Finite Group Databasebeta

← C2×#17986740 #22602458

Subgroups — C2×#17986740 #22602458

Order 9088.

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#17986740C2116yesmaximal, direct product factor
2C2×#14556125C211yesmaximal, characteristic, complemented
3C2×#14556122C211yesmaximal, characteristic, complemented
4C2×#14556113C213yesmaximal, complemented
5C2×#14556103C213yesmaximal, complemented
6C2×#14556101C213yesmaximal, complemented
7C2×#14556098C213yesmaximal, complemented
8C2×#126×C71C211yesmaximal, characteristic, complemented
9C24×D71C2211yescharacteristic
10C22×D284C2213yes—
11C22×D284C2213yes—
12C22×D284C2213yescomplemented
13C22×C22⋊D71C2213yes—
14C22×C4×D71C2213yescomplemented
15#14556125C2218yescomplemented
16#14556122C2218yescomplemented
17#14556113C22124yescomplemented
18#14556103C22124yescomplemented
19#14556101C22124yescomplemented
20#14556098C22124yescomplemented
21C2×Dic71⋊C4C2213yescomplemented
22C2×(C2×D71)⋊C4C2213yes—
23C2×(C2×D71)⋊C4C2213yes—
24C2×(C2×D71)⋊C4C2213yescomplemented
25#126×C71C2218yescomplemented
26C2×C4⋊C4×C71C2213yescomplemented
27C2×C22⋊C4×C71C2213yes—
28C2×C42×C71C2211yescharacteristic, complemented
29C23×D71—46no—
30C23×D71C2313yes—
31C23×D71C2311yescharacteristic
32C23×D71C2311yescharacteristic
33C23×D71C2314yes—
34C2×D284—46no—
35C2×D284—46no—
36C2×D284—46no—
37C2×D284C23112yes—
38C2×D284C23112yes—
39C2×D284C23112yescomplemented
40C2×C22⋊D71—46no—
41C2×C22⋊D71C23112yes—
42C22×Dic71C2313yes—
43C2×C4×D71—46no—
44C2×C4×D71C23112yescomplemented
45Dic71⋊C4C23112yescomplemented
46(C2×D71)⋊C4C23112yes—
47(C2×D71)⋊C4C23112yes—
48(C2×D71)⋊C4C23112yescomplemented
49C24×C71C2311yescharacteristic
50C22×C4×C71C2313yes—
51C22×C4×C71C2313yes—
52C4⋊C4×C71C23112yescomplemented
53C22⋊C4×C71C23112yes—
54C42×C71C2314yescomplemented
55C23×C71—26no—
56C23×C71C2412yes—
57C23×C71C2411yescharacteristic
58C22×D71—26no—
59C22×D71—26no—
60C22×D71—26no—
61C22×D71—26no—
62C22×D71—26no—
63C22×D71—412no—
64C22×D71—416no—
65C22×D71C2412yes—
66C22×D71C2412yes—
67C22×D71C2416yes—
68C2×C4×C71—26no—
69C2×C4×C71—26no—
70C2×C4×C71C2416yes—
71C2×C4×C71C2416yes—
72C22⋊D71—412no—
73C2×Dic71—26no—
74C2×Dic71C2416yes—
75D284—412no—
76D284—412no—
77D284—412no—
78C4×D71—412no—
79C22×C71—26no—
80C22×C71—44no—
81C22×C71H(5,𝔽2)16yes—
82C22×C71C2511yescharacteristic
83C2×D71—26no—
84C2×D71—26no—
85C2×D71—26no—
86C2×D71—26no—
87C2×D71—26no—
88C2×D71—416no—
89C2×D71—412no—
90C2×D71—44no—
91C2×D71—44no—
92C4×C71—26no—
93C4×C71—26no—
94Dic71—26no—
95C2×C71—42no—
96C2×C71#56014yescomplemented
97C2×C71C2×H(5,𝔽2)13yes—
98D71—46no—
99D71—42no—
100D71—42no—
101C2×#560—711nomaximal, Sylow
102C71C2×#56011yescharacteristic, complemented, Sylow
103#560—7116no—
104C2×#127—711no—
105C2×#126—711no—
106C2×#126D7111yescharacteristic, complemented
107C2×#123—713no—
108C2×#121—713no—
109C2×#121—713no—
110C2×#120—713no—
111C25—711no—
112C22×D4—713no—
113C22×D4—713no—
114C22×D4—713no—
115C22×D4—713no—
116C23×C4—713no—
117#127—718no—
118#126—718no—
119#126C2×D7118yescomplemented
120#123—7124no—
121#121—7124no—
122#121—7124no—
123#120—7124no—
124C2×C4⋊C4—713no—
125C2×C4⋊C4C2×D7113yescomplemented
126C2×C22⋊C4—713no—
127C2×C22⋊C4—713no—
128C2×C22⋊C4—713no—
129C2×C22⋊C4C2×D7113yes—
130C2×C42C2×D7111yescharacteristic, complemented
131C24—2846no—
132C24—714no—
133C24—711no—
134C24—711no—
135C24—713no—
136C24C22×D7111yescharacteristic
137C2×D4—2846no—
138C2×D4—2846no—
139C2×D4—2846no—
140C2×D4—2846no—
141C2×D4—7112no—
142C2×D4—7112no—
143C2×D4—7112no—
144C2×D4—7112no—
145C22×C4—2846no—
146C22×C4—7112no—
147C22×C4—713no—
148C22×C4C22×D7113yes—
149C22×C4C22×D7113yes—
150C4⋊C4—7112no—
151C4⋊C4C22×D71112yescomplemented
152C22⋊C4—7112no—
153C22⋊C4—7112no—
154C22⋊C4—7112no—
155C22⋊C4C22×D71112yes—
156C42C22×D7114yescomplemented
157C23—26no—
158C23—28412no—
159C23—28416no—
160C23—712no—
161C23—712no—
162C23—716no—
163C23—1426no—
164C23—1426no—
165C23—1426no—
166C23—1426no—
167C23—1426no—
168C23C23×D7112yes—
169C23C23×D7111yescharacteristic
170D4—28412no—
171D4—28412no—
172D4—28412no—
173D4—28412no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—28412no—
177C2×C4—716no—
178C2×C4—1426no—
179C2×C4C23×D7116yes—
180C2×C4C23×D7116yes—
181C22—26no—
182C22—44no—
183C22—28416no—
184C22—2844no—
185C22—2844no—
186C22—28412no—
187C22—1426no—
188C22—1426no—
189C22—1426no—
190C22—1426no—
191C22—1426no—
192C22#1455617416yes—
193C22C24×D7111yescharacteristic
194C4—26no—
195C4—26no—
196C4—1426no—
197C2—42no—
198C2—2842no—
199C2—2842no—
200C2—2846no—
201C2#1798674014yesdirect product factor
202C2C2×#1455617413yes—
subgroupG/H
centerC23C23×D71
commutatorC22×C71C25
FrattiniC22C24×D71
FittingC2×#126×C71C2
socleC23×C71C24
hypercenterC2×#126D71
hypocenterC71C2×#560
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #17986740C2direct (N × H)
2. C2×#14556125C2semidirect
3. C2×#14556122C2semidirect
4. C2×#14556113C2semidirect
5. C2×#14556103C2semidirect
6. C2×#14556101C2semidirect
7. C2×#14556098C2semidirect
8. C2×#126×C71C2semidirect
12. C22×D284C22semidirect
14. C22×C4×D71C22semidirect
15. #14556125C22semidirect
16. #14556122C22semidirect
17. #14556113C22semidirect
18. #14556103C22semidirect
19. #14556101C22semidirect
20. #14556098C22semidirect
21. C2×Dic71⋊C4C22semidirect
24. C2×(C2×D71)⋊C4C22semidirect
25. #126×C71C22semidirect
26. C2×C4⋊C4×C71C22semidirect
28. C2×C42×C71C22semidirect
39. C2×D284C23semidirect
44. C2×C4×D71C23semidirect
45. Dic71⋊C4C23semidirect
48. (C2×D71)⋊C4C23semidirect
52. C4⋊C4×C71C23semidirect
54. C42×C71C23semidirect
96. C2×C71#560semidirect
102. C71C2×#560semidirect
106. C2×#126D71semidirect
119. #126C2×D71semidirect
125. C2×C4⋊C4C2×D71semidirect
130. C2×C42C2×D71semidirect
151. C4⋊C4C22×D71semidirect
156. C42C22×D71semidirect
201. C2#17986740direct (N × H)