Finite Group Databasebeta

← C2×#126×D71 #22601594

Subgroups — C2×#126×D71 #22601594

Order 9088.

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×#14556123C213yesmaximal, complemented
2C2×#14556122C211yesmaximal, characteristic, complemented
3C2×#14556121C213yesmaximal, complemented
4#126×D71C2116yesmaximal, direct product factor
5C2×C4⋊C4×D71C213yesmaximal, complemented
6C2×C22⋊C4×D71C213yesmaximal
7C2×C42×D71C211yesmaximal, characteristic, complemented
8C2×#126×C71C211yesmaximal, characteristic, complemented
9C24×D71C2211yescharacteristic
10C22×C4×D71C2213yes—
11C22×C4×D71C2213yes—
12#14556123C22124yescomplemented
13#14556122C2218yescomplemented
14#14556121C22124yescomplemented
15C2×C4⋊Dic71C2213yescomplemented
16C2×Dic71⋊C4C2213yescomplemented
17C2×Dic71⋊C4C2213yescomplemented
18C4⋊C4×D71C22124yescomplemented
19C2×C22⋊Dic71C2213yes—
20C2×(C2×D71)⋊C4C2213yes—
21C2×(C2×D71)⋊C4C2213yes—
22C22⋊C4×D71C22124yes—
23C2×C4×Dic71C2213yescomplemented
24C42×D71C2218yescomplemented
25#126×C71C2218yescomplemented
26C2×C4⋊C4×C71C2213yescomplemented
27C2×C22⋊C4×C71C2213yes—
28C2×C42×C71C2211yescharacteristic, complemented
29C23×D71—212no—
30C23×D71C2314yes—
31C23×D71C2311yescharacteristic
32C23×D71C2311yescharacteristic
33C22×Dic71C2313yes—
34C22×Dic71C2313yes—
35C2×C4×D71—212no—
36C2×C4×D71—212no—
37C2×C4×D71C23112yes—
38C2×C4×D71C23112yes—
39C4⋊Dic71C23112yescomplemented
40Dic71⋊C4C23112yescomplemented
41Dic71⋊C4C23112yescomplemented
42C22⋊Dic71C23112yes—
43(C2×D71)⋊C4C23112yes—
44(C2×D71)⋊C4C23112yes—
45C4×Dic71C23112yescomplemented
46C24×C71C2311yescharacteristic
47C22×C4×C71C2313yes—
48C22×C4×C71C2313yes—
49C4⋊C4×C71C23112yescomplemented
50C22⋊C4×C71C23112yes—
51C42×C71C2314yescomplemented
52C23×C71—26no—
53C23×C71C2412yes—
54C23×C71C2411yescharacteristic
55C22×D71—26no—
56C22×D71—224no—
57C22×D71—416no—
58C22×D71#41112yes—
59C22×D71C2412yes—
60C22×D71C2412yes—
61C2×C4×C71—26no—
62C2×C4×C71—26no—
63C2×C4×C71C2416yes—
64C2×C4×C71C2416yes—
65C2×Dic71—26no—
66C2×Dic71—26no—
67C2×Dic71C2416yes—
68C2×Dic71C2416yes—
69C4×D71—224no—
70C4×D71—224no—
71C22×C71—26no—
72C22×C71—44no—
73C22×C71C2×#4116yes—
74C22×C71C2511yescharacteristic
75C2×D71—26no—
76C2×D71—44no—
77C2×D71—416no—
78C2×D71#126116yesdirect product factor
79C2×D71C2×#41112yes—
80C4×C71—26no—
81C4×C71—26no—
82Dic71—26no—
83Dic71—26no—
84C2×C71—42no—
85C2×C71C2×#12614yescomplemented
86C2×C71C22×#4113yes—
87D71—42no—
88D71C2×#12618yesdirect product factor
89C22×#126—711nomaximal, Sylow
90C71C22×#12611yescharacteristic, complemented, Sylow
91C2×#126—7116no—
92C2×#126—713no—
93C2×#126—713no—
94C2×#126—711no—
95C2×#126D7111yescharacteristic, direct product factor
96C22×C4⋊C4—713no—
97C22×C22⋊C4—713no—
98C22×C42—711no—
99C25—711no—
100C23×C4—713no—
101C23×C4—713no—
102#126—7124no—
103#126—7124no—
104#126—718no—
105#126C2×D7118yesdirect product factor
106C2×C4⋊C4—713no—
107C2×C4⋊C4—713no—
108C2×C4⋊C4—713no—
109C2×C4⋊C4—7124no—
110C2×C4⋊C4C2×D7113yescomplemented
111C2×C22⋊C4—713no—
112C2×C22⋊C4—713no—
113C2×C22⋊C4—713no—
114C2×C22⋊C4—7124no—
115C2×C22⋊C4C2×D7113yes—
116C2×C42—713no—
117C2×C42—718no—
118C2×C42C2×D7111yescharacteristic, complemented
119C24—711no—
120C24—714no—
121C24—711no—
122C24—14212no—
123C24C22×D7111yescharacteristic
124C22×C4—713no—
125C22×C4—713no—
126C22×C4—7112no—
127C22×C4—7112no—
128C22×C4—14212no—
129C22×C4—14212no—
130C22×C4C22×D7113yes—
131C22×C4C22×D7113yes—
132C4⋊C4—7112no—
133C4⋊C4—7112no—
134C4⋊C4—7112no—
135C4⋊C4C22×D71112yescomplemented
136C22⋊C4—7112no—
137C22⋊C4—7112no—
138C22⋊C4—7112no—
139C22⋊C4C22×D71112yes—
140C42—7112no—
141C42C22×D7114yescomplemented
142C23—26no—
143C23—28416no—
144C23—7112no—
145C23—712no—
146C23—712no—
147C23—14224no—
148C23—1426no—
149C23C23×D7112yes—
150C23C23×D7111yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—716no—
154C2×C4—716no—
155C2×C4—14224no—
156C2×C4—14224no—
157C2×C4—1426no—
158C2×C4—1426no—
159C2×C4C23×D7116yes—
160C2×C4C23×D7116yes—
161C22—26no—
162C22—44no—
163C22—28416no—
164C22—2844no—
165C22—7112no—
166C22—7116no—
167C22—1426no—
168C22#41×D7116yes—
169C22C24×D7111yescharacteristic
170C4—26no—
171C4—26no—
172C4—1426no—
173C4—1426no—
174C2—42no—
175C2—2842no—
176C2—718no—
177C2#126×D7114yesdirect product factor
178C2C2×#41×D7113yes—
subgroupG/H
centerC23C23×D71
commutatorC22×C71C25
FrattiniC22C24×D71
FittingC2×#126×C71C2
socleC23×C71C24
hypercenterC2×#126D71
hypocenterC71C22×#126
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14556123C2semidirect
2. C2×#14556122C2semidirect
3. C2×#14556121C2semidirect
4. #126×D71C2direct (N × H)
5. C2×C4⋊C4×D71C2semidirect
7. C2×C42×D71C2semidirect
8. C2×#126×C71C2semidirect
12. #14556123C22semidirect
13. #14556122C22semidirect
14. #14556121C22semidirect
15. C2×C4⋊Dic71C22semidirect
16. C2×Dic71⋊C4C22semidirect
17. C2×Dic71⋊C4C22semidirect
18. C4⋊C4×D71C22semidirect
23. C2×C4×Dic71C22semidirect
24. C42×D71C22semidirect
25. #126×C71C22semidirect
26. C2×C4⋊C4×C71C22semidirect
28. C2×C42×C71C22semidirect
39. C4⋊Dic71C23semidirect
40. Dic71⋊C4C23semidirect
41. Dic71⋊C4C23semidirect
45. C4×Dic71C23semidirect
49. C4⋊C4×C71C23semidirect
51. C42×C71C23semidirect
78. C2×D71#126direct (N × H)
85. C2×C71C2×#126semidirect
88. D71C2×#126direct (N × H)
90. C71C22×#126semidirect
95. C2×#126D71direct (N × H)
105. #126C2×D71direct (N × H)
110. C2×C4⋊C4C2×D71semidirect
118. C2×C42C2×D71semidirect
135. C4⋊C4C22×D71semidirect
141. C42C22×D71semidirect
177. C2#126×D71direct (N × H)