Finite Group Databasebeta

← C2×#126×D31 #16858965

Subgroups — C2×#126×D31 #16858965

Order 3968.

1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 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
1C2×#11758475C213yesmaximal, complemented
2C2×#11758453C211yesmaximal, characteristic, complemented
3C2×#11758456C213yesmaximal, complemented
4#126×D31C2116yesmaximal, direct product factor
5C2×C4⋊C4×D31C213yesmaximal, complemented
6C2×C22⋊C4×D31C213yesmaximal
7C2×C42×D31C211yesmaximal, characteristic, complemented
8C2×C31×#126C211yesmaximal, characteristic, complemented
9C24×D31C2211yescharacteristic
10C22×C4×D31C2213yes—
11C22×C4×D31C2213yes—
12C31×#126C2218yescomplemented
13C2×C4⋊C4×C31C2213yescomplemented
14C2×C22⋊C4×C31C2213yes—
15C2×C42×C31C2211yescharacteristic, complemented
16C2×C22⋊Dic31C2213yes—
17C2×(C2×D31)⋊C4C2213yes—
18C2×(C2×D31)⋊C4C2213yes—
19C2×C4⋊Dic31C2213yescomplemented
20C2×Dic31⋊C4C2213yescomplemented
21C2×Dic31⋊C4C2213yescomplemented
22C2×C4×Dic31C2213yescomplemented
23#11758475C22124yescomplemented
24C4⋊C4×D31C22124yescomplemented
25C22⋊C4×D31C22124yes—
26#11758456C22124yescomplemented
27#11758453C2218yescomplemented
28C42×D31C2218yescomplemented
29C24×C31C2311yescharacteristic
30C23×D31—212no—
31C23×D31C2314yes—
32C23×D31C2311yescharacteristic
33C23×D31C2311yescharacteristic
34C22×C4×C31C2313yes—
35C22×C4×C31C2313yes—
36C22×Dic31C2313yes—
37C22×Dic31C2313yes—
38C2×C4×D31—212no—
39C2×C4×D31—212no—
40C2×C4×D31C23112yes—
41C2×C4×D31C23112yes—
42C4⋊C4×C31C23112yescomplemented
43C22⋊C4×C31C23112yes—
44C42×C31C2314yescomplemented
45C22⋊Dic31C23112yes—
46(C2×D31)⋊C4C23112yes—
47(C2×D31)⋊C4C23112yes—
48C4⋊Dic31C23112yescomplemented
49Dic31⋊C4C23112yescomplemented
50Dic31⋊C4C23112yescomplemented
51C4×Dic31C23112yescomplemented
52C23×C31—26no—
53C23×C31C2412yes—
54C23×C31C2411yescharacteristic
55C22×D31—26no—
56C22×D31—224no—
57C22×D31—416no—
58C22×D31#41112yes—
59C22×D31C2412yes—
60C22×D31C2412yes—
61C2×C4×C31—26no—
62C2×C4×C31—26no—
63C2×C4×C31C2416yes—
64C2×C4×C31C2416yes—
65C2×Dic31—26no—
66C2×Dic31—26no—
67C2×Dic31C2416yes—
68C2×Dic31C2416yes—
69C4×D31—224no—
70C4×D31—224no—
71C22×#126—311nomaximal, Sylow
72C22×C31—26no—
73C22×C31—44no—
74C22×C31C2×#4116yes—
75C22×C31C2511yescharacteristic
76C2×D31—26no—
77C2×D31—44no—
78C2×D31—416no—
79C2×D31#126116yesdirect product factor
80C2×D31C2×#41112yes—
81C4×C31—26no—
82C4×C31—26no—
83Dic31—26no—
84Dic31—26no—
85C2×#126—3116no—
86C2×#126—313no—
87C2×#126—313no—
88C2×#126—311no—
89C2×#126D3111yescharacteristic, direct product factor
90C22×C4⋊C4—313no—
91C22×C22⋊C4—313no—
92C22×C42—311no—
93C2×C31—42no—
94C2×C31C2×#12614yescomplemented
95C2×C31C22×#4113yes—
96D31—42no—
97D31C2×#12618yesdirect product factor
98C25—311no—
99C23×C4—313no—
100C23×C4—313no—
101#126—3124no—
102#126—3124no—
103#126—318no—
104#126C2×D3118yesdirect product factor
105C2×C4⋊C4—313no—
106C2×C4⋊C4—313no—
107C2×C4⋊C4—313no—
108C2×C4⋊C4—3124no—
109C2×C4⋊C4C2×D3113yescomplemented
110C2×C22⋊C4—313no—
111C2×C22⋊C4—313no—
112C2×C22⋊C4—313no—
113C2×C22⋊C4—3124no—
114C2×C22⋊C4C2×D3113yes—
115C2×C42—313no—
116C2×C42—318no—
117C2×C42C2×D3111yescharacteristic, complemented
118C31C22×#12611yescharacteristic, complemented, Sylow
119C24—311no—
120C24—314no—
121C24—311no—
122C24—6212no—
123C24C22×D3111yescharacteristic
124C22×C4—313no—
125C22×C4—313no—
126C22×C4—3112no—
127C22×C4—3112no—
128C22×C4—6212no—
129C22×C4—6212no—
130C22×C4C22×D3113yes—
131C22×C4C22×D3113yes—
132C4⋊C4—3112no—
133C4⋊C4—3112no—
134C4⋊C4—3112no—
135C4⋊C4C22×D31112yescomplemented
136C22⋊C4—3112no—
137C22⋊C4—3112no—
138C22⋊C4—3112no—
139C22⋊C4C22×D31112yes—
140C42—3112no—
141C42C22×D3114yescomplemented
142C23—26no—
143C23—3112no—
144C23—312no—
145C23—312no—
146C23—6224no—
147C23—626no—
148C23—12416no—
149C23C23×D3112yes—
150C23C23×D3111yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—316no—
154C2×C4—316no—
155C2×C4—6224no—
156C2×C4—6224no—
157C2×C4—626no—
158C2×C4—626no—
159C2×C4C23×D3116yes—
160C2×C4C23×D3116yes—
161C22—26no—
162C22—44no—
163C22—3112no—
164C22—3116no—
165C22—626no—
166C22—12416no—
167C22—1244no—
168C22#41×D3116yes—
169C22C24×D3111yescharacteristic
170C4—26no—
171C4—26no—
172C4—626no—
173C4—626no—
174C2—42no—
175C2—318no—
176C2—1242no—
177C2#126×D3114yesdirect product factor
178C2C2×#41×D3113yes—
subgroupG/H
centerC23C23×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC2×C31×#126C2
socleC23×C31C24
prime radicalC24×D31C22
hypercenterC2×#126D31
hypocenterC31C22×#126
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#11758475C2semidirect
2. C2×#11758453C2semidirect
3. C2×#11758456C2semidirect
4. #126×D31C2direct (N × H)
5. C2×C4⋊C4×D31C2semidirect
7. C2×C42×D31C2semidirect
8. C2×C31×#126C2semidirect
12. C31×#126C22semidirect
13. C2×C4⋊C4×C31C22semidirect
15. C2×C42×C31C22semidirect
19. C2×C4⋊Dic31C22semidirect
20. C2×Dic31⋊C4C22semidirect
21. C2×Dic31⋊C4C22semidirect
22. C2×C4×Dic31C22semidirect
23. #11758475C22semidirect
24. C4⋊C4×D31C22semidirect
26. #11758456C22semidirect
27. #11758453C22semidirect
28. C42×D31C22semidirect
42. C4⋊C4×C31C23semidirect
44. C42×C31C23semidirect
48. C4⋊Dic31C23semidirect
49. Dic31⋊C4C23semidirect
50. Dic31⋊C4C23semidirect
51. C4×Dic31C23semidirect
79. C2×D31#126direct (N × H)
89. C2×#126D31direct (N × H)
94. C2×C31C2×#126semidirect
97. D31C2×#126direct (N × H)
104. #126C2×D31direct (N × H)
109. C2×C4⋊C4C2×D31semidirect
117. C2×C42C2×D31semidirect
118. C31C22×#126semidirect
135. C4⋊C4C22×D31semidirect
141. C42C22×D31semidirect
177. C2#126×D31direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117584752
C2×#117584532
C2×#117584562
#126×D312
C2×C4⋊C4×D312
C2×C22⋊C4×D312
C2×C42×D312
C2×C31×#1262
C22×#12631