Finite Group Databasebeta

← C2×#126×D53 #20287969

Subgroups — C2×#126×D53 #20287969

Order 6784.

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×#13133212C213yesmaximal, complemented
2C2×#13133211C211yesmaximal, characteristic, complemented
3C2×#13133210C213yesmaximal, complemented
4#126×D53C2116yesmaximal, direct product factor
5C2×C4⋊C4×D53C213yesmaximal, complemented
6C2×C22⋊C4×D53C213yesmaximal
7C2×C42×D53C211yesmaximal, characteristic, complemented
8C2×#126×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C22×C4×D53C2213yes—
11C22×C4×D53C2213yes—
12#13133212C22124yescomplemented
13#13133211C2218yescomplemented
14#13133210C22124yescomplemented
15C2×C4⋊Dic53C2213yescomplemented
16C2×#11719919C2213yescomplemented
17C2×#11719919C2213yescomplemented
18C4⋊C4×D53C22124yescomplemented
19C2×C22⋊Dic53C2213yes—
20C2×#11719921C2213yes—
21C2×#11719921C2213yes—
22C22⋊C4×D53C22124yes—
23C2×C4×Dic53C2213yescomplemented
24C42×D53C2218yescomplemented
25#126×C53C2218yescomplemented
26C2×C4⋊C4×C53C2213yescomplemented
27C2×C22⋊C4×C53C2213yes—
28C2×C42×C53C2211yescharacteristic, complemented
29C24×C53C2311yescharacteristic
30C23×D53—212no—
31C23×D53C2314yes—
32C23×D53C2311yescharacteristic
33C23×D53C2311yescharacteristic
34C22×C4×C53C2313yes—
35C22×C4×C53C2313yes—
36C22×Dic53C2313yes—
37C22×Dic53C2313yes—
38C2×C4×D53—212no—
39C2×C4×D53—212no—
40C2×C4×D53C23112yes—
41C2×C4×D53C23112yes—
42C4⋊C4×C53C23112yescomplemented
43C22⋊C4×C53C23112yes—
44C42×C53C2314yescomplemented
45C22⋊Dic53C23112yes—
46#11719921C23112yes—
47#11719921C23112yes—
48C4⋊Dic53C23112yescomplemented
49#11719919C23112yescomplemented
50#11719919C23112yescomplemented
51C4×Dic53C23112yescomplemented
52C23×C53—26no—
53C23×C53C2412yes—
54C23×C53C2411yescharacteristic
55C22×D53—26no—
56C22×D53—224no—
57C22×D53—416no—
58C22×D53#41112yes—
59C22×D53C2412yes—
60C22×D53C2412yes—
61C2×C4×C53—26no—
62C2×C4×C53—26no—
63C2×C4×C53C2416yes—
64C2×C4×C53C2416yes—
65C2×Dic53—26no—
66C2×Dic53—26no—
67C2×Dic53C2416yes—
68C2×Dic53C2416yes—
69C4×D53—224no—
70C4×D53—224no—
71C22×C53—26no—
72C22×C53—44no—
73C22×C53C2×#4116yes—
74C22×C53C2511yescharacteristic
75C2×D53—26no—
76C2×D53—44no—
77C2×D53—416no—
78C2×D53#126116yesdirect product factor
79C2×D53C2×#41112yes—
80C4×C53—26no—
81C4×C53—26no—
82Dic53—26no—
83Dic53—26no—
84C22×#126—531nomaximal, Sylow
85C2×C53—42no—
86C2×C53C2×#12614yescomplemented
87C2×C53C22×#4113yes—
88D53—42no—
89D53C2×#12618yesdirect product factor
90C2×#126—5316no—
91C2×#126—533no—
92C2×#126—533no—
93C2×#126—531no—
94C2×#126D5311yescharacteristic, direct product factor
95C22×C4⋊C4—533no—
96C22×C22⋊C4—533no—
97C22×C42—531no—
98C53C22×#12611yescharacteristic, complemented, Sylow
99C25—531no—
100C23×C4—533no—
101C23×C4—533no—
102#126—5324no—
103#126—5324no—
104#126—538no—
105#126C2×D5318yesdirect product factor
106C2×C4⋊C4—533no—
107C2×C4⋊C4—533no—
108C2×C4⋊C4—533no—
109C2×C4⋊C4—5324no—
110C2×C4⋊C4C2×D5313yescomplemented
111C2×C22⋊C4—533no—
112C2×C22⋊C4—533no—
113C2×C22⋊C4—533no—
114C2×C22⋊C4—5324no—
115C2×C22⋊C4C2×D5313yes—
116C2×C42—533no—
117C2×C42—538no—
118C2×C42C2×D5311yescharacteristic, complemented
119C24—531no—
120C24—534no—
121C24—531no—
122C24—10612no—
123C24C22×D5311yescharacteristic
124C22×C4—533no—
125C22×C4—533no—
126C22×C4—5312no—
127C22×C4—5312no—
128C22×C4—10612no—
129C22×C4—10612no—
130C22×C4C22×D5313yes—
131C22×C4C22×D5313yes—
132C4⋊C4—5312no—
133C4⋊C4—5312no—
134C4⋊C4—5312no—
135C4⋊C4C22×D53112yescomplemented
136C22⋊C4—5312no—
137C22⋊C4—5312no—
138C22⋊C4—5312no—
139C22⋊C4C22×D53112yes—
140C42—5312no—
141C42C22×D5314yescomplemented
142C23—26no—
143C23—5312no—
144C23—532no—
145C23—532no—
146C23—10624no—
147C23—1066no—
148C23—21216no—
149C23C23×D5312yes—
150C23C23×D5311yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—536no—
154C2×C4—536no—
155C2×C4—10624no—
156C2×C4—10624no—
157C2×C4—1066no—
158C2×C4—1066no—
159C2×C4C23×D5316yes—
160C2×C4C23×D5316yes—
161C22—26no—
162C22—44no—
163C22—5316no—
164C22—5312no—
165C22—1066no—
166C22—21216no—
167C22—2124no—
168C22#41×D5316yes—
169C22C24×D5311yescharacteristic
170C4—26no—
171C4—26no—
172C4—1066no—
173C4—1066no—
174C2—42no—
175C2—538no—
176C2—2122no—
177C2#126×D5314yesdirect product factor
178C2C2×#41×D5313yes—
subgroupG/H
centerC23C23×D53
commutatorC22×C53C25
FrattiniC22C24×D53
FittingC2×#126×C53C2
socleC23×C53C24
hypercenterC2×#126D53
hypocenterC53C22×#126
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
2. C2×#13133211C2semidirect
3. C2×#13133210C2semidirect
4. #126×D53C2direct (N × H)
5. C2×C4⋊C4×D53C2semidirect
7. C2×C42×D53C2semidirect
8. C2×#126×C53C2semidirect
12. #13133212C22semidirect
13. #13133211C22semidirect
14. #13133210C22semidirect
15. C2×C4⋊Dic53C22semidirect
16. C2×#11719919C22semidirect
17. C2×#11719919C22semidirect
18. C4⋊C4×D53C22semidirect
23. C2×C4×Dic53C22semidirect
24. C42×D53C22semidirect
25. #126×C53C22semidirect
26. C2×C4⋊C4×C53C22semidirect
28. C2×C42×C53C22semidirect
42. C4⋊C4×C53C23semidirect
44. C42×C53C23semidirect
48. C4⋊Dic53C23semidirect
49. #11719919C23semidirect
50. #11719919C23semidirect
51. C4×Dic53C23semidirect
78. C2×D53#126direct (N × H)
86. C2×C53C2×#126semidirect
89. D53C2×#126direct (N × H)
94. C2×#126D53direct (N × H)
98. C53C22×#126semidirect
105. #126C2×D53direct (N × H)
110. C2×C4⋊C4C2×D53semidirect
118. C2×C42C2×D53semidirect
135. C4⋊C4C22×D53semidirect
141. C42C22×D53semidirect
177. C2#126×D53direct (N × H)