Finite Group Databasebeta

← C2×#16864137 #22574327

Subgroups — C2×#16864137 #22574327

Order 8576.

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#16864137C2116yesmaximal, direct product factor
2C2×#14524040C211yesmaximal, characteristic, complemented
3C2×#14524037C211yesmaximal, characteristic, complemented
4C2×#14524028C213yesmaximal, complemented
5C2×#14524018C213yesmaximal, complemented
6C2×#14524016C213yesmaximal, complemented
7C2×#14524013C213yesmaximal, complemented
8C2×#126×C67C211yesmaximal, characteristic, complemented
9C24×D67C2211yescharacteristic
10C22×D268C2213yes—
11C22×D268C2213yes—
12C22×D268C2213yescomplemented
13C22×C22⋊D67C2213yes—
14C22×C4×D67C2213yescomplemented
15#14524040C2218yescomplemented
16#14524037C2218yescomplemented
17#14524028C22124yescomplemented
18#14524018C22124yescomplemented
19#14524016C22124yescomplemented
20#14524013C22124yescomplemented
21C2×Dic67⋊C4C2213yescomplemented
22C2×(C2×D67)⋊C4C2213yes—
23C2×(C2×D67)⋊C4C2213yes—
24C2×(C2×D67)⋊C4C2213yescomplemented
25#126×C67C2218yescomplemented
26C2×C4⋊C4×C67C2213yescomplemented
27C2×C22⋊C4×C67C2213yes—
28C2×C42×C67C2211yescharacteristic, complemented
29C23×D67—46no—
30C23×D67C2311yescharacteristic
31C23×D67C2311yescharacteristic
32C23×D67C2313yes—
33C23×D67C2314yes—
34C2×D268—46no—
35C2×D268—46no—
36C2×D268—46no—
37C2×D268C23112yes—
38C2×D268C23112yes—
39C2×D268C23112yescomplemented
40C2×C22⋊D67—46no—
41C2×C22⋊D67C23112yes—
42C22×Dic67C2313yes—
43C2×C4×D67—46no—
44C2×C4×D67C23112yescomplemented
45Dic67⋊C4C23112yescomplemented
46(C2×D67)⋊C4C23112yes—
47(C2×D67)⋊C4C23112yes—
48(C2×D67)⋊C4C23112yescomplemented
49C24×C67C2311yescharacteristic
50C22×C4×C67C2313yes—
51C22×C4×C67C2313yes—
52C4⋊C4×C67C23112yescomplemented
53C22⋊C4×C67C23112yes—
54C42×C67C2314yescomplemented
55C23×C67—26no—
56C23×C67C2412yes—
57C23×C67C2411yescharacteristic
58C22×D67—26no—
59C22×D67—26no—
60C22×D67—26no—
61C22×D67—26no—
62C22×D67—26no—
63C22×D67—412no—
64C22×D67—416no—
65C22×D67C2412yes—
66C22×D67C2412yes—
67C22×D67C2416yes—
68C2×C4×C67—26no—
69C2×C4×C67—26no—
70C2×C4×C67C2416yes—
71C2×C4×C67C2416yes—
72C22⋊D67—412no—
73C2×Dic67—26no—
74C2×Dic67C2416yes—
75D268—412no—
76D268—412no—
77D268—412no—
78C4×D67—412no—
79C22×C67—26no—
80C22×C67—44no—
81C22×C67H(5,𝔽2)16yes—
82C22×C67C2511yescharacteristic
83C2×D67—26no—
84C2×D67—26no—
85C2×D67—26no—
86C2×D67—26no—
87C2×D67—26no—
88C2×D67—416no—
89C2×D67—412no—
90C2×D67—44no—
91C2×D67—44no—
92C4×C67—26no—
93C4×C67—26no—
94Dic67—26no—
95C2×C67—42no—
96C2×C67#56014yescomplemented
97C2×C67C2×H(5,𝔽2)13yes—
98D67—46no—
99D67—42no—
100D67—42no—
101C2×#560—671nomaximal, Sylow
102C67C2×#56011yescharacteristic, complemented, Sylow
103#560—6716no—
104C2×#127—671no—
105C2×#126—671no—
106C2×#126D6711yescharacteristic, complemented
107C2×#123—673no—
108C2×#121—673no—
109C2×#121—673no—
110C2×#120—673no—
111C25—671no—
112C22×D4—673no—
113C22×D4—673no—
114C22×D4—673no—
115C22×D4—673no—
116C23×C4—673no—
117#127—678no—
118#126—678no—
119#126C2×D6718yescomplemented
120#123—6724no—
121#121—6724no—
122#121—6724no—
123#120—6724no—
124C2×C4⋊C4—673no—
125C2×C4⋊C4C2×D6713yescomplemented
126C2×C22⋊C4—673no—
127C2×C22⋊C4—673no—
128C2×C22⋊C4—673no—
129C2×C22⋊C4C2×D6713yes—
130C2×C42C2×D6711yescharacteristic, complemented
131C24—2686no—
132C24—674no—
133C24—671no—
134C24—671no—
135C24—673no—
136C24C22×D6711yescharacteristic
137C2×D4—2686no—
138C2×D4—2686no—
139C2×D4—2686no—
140C2×D4—2686no—
141C2×D4—6712no—
142C2×D4—6712no—
143C2×D4—6712no—
144C2×D4—6712no—
145C22×C4—2686no—
146C22×C4—6712no—
147C22×C4—673no—
148C22×C4C22×D6713yes—
149C22×C4C22×D6713yes—
150C4⋊C4—6712no—
151C4⋊C4C22×D67112yescomplemented
152C22⋊C4—6712no—
153C22⋊C4—6712no—
154C22⋊C4—6712no—
155C22⋊C4C22×D67112yes—
156C42C22×D6714yescomplemented
157C23—26no—
158C23—26812no—
159C23—26816no—
160C23—672no—
161C23—672no—
162C23—676no—
163C23—1346no—
164C23—1346no—
165C23—1346no—
166C23—1346no—
167C23—1346no—
168C23C23×D6712yes—
169C23C23×D6711yescharacteristic
170D4—26812no—
171D4—26812no—
172D4—26812no—
173D4—26812no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—26812no—
177C2×C4—676no—
178C2×C4—1346no—
179C2×C4C23×D6716yes—
180C2×C4C23×D6716yes—
181C22—26no—
182C22—44no—
183C22—26816no—
184C22—2684no—
185C22—2684no—
186C22—26812no—
187C22—1346no—
188C22—1346no—
189C22—1346no—
190C22—1346no—
191C22—1346no—
192C22#1452408916yes—
193C22C24×D6711yescharacteristic
194C4—26no—
195C4—26no—
196C4—1346no—
197C2—42no—
198C2—2682no—
199C2—2682no—
200C2—2686no—
201C2#1686413714yesdirect product factor
202C2C2×#1452408913yes—
subgroupG/H
centerC23C23×D67
commutatorC22×C67C25
FrattiniC22C24×D67
FittingC2×#126×C67C2
socleC23×C67C24
hypercenterC2×#126D67
hypocenterC67C2×#560
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #16864137C2direct (N × H)
2. C2×#14524040C2semidirect
3. C2×#14524037C2semidirect
4. C2×#14524028C2semidirect
5. C2×#14524018C2semidirect
6. C2×#14524016C2semidirect
7. C2×#14524013C2semidirect
8. C2×#126×C67C2semidirect
12. C22×D268C22semidirect
14. C22×C4×D67C22semidirect
15. #14524040C22semidirect
16. #14524037C22semidirect
17. #14524028C22semidirect
18. #14524018C22semidirect
19. #14524016C22semidirect
20. #14524013C22semidirect
21. C2×Dic67⋊C4C22semidirect
24. C2×(C2×D67)⋊C4C22semidirect
25. #126×C67C22semidirect
26. C2×C4⋊C4×C67C22semidirect
28. C2×C42×C67C22semidirect
39. C2×D268C23semidirect
44. C2×C4×D67C23semidirect
45. Dic67⋊C4C23semidirect
48. (C2×D67)⋊C4C23semidirect
52. C4⋊C4×C67C23semidirect
54. C42×C67C23semidirect
96. C2×C67#560semidirect
102. C67C2×#560semidirect
106. C2×#126D67semidirect
119. #126C2×D67semidirect
125. C2×C4⋊C4C2×D67semidirect
130. C2×C42C2×D67semidirect
151. C4⋊C4C22×D67semidirect
156. C42C22×D67semidirect
201. C2#16864137direct (N × H)