Finite Group Databasebeta

← C2×#20265918 #27185250

Subgroups — C2×#20265918 #27185250

Order 12928.

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#20265918C2116yesmaximal, direct product factor
2C2×#15692252C211yesmaximal, characteristic, complemented
3C2×#15692249C211yesmaximal, characteristic, complemented
4C2×#15692240C213yesmaximal, complemented
5C2×#15692230C213yesmaximal, complemented
6C2×#15692228C213yesmaximal, complemented
7C2×#15692225C213yesmaximal, complemented
8C2×#126×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C22×D404C2213yes—
11C22×D404C2213yes—
12C22×D404C2213yescomplemented
13C22×C22⋊D101C2213yes—
14C22×C4×D101C2213yescomplemented
15#15692252C2218yescomplemented
16#15692249C2218yescomplemented
17#15692240C22124yescomplemented
18#15692230C22124yescomplemented
19#15692228C22124yescomplemented
20#15692225C22124yescomplemented
21C2×#13108053C2213yescomplemented
22C2×#13108051C2213yes—
23C2×#13108051C2213yes—
24C2×#13108051C2213yescomplemented
25#126×C101C2218yescomplemented
26C2×C4⋊C4×C101C2213yescomplemented
27C2×C22⋊C4×C101C2213yes—
28C2×C42×C101C2211yescharacteristic, complemented
29C23×D101—46no—
30C23×D101C2313yes—
31C23×D101C2311yescharacteristic
32C23×D101C2311yescharacteristic
33C23×D101C2314yes—
34C2×D404—46no—
35C2×D404—46no—
36C2×D404—46no—
37C2×D404C23112yes—
38C2×D404C23112yes—
39C2×D404C23112yescomplemented
40C2×C22⋊D101—46no—
41C2×C22⋊D101C23112yes—
42C22×Dic101C2313yes—
43C2×C4×D101—46no—
44C2×C4×D101C23112yescomplemented
45#13108053C23112yescomplemented
46#13108051C23112yes—
47#13108051C23112yes—
48#13108051C23112yescomplemented
49C24×C101C2311yescharacteristic
50C22×C4×C101C2313yes—
51C22×C4×C101C2313yes—
52C4⋊C4×C101C23112yescomplemented
53C22⋊C4×C101C23112yes—
54C42×C101C2314yescomplemented
55C23×C101—26no—
56C23×C101C2412yes—
57C23×C101C2411yescharacteristic
58C22×D101—26no—
59C22×D101—26no—
60C22×D101—26no—
61C22×D101—26no—
62C22×D101—26no—
63C22×D101—412no—
64C22×D101—416no—
65C22×D101C2412yes—
66C22×D101C2412yes—
67C22×D101C2416yes—
68C2×C4×C101—26no—
69C2×C4×C101—26no—
70C2×C4×C101C2416yes—
71C2×C4×C101C2416yes—
72C22⋊D101—412no—
73C2×Dic101—26no—
74C2×Dic101C2416yes—
75D404—412no—
76D404—412no—
77D404—412no—
78C4×D101—412no—
79C22×C101—26no—
80C22×C101—44no—
81C22×C101H(5,𝔽2)16yes—
82C22×C101C2511yescharacteristic
83C2×D101—26no—
84C2×D101—26no—
85C2×D101—26no—
86C2×D101—26no—
87C2×D101—26no—
88C2×D101—416no—
89C2×D101—412no—
90C2×D101—44no—
91C2×D101—44no—
92C4×C101—26no—
93C4×C101—26no—
94Dic101—26no—
95C2×C101—42no—
96C2×C101#56014yescomplemented
97C2×C101C2×H(5,𝔽2)13yes—
98D101—46no—
99D101—42no—
100D101—42no—
101C2×#560—1011nomaximal, Sylow
102C101C2×#56011yescharacteristic, complemented, Sylow
103#560—10116no—
104C2×#127—1011no—
105C2×#126—1011no—
106C2×#126D10111yescharacteristic, complemented
107C2×#123—1013no—
108C2×#121—1013no—
109C2×#121—1013no—
110C2×#120—1013no—
111C25—1011no—
112C22×D4—1013no—
113C22×D4—1013no—
114C22×D4—1013no—
115C22×D4—1013no—
116C23×C4—1013no—
117#127—1018no—
118#126—1018no—
119#126C2×D10118yescomplemented
120#123—10124no—
121#121—10124no—
122#121—10124no—
123#120—10124no—
124C2×C4⋊C4—1013no—
125C2×C4⋊C4C2×D10113yescomplemented
126C2×C22⋊C4—1013no—
127C2×C22⋊C4—1013no—
128C2×C22⋊C4—1013no—
129C2×C22⋊C4C2×D10113yes—
130C2×C42C2×D10111yescharacteristic, complemented
131C24—1014no—
132C24—1011no—
133C24—1011no—
134C24—1013no—
135C24—4046no—
136C24C22×D10111yescharacteristic
137C2×D4—10112no—
138C2×D4—10112no—
139C2×D4—10112no—
140C2×D4—10112no—
141C2×D4—4046no—
142C2×D4—4046no—
143C2×D4—4046no—
144C2×D4—4046no—
145C22×C4—1013no—
146C22×C4—10112no—
147C22×C4—4046no—
148C22×C4C22×D10113yes—
149C22×C4C22×D10113yes—
150C4⋊C4—10112no—
151C4⋊C4C22×D101112yescomplemented
152C22⋊C4—10112no—
153C22⋊C4—10112no—
154C22⋊C4—10112no—
155C22⋊C4C22×D101112yes—
156C42C22×D10114yescomplemented
157C23—26no—
158C23—1012no—
159C23—1012no—
160C23—1016no—
161C23—40412no—
162C23—40416no—
163C23—2026no—
164C23—2026no—
165C23—2026no—
166C23—2026no—
167C23—2026no—
168C23C23×D10112yes—
169C23C23×D10111yescharacteristic
170D4—40412no—
171D4—40412no—
172D4—40412no—
173D4—40412no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—1016no—
177C2×C4—40412no—
178C2×C4—2026no—
179C2×C4C23×D10116yes—
180C2×C4C23×D10116yes—
181C22—26no—
182C22—44no—
183C22—40416no—
184C22—4044no—
185C22—4044no—
186C22—40412no—
187C22—2026no—
188C22—2026no—
189C22—2026no—
190C22—2026no—
191C22—2026no—
192C22#1569230116yes—
193C22C24×D10111yescharacteristic
194C4—26no—
195C4—26no—
196C4—2026no—
197C2—42no—
198C2—4042no—
199C2—4042no—
200C2—4046no—
201C2#2026591814yesdirect product factor
202C2C2×#1569230113yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#126×C101C2
socleC23×C101C24
hypercenterC2×#126D101
hypocenterC101C2×#560
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20265918C2direct (N × H)
2. C2×#15692252C2semidirect
3. C2×#15692249C2semidirect
4. C2×#15692240C2semidirect
5. C2×#15692230C2semidirect
6. C2×#15692228C2semidirect
7. C2×#15692225C2semidirect
8. C2×#126×C101C2semidirect
12. C22×D404C22semidirect
14. C22×C4×D101C22semidirect
15. #15692252C22semidirect
16. #15692249C22semidirect
17. #15692240C22semidirect
18. #15692230C22semidirect
19. #15692228C22semidirect
20. #15692225C22semidirect
21. C2×#13108053C22semidirect
24. C2×#13108051C22semidirect
25. #126×C101C22semidirect
26. C2×C4⋊C4×C101C22semidirect
28. C2×C42×C101C22semidirect
39. C2×D404C23semidirect
44. C2×C4×D101C23semidirect
45. #13108053C23semidirect
48. #13108051C23semidirect
52. C4⋊C4×C101C23semidirect
54. C42×C101C23semidirect
96. C2×C101#560semidirect
102. C101C2×#560semidirect
106. C2×#126D101semidirect
119. #126C2×D101semidirect
125. C2×C4⋊C4C2×D101semidirect
130. C2×C42C2×D101semidirect
151. C4⋊C4C22×D101semidirect
156. C42C22×D101semidirect
201. C2#20265918direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202659182
C2×#156922522
C2×#156922492
C2×#156922402
C2×#156922302
C2×#156922282
C2×#156922252
C2×#126×C1012
C2×#560101