Finite Group Databasebeta

← C2×#16811364 #20288833

Subgroups — C2×#16811364 #20288833

Order 6784.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#16811364C2116yesmaximal, direct product factor
2C2×#13133214C211yesmaximal, characteristic, complemented
3C2×#13133211C211yesmaximal, characteristic, complemented
4C2×#13133202C213yesmaximal, complemented
5C2×#13133192C213yesmaximal, complemented
6C2×#13133190C213yesmaximal, complemented
7C2×#13133187C213yesmaximal, complemented
8C2×#126×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C22×D212C2213yes—
11C22×D212C2213yes—
12C22×D212C2213yescomplemented
13C22×C22⋊D53C2213yes—
14C22×C4×D53C2213yescomplemented
15#13133214C2218yescomplemented
16#13133211C2218yescomplemented
17#13133202C22124yescomplemented
18#13133192C22124yescomplemented
19#13133190C22124yescomplemented
20#13133187C22124yescomplemented
21C2×#11719919C2213yescomplemented
22C2×#11719921C2213yes—
23C2×#11719921C2213yes—
24C2×#11719921C2213yescomplemented
25#126×C53C2218yescomplemented
26C2×C4⋊C4×C53C2213yescomplemented
27C2×C22⋊C4×C53C2213yes—
28C2×C42×C53C2211yescharacteristic, complemented
29C24×C53C2311yescharacteristic
30C23×D53—46no—
31C23×D53C2311yescharacteristic
32C23×D53C2311yescharacteristic
33C23×D53C2313yes—
34C23×D53C2314yes—
35C22×C4×C53C2313yes—
36C22×C4×C53C2313yes—
37C2×C22⋊D53—46no—
38C2×C22⋊D53C23112yes—
39C22×Dic53C2313yes—
40C2×D212—46no—
41C2×D212—46no—
42C2×D212—46no—
43C2×D212C23112yes—
44C2×D212C23112yes—
45C2×D212C23112yescomplemented
46C2×C4×D53—46no—
47C2×C4×D53C23112yescomplemented
48C4⋊C4×C53C23112yescomplemented
49C22⋊C4×C53C23112yes—
50C42×C53C2314yescomplemented
51#11719921C23112yes—
52#11719921C23112yes—
53#11719921C23112yescomplemented
54#11719919C23112yescomplemented
55C23×C53—26no—
56C23×C53C2412yes—
57C23×C53C2411yescharacteristic
58C22×D53—26no—
59C22×D53—26no—
60C22×D53—26no—
61C22×D53—26no—
62C22×D53—26no—
63C22×D53—412no—
64C22×D53—416no—
65C22×D53C2416yes—
66C22×D53C2412yes—
67C22×D53C2412yes—
68C2×C4×C53—26no—
69C2×C4×C53—26no—
70C2×C4×C53C2416yes—
71C2×C4×C53C2416yes—
72C22⋊D53—412no—
73C2×Dic53—26no—
74C2×Dic53C2416yes—
75D212—412no—
76D212—412no—
77D212—412no—
78C4×D53—412no—
79C22×C53—26no—
80C22×C53—44no—
81C22×C53H(5,𝔽2)16yes—
82C22×C53C2511yescharacteristic
83C2×D53—26no—
84C2×D53—26no—
85C2×D53—26no—
86C2×D53—26no—
87C2×D53—26no—
88C2×D53—416no—
89C2×D53—412no—
90C2×D53—44no—
91C2×D53—44no—
92C4×C53—26no—
93C4×C53—26no—
94Dic53—26no—
95C2×#560—531nomaximal, Sylow
96C2×C53—42no—
97C2×C53#56014yescomplemented
98C2×C53C2×H(5,𝔽2)13yes—
99D53—46no—
100D53—42no—
101D53—42no—
102#560—5316no—
103C2×#127—531no—
104C2×#126—531no—
105C2×#126D5311yescharacteristic, complemented
106C2×#123—533no—
107C2×#121—533no—
108C2×#121—533no—
109C2×#120—533no—
110C53C2×#56011yescharacteristic, complemented, Sylow
111C25—531no—
112C22×D4—533no—
113C22×D4—533no—
114C22×D4—533no—
115C22×D4—533no—
116C23×C4—533no—
117#127—538no—
118#126—538no—
119#126C2×D5318yescomplemented
120#123—5324no—
121#121—5324no—
122#121—5324no—
123#120—5324no—
124C2×C4⋊C4—533no—
125C2×C4⋊C4C2×D5313yescomplemented
126C2×C22⋊C4—533no—
127C2×C22⋊C4—533no—
128C2×C22⋊C4—533no—
129C2×C22⋊C4C2×D5313yes—
130C2×C42C2×D5311yescharacteristic, complemented
131C24—531no—
132C24—531no—
133C24—533no—
134C24—534no—
135C24—2126no—
136C24C22×D5311yescharacteristic
137C2×D4—5312no—
138C2×D4—5312no—
139C2×D4—5312no—
140C2×D4—5312no—
141C2×D4—2126no—
142C2×D4—2126no—
143C2×D4—2126no—
144C2×D4—2126no—
145C22×C4—533no—
146C22×C4—5312no—
147C22×C4—2126no—
148C22×C4C22×D5313yes—
149C22×C4C22×D5313yes—
150C4⋊C4—5312no—
151C4⋊C4C22×D53112yescomplemented
152C22⋊C4—5312no—
153C22⋊C4—5312no—
154C22⋊C4—5312no—
155C22⋊C4C22×D53112yes—
156C42C22×D5314yescomplemented
157C23—26no—
158C23—532no—
159C23—532no—
160C23—536no—
161C23—1066no—
162C23—1066no—
163C23—1066no—
164C23—1066no—
165C23—1066no—
166C23—21212no—
167C23—21216no—
168C23C23×D5312yes—
169C23C23×D5311yescharacteristic
170D4—21212no—
171D4—21212no—
172D4—21212no—
173D4—21212no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—536no—
177C2×C4—1066no—
178C2×C4—21212no—
179C2×C4C23×D5316yes—
180C2×C4C23×D5316yes—
181C22—26no—
182C22—44no—
183C22—1066no—
184C22—1066no—
185C22—1066no—
186C22—1066no—
187C22—1066no—
188C22—21216no—
189C22—2124no—
190C22—2124no—
191C22—21212no—
192C22#1313326316yes—
193C22C24×D5311yescharacteristic
194C4—26no—
195C4—26no—
196C4—1066no—
197C2—42no—
198C2—2122no—
199C2—2122no—
200C2—2126no—
201C2#1681136414yesdirect product factor
202C2C2×#1313326313yes—
subgroupG/H
centerC23C23×D53
commutatorC22×C53C25
FrattiniC22C24×D53
FittingC2×#126×C53C2
socleC23×C53C24
hypercenterC2×#126D53
hypocenterC53C2×#560
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #16811364C2direct (N × H)
2. C2×#13133214C2semidirect
3. C2×#13133211C2semidirect
4. C2×#13133202C2semidirect
5. C2×#13133192C2semidirect
6. C2×#13133190C2semidirect
7. C2×#13133187C2semidirect
8. C2×#126×C53C2semidirect
12. C22×D212C22semidirect
14. C22×C4×D53C22semidirect
15. #13133214C22semidirect
16. #13133211C22semidirect
17. #13133202C22semidirect
18. #13133192C22semidirect
19. #13133190C22semidirect
20. #13133187C22semidirect
21. C2×#11719919C22semidirect
24. C2×#11719921C22semidirect
25. #126×C53C22semidirect
26. C2×C4⋊C4×C53C22semidirect
28. C2×C42×C53C22semidirect
45. C2×D212C23semidirect
47. C2×C4×D53C23semidirect
48. C4⋊C4×C53C23semidirect
50. C42×C53C23semidirect
53. #11719921C23semidirect
54. #11719919C23semidirect
97. C2×C53#560semidirect
105. C2×#126D53semidirect
110. C53C2×#560semidirect
119. #126C2×D53semidirect
125. C2×C4⋊C4C2×D53semidirect
130. C2×C42C2×D53semidirect
151. C4⋊C4C22×D53semidirect
156. C42C22×D53semidirect
201. C2#16811364direct (N × H)