2×C42×C31C22semidirect
16. #11758517C22semidirect
17. C2×C4×Dic31C22semidirect
18. #11758451C22semidirect
19. C42×D31C22semidirect
26. C2×D4×C31C23semidirect
29. C2×C22⋊D31C23semidirect
33. C2×D124C23semidirect
34. C2×C4×D31D4semidirect
36. C42×C31C23semidirect
37. C4×Dic31C23semidirect
48. C2×C4×C31C2×D4semidirect
51. C2×Dic31C2×D4semidirect
54. C4×D31C2×D4semidirect
63. C2×D31#127direct (N × H)
65. C4×C31C22×D4semidirect
66. Dic31C22×D4semidirect
71. C2×#127D31direct (N × H)
74. C2×C31C2×#127semidirect
77. D31C2×#127direct (N × H)
83. C22×D4C2×D31semidirect
87. #127C2×D31direct (N × H)
90. C2×C42C2×D31semidirect
91. C31C22×#127semidirect
103. C2×D4C22×D31semidirect
109. C42C22×D31semidirect
125. C2×C4D4×D31semidirect
137. C4C2×D4×D31semidirect
141. C2#127×D31direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117584512
C2×#117585172
#127×D312
C2×C42×D312
C2×C31×#1272
C22×#12731