Finite Group Databasebeta

← C2×#20261476 #27155854

Subgroups — C2×#20261476 #27155854

Order 12416.

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#20261476C2116yesmaximal, direct product factor
2C2×#15690807C211yesmaximal, characteristic, complemented
3C2×#15690804C211yesmaximal, characteristic, complemented
4C2×#15690795C213yesmaximal, complemented
5C2×#15690785C213yesmaximal, complemented
6C2×#15690783C213yesmaximal, complemented
7C2×#15690780C213yesmaximal, complemented
8C2×#126×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×D388C2213yes—
11C22×D388C2213yes—
12C22×D388C2213yescomplemented
13C22×C22⋊D97C2213yes—
14C22×C4×D97C2213yescomplemented
15#15690807C2218yescomplemented
16#15690804C2218yescomplemented
17#15690795C22124yescomplemented
18#15690785C22124yescomplemented
19#15690783C22124yescomplemented
20#15690780C22124yescomplemented
21C2×#13095490C2213yescomplemented
22C2×#13095488C2213yes—
23C2×#13095488C2213yes—
24C2×#13095488C2213yescomplemented
25#126×C97C2218yescomplemented
26C2×C4⋊C4×C97C2213yescomplemented
27C2×C22⋊C4×C97C2213yes—
28C2×C42×C97C2211yescharacteristic, complemented
29C23×D97—46no—
30C23×D97C2311yescharacteristic
31C23×D97C2311yescharacteristic
32C23×D97C2313yes—
33C23×D97C2314yes—
34C2×D388—46no—
35C2×D388—46no—
36C2×D388—46no—
37C2×D388C23112yes—
38C2×D388C23112yes—
39C2×D388C23112yescomplemented
40C2×C22⋊D97—46no—
41C2×C22⋊D97C23112yes—
42C22×Dic97C2313yes—
43C2×C4×D97—46no—
44C2×C4×D97C23112yescomplemented
45#13095490C23112yescomplemented
46#13095488C23112yes—
47#13095488C23112yes—
48#13095488C23112yescomplemented
49C24×C97C2311yescharacteristic
50C22×C4×C97C2313yes—
51C22×C4×C97C2313yes—
52C4⋊C4×C97C23112yescomplemented
53C22⋊C4×C97C23112yes—
54C42×C97C2314yescomplemented
55C23×C97—26no—
56C23×C97C2412yes—
57C23×C97C2411yescharacteristic
58C22×D97—26no—
59C22×D97—26no—
60C22×D97—26no—
61C22×D97—26no—
62C22×D97—26no—
63C22×D97—412no—
64C22×D97—416no—
65C22×D97C2412yes—
66C22×D97C2412yes—
67C22×D97C2416yes—
68C2×C4×C97—26no—
69C2×C4×C97—26no—
70C2×C4×C97C2416yes—
71C2×C4×C97C2416yes—
72C22⋊D97—412no—
73C2×Dic97—26no—
74C2×Dic97C2416yes—
75D388—412no—
76D388—412no—
77D388—412no—
78C4×D97—412no—
79C22×C97—26no—
80C22×C97—44no—
81C22×C97H(5,𝔽2)16yes—
82C22×C97C2511yescharacteristic
83C2×D97—26no—
84C2×D97—26no—
85C2×D97—26no—
86C2×D97—26no—
87C2×D97—26no—
88C2×D97—416no—
89C2×D97—412no—
90C2×D97—44no—
91C2×D97—44no—
92C4×C97—26no—
93C4×C97—26no—
94Dic97—26no—
95C2×C97—42no—
96C2×C97#56014yescomplemented
97C2×C97C2×H(5,𝔽2)13yes—
98D97—46no—
99D97—42no—
100D97—42no—
101C2×#560—971nomaximal, Sylow
102C97C2×#56011yescharacteristic, complemented, Sylow
103#560—9716no—
104C2×#127—971no—
105C2×#126—971no—
106C2×#126D9711yescharacteristic, complemented
107C2×#123—973no—
108C2×#121—973no—
109C2×#121—973no—
110C2×#120—973no—
111C25—971no—
112C22×D4—973no—
113C22×D4—973no—
114C22×D4—973no—
115C22×D4—973no—
116C23×C4—973no—
117#127—978no—
118#126—978no—
119#126C2×D9718yescomplemented
120#123—9724no—
121#121—9724no—
122#121—9724no—
123#120—9724no—
124C2×C4⋊C4—973no—
125C2×C4⋊C4C2×D9713yescomplemented
126C2×C22⋊C4—973no—
127C2×C22⋊C4—973no—
128C2×C22⋊C4—973no—
129C2×C22⋊C4C2×D9713yes—
130C2×C42C2×D9711yescharacteristic, complemented
131C24—974no—
132C24—971no—
133C24—971no—
134C24—973no—
135C24—3886no—
136C24C22×D9711yescharacteristic
137C2×D4—9712no—
138C2×D4—9712no—
139C2×D4—9712no—
140C2×D4—9712no—
141C2×D4—3886no—
142C2×D4—3886no—
143C2×D4—3886no—
144C2×D4—3886no—
145C22×C4—973no—
146C22×C4—9712no—
147C22×C4—3886no—
148C22×C4C22×D9713yes—
149C22×C4C22×D9713yes—
150C4⋊C4—9712no—
151C4⋊C4C22×D97112yescomplemented
152C22⋊C4—9712no—
153C22⋊C4—9712no—
154C22⋊C4—9712no—
155C22⋊C4C22×D97112yes—
156C42C22×D9714yescomplemented
157C23—26no—
158C23—972no—
159C23—972no—
160C23—976no—
161C23—38812no—
162C23—38816no—
163C23—1946no—
164C23—1946no—
165C23—1946no—
166C23—1946no—
167C23—1946no—
168C23C23×D9712yes—
169C23C23×D9711yescharacteristic
170D4—38812no—
171D4—38812no—
172D4—38812no—
173D4—38812no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—976no—
177C2×C4—38812no—
178C2×C4—1946no—
179C2×C4C23×D9716yes—
180C2×C4C23×D9716yes—
181C22—26no—
182C22—44no—
183C22—38816no—
184C22—3884no—
185C22—3884no—
186C22—38812no—
187C22—1946no—
188C22—1946no—
189C22—1946no—
190C22—1946no—
191C22—1946no—
192C22#1569085616yes—
193C22C24×D9711yescharacteristic
194C4—26no—
195C4—26no—
196C4—1946no—
197C2—42no—
198C2—3882no—
199C2—3882no—
200C2—3886no—
201C2#2026147614yesdirect product factor
202C2C2×#1569085613yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#126×C97C2
socleC23×C97C24
hypercenterC2×#126D97
hypocenterC97C2×#560
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20261476C2direct (N × H)
2. C2×#15690807C2semidirect
3. C2×#15690804C2semidirect
4. C2×#15690795C2semidirect
5. C2×#15690785C2semidirect
6. C2×#15690783C2semidirect
7. C2×#15690780C2semidirect
8. C2×#126×C97C2semidirect
12. C22×D388C22semidirect
14. C22×C4×D97C22semidirect
15. #15690807C22semidirect
16. #15690804C22semidirect
17. #15690795C22semidirect
18. #15690785C22semidirect
19. #15690783C22semidirect
20. #15690780C22semidirect
21. C2×#13095490C22semidirect
24. C2×#13095488C22semidirect
25. #126×C97C22semidirect
26. C2×C4⋊C4×C97C22semidirect
28. C2×C42×C97C22semidirect
39. C2×D388C23semidirect
44. C2×C4×D97C23semidirect
45. #13095490C23semidirect
48. #13095488C23semidirect
52. C4⋊C4×C97C23semidirect
54. C42×C97C23semidirect
96. C2×C97#560semidirect
102. C97C2×#560semidirect
106. C2×#126D97semidirect
119. #126C2×D97semidirect
125. C2×C4⋊C4C2×D97semidirect
130. C2×C42C2×D97semidirect
151. C4⋊C4C22×D97semidirect
156. C42C22×D97semidirect
201. C2#20261476direct (N × H)

Maximal subgroups

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

subgroupindex
#202614762
C2×#156908072
C2×#156908042
C2×#156907952
C2×#156907852
C2×#156907832
C2×#156907802
C2×#126×C972
C2×#56097