Finite Group Databasebeta

← C2×#20261472 #27155850

Subgroups — C2×#20261472 #27155850

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#20261472C2116yesmaximal, direct product factor
2C2×#15690804C212yesmaximal, complemented
3C2×#15690791C213yesmaximal, complemented
4C2×#15690782C216yesmaximal, complemented
5C2×#15690778C213yesmaximal, complemented
6C2×#127×C97C211yesmaximal, characteristic, complemented
7C24×D97C2211yescharacteristic
8C22×C22⋊D97C2216yes—
9C23×Dic97C2213yescomplemented
10#15690804C22116yescomplemented
11#15690791C22124yescomplemented
12#15690782C22148yescomplemented
13#15690778C22124yescomplemented
14C2×#13095490C2216yescomplemented
15C2×C22⋊Dic97C2213yes—
16C2×C22⋊Dic97C2213yescomplemented
17C2×#13095488C2216yes—
18C22×D4×C97C2213yes—
19C22×D4×C97C2213yescomplemented
20#127×C97C2218yescomplemented
21C2×C42×C97C2211yescharacteristic, complemented
22C23×D97—46no—
23C23×D97C2312yes—
24C23×D97C2314yes—
25C2×C22⋊D97—412no—
26C2×C22⋊D97C23124yes—
27C22×Dic97—46no—
28C22×Dic97C2316yes—
29C22×Dic97C23112yescomplemented
30#13095490C23124yescomplemented
31C22⋊Dic97C23112yes—
32C22⋊Dic97C23112yescomplemented
33#13095488C23124yes—
34C24×C97C2311yescharacteristic
35C24×C97C2313yes—
36C2×D4×C97—46no—
37C2×D4×C97—46no—
38C2×D4×C97C23112yes—
39C2×D4×C97C23112yescomplemented
40C22×C4×C97C2313yes—
41C42×C97C2314yescomplemented
42C23×C97—26no—
43C23×C97—212no—
44C23×C97—26no—
45C23×C97C2412yes—
46C23×C97C2416yes—
47C23×C97C2411yescharacteristic
48C22×D97—212no—
49C22×D97—412no—
50C22×D97—416no—
51C22×D97C2414yes—
52D4×C97—412no—
53D4×C97—412no—
54C2×C4×C97—26no—
55C2×C4×C97C2416yes—
56C22⋊D97—424no—
57C2×Dic97—212no—
58C2×Dic97—412no—
59C2×Dic97C24112yes—
60C22×C97—26no—
61C22×C97—212no—
62C22×C97—26no—
63C22×C97—44no—
64C22×C97—412no—
65C22×C97H(5,𝔽2)16yes—
66C22×C97C2511yescharacteristic
67C2×D97—212no—
68C2×D97—416no—
69C2×D97—48no—
70C4×C97—26no—
71Dic97—212no—
72C2×C97—42no—
73C2×C97—46no—
74C2×C97#56014yescomplemented
75C2×C97C2×H(5,𝔽2)13yes—
76D97—44no—
77C2×#560—971nomaximal, Sylow
78C97C2×#56011yescharacteristic, complemented, Sylow
79#560—9716no—
80C2×#127D9711yescharacteristic, complemented
81C2×#126—972no—
82C2×#123—973no—
83C2×#121—976no—
84C2×#120—973no—
85C25—971no—
86C22×D4—976no—
87C22×D4C2×D9713yes—
88C22×D4C2×D9713yescomplemented
89C23×C4—973no—
90#127C2×D9718yescomplemented
91#126—9716no—
92#123—9724no—
93#121—9748no—
94#120—9724no—
95C2×C4⋊C4—976no—
96C2×C22⋊C4—973no—
97C2×C22⋊C4—973no—
98C2×C22⋊C4—976no—
99C2×C42C2×D9711yescharacteristic, complemented
100C24—974no—
101C24—972no—
102C24—3886no—
103C24C22×D9711yescharacteristic
104C24C22×D9713yes—
105C2×D4—46no—
106C2×D4—46no—
107C2×D4—9724no—
108C2×D4—38812no—
109C2×D4C22×D97112yes—
110C2×D4C22×D97112yescomplemented
111C22×C4—9712no—
112C22×C4—976no—
113C22×C4—3886no—
114C22×C4C22×D9713yes—
115C4⋊C4—9724no—
116C22⋊C4—9712no—
117C22⋊C4—9712no—
118C22⋊C4—9724no—
119C42C22×D9714yescomplemented
120C23—26no—
121C23—212no—
122C23—26no—
123C23—974no—
124C23—38812no—
125C23—38816no—
126C23—19412no—
127C23C23×D9712yes—
128C23C23×D9716yes—
129C23C23×D9711yescharacteristic
130D4—412no—
131D4—412no—
132D4—38824no—
133C2×C4—26no—
134C2×C4—9712no—
135C2×C4—38812no—
136C2×C4—19412no—
137C2×C4C23×D9716yes—
138C22—26no—
139C22—212no—
140C22—26no—
141C22—44no—
142C22—412no—
143C22—38816no—
144C22—3888no—
145C22—19412no—
146C22#1569085516yes—
147C22C24×D9711yescharacteristic
148C4—26no—
149C4—19412no—
150C2—42no—
151C2—46no—
152C2—3884no—
153C2#2026147214yesdirect product factor
154C2C2×#1569085513yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#127×C97C2
socleC23×C97C24
hypercenterC2×#127D97
hypocenterC97C2×#560
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20261472C2direct (N × H)
2. C2×#15690804C2semidirect
3. C2×#15690791C2semidirect
4. C2×#15690782C2semidirect
5. C2×#15690778C2semidirect
6. C2×#127×C97C2semidirect
9. C23×Dic97C22semidirect
10. #15690804C22semidirect
11. #15690791C22semidirect
12. #15690782C22semidirect
13. #15690778C22semidirect
14. C2×#13095490C22semidirect
16. C2×C22⋊Dic97C22semidirect
19. C22×D4×C97C22semidirect
20. #127×C97C22semidirect
21. C2×C42×C97C22semidirect
29. C22×Dic97C23semidirect
30. #13095490C23semidirect
32. C22⋊Dic97C23semidirect
39. C2×D4×C97C23semidirect
41. C42×C97C23semidirect
74. C2×C97#560semidirect
78. C97C2×#560semidirect
80. C2×#127D97semidirect
88. C22×D4C2×D97semidirect
90. #127C2×D97semidirect
99. C2×C42C2×D97semidirect
110. C2×D4C22×D97semidirect
119. C42C22×D97semidirect
153. C2#20261472direct (N × H)

Maximal subgroups

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

subgroupindex
#202614722
C2×#156908042
C2×#156907912
C2×#156907822
C2×#156907782
C2×#127×C972
C2×#56097