Finite Group Databasebeta

← C2×#20261478 #27155857

Subgroups — C2×#20261478 #27155857

Order 12416.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 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#20261478C2116yesmaximal, direct product factor
2C2×#15690797C213yesmaximal, complemented
3C2×#15690796C216yesmaximal, complemented
4C2×#15690779C212yesmaximal, complemented
5C2×#15690778C213yesmaximal, complemented
6C2×#120×C97C211yesmaximal, characteristic, complemented
7C24×D97C2211yescharacteristic, complemented
8C22×Dic194C2213yescomplemented
9C22×C22⋊D97C2216yes—
10#15690797C22124yescomplemented
11#15690796C22148yescomplemented
12#15690779C22116yescomplemented
13#15690778C22124yescomplemented
14C2×C22⋊Dic97C2213yescomplemented
15C2×C22⋊Dic97C2216yescomplemented
16C2×#13095488C2216yescomplemented
17C2×C4×Dic97C2213yescomplemented
18C25×C97C2211yescharacteristic, complemented
19C22×D4×C97C2213yes—
20#120×C97C2218yescomplemented
21C2×C22⋊C4×C97C2213yescomplemented
22C23×D97—46no—
23C23×D97C2312yes—
24C23×D97C2314yescomplemented
25C2×Dic194—46no—
26C2×Dic194C23112yescomplemented
27C2×C22⋊D97—412no—
28C2×C22⋊D97C23124yes—
29C22×Dic97C2316yes—
30C22⋊Dic97C23112yescomplemented
31C22⋊Dic97C23124yescomplemented
32#13095488C23124yescomplemented
33C4×Dic97C23112yescomplemented
34C24×C97—46no—
35C24×C97C2311yescharacteristic
36C24×C97C2313yes—
37C24×C97C2314yescomplemented
38C2×D4×C97—46no—
39C2×D4×C97C23112yes—
40C22×C4×C97C2313yes—
41C22⋊C4×C97C23112yescomplemented
42C23×C97—26no—
43C23×C97—26no—
44C23×C97—212no—
45C23×C97—412no—
46C23×C97—416no—
47C23×C97C2416yes—
48C23×C97C2412yes—
49C23×C97C2411yescharacteristic
50C22×D97—212no—
51C22×D97—412no—
52C22×D97—416no—
53C22×D97C2414yes—
54D4×C97—412no—
55C2×C4×C97—26no—
56C2×C4×C97C2416yes—
57C22⋊D97—424no—
58C2×Dic97—212no—
59C2×Dic97C24112yes—
60Dic194—412no—
61C22×C97—212no—
62C22×C97—26no—
63C22×C97—26no—
64C22×C97—416no—
65C22×C97—44no—
66C22×C97—412no—
67C22×C97H(5,𝔽2)16yes—
68C22×C97C2511yescharacteristic
69C2×D97—212no—
70C2×D97—416no—
71C2×D97—48no—
72C4×C97—26no—
73Dic97—212no—
74C2×C97—42no—
75C2×C97—46no—
76C2×C97H(3,𝔽4)14yescomplemented
77C2×C97C2×H(5,𝔽2)13yes—
78D97—44no—
79C2×H(3,𝔽4)—971nomaximal, Sylow
80C97C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
81H(3,𝔽4)—9716no—
82C2×#124—973no—
83C2×#124—976no—
84C2×#120—972no—
85C2×#120—973no—
86C2×#120D9711yescharacteristic, complemented
87C25—971no—
88C25C2×D9711yescharacteristic, complemented
89C22×Q8—973no—
90C22×D4—976no—
91C22×D4C2×D9713yes—
92#124—9724no—
93#124—9748no—
94#120—9716no—
95#120—9724no—
96#120C2×D9718yescomplemented
97C2×C22⋊C4—973no—
98C2×C22⋊C4—976no—
99C2×C22⋊C4—976no—
100C2×C22⋊C4C2×D9713yescomplemented
101C2×C42—973no—
102C24—46no—
103C24—974no—
104C24—972no—
105C24—3886no—
106C24C22×D9711yescharacteristic
107C24C22×D9713yes—
108C24C22×D9714yescomplemented
109C2×Q8—9712no—
110C2×Q8—3886no—
111C2×D4—46no—
112C2×D4—9724no—
113C2×D4—38812no—
114C2×D4C22×D97112yes—
115C22×C4—976no—
116C22×C4C22×D9713yes—
117C22⋊C4—9712no—
118C22⋊C4—9724no—
119C22⋊C4—9724no—
120C22⋊C4C22×D97112yescomplemented
121C42—9712no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—974no—
128C23—38812no—
129C23—38816no—
130C23—19412no—
131C23C23×D9712yes—
132C23C23×D9716yes—
133C23C23×D9711yescharacteristic
134Q8—38812no—
135D4—412no—
136D4—38824no—
137C2×C4—26no—
138C2×C4—9712no—
139C2×C4—19412no—
140C2×C4C23×D9716yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—38816no—
148C22—3888no—
149C22—19412no—
150C22#1569085516yes—
151C22C24×D9711yescharacteristic
152C4—26no—
153C4—19412no—
154C2—46no—
155C2—42no—
156C2—3884no—
157C2#2026147814yesdirect product factor
158C2C2×#1569085513yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#120×C97C2
socleC23×C97C24
hypercenterC2×#120D97
hypocenterC97C2×H(3,𝔽4)
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #20261478C2direct (N × H)
2. C2×#15690797C2semidirect
3. C2×#15690796C2semidirect
4. C2×#15690779C2semidirect
5. C2×#15690778C2semidirect
6. C2×#120×C97C2semidirect
7. C24×D97C22semidirect
8. C22×Dic194C22semidirect
10. #15690797C22semidirect
11. #15690796C22semidirect
12. #15690779C22semidirect
13. #15690778C22semidirect
14. C2×C22⋊Dic97C22semidirect
15. C2×C22⋊Dic97C22semidirect
16. C2×#13095488C22semidirect
17. C2×C4×Dic97C22semidirect
18. C25×C97C22semidirect
20. #120×C97C22semidirect
21. C2×C22⋊C4×C97C22semidirect
24. C23×D97C23semidirect
26. C2×Dic194C23semidirect
30. C22⋊Dic97C23semidirect
31. C22⋊Dic97C23semidirect
32. #13095488C23semidirect
33. C4×Dic97C23semidirect
37. C24×C97C23semidirect
41. C22⋊C4×C97C23semidirect
76. C2×C97H(3,𝔽4)semidirect
80. C97C2×H(3,𝔽4)semidirect
86. C2×#120D97semidirect
88. C25C2×D97semidirect
96. #120C2×D97semidirect
100. C2×C22⋊C4C2×D97semidirect
108. C24C22×D97semidirect
120. C22⋊C4C22×D97semidirect
157. C2#20261478direct (N × H)

Maximal subgroups

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

subgroupindex
#202614782
C2×#156907972
C2×#156907962
C2×#156907792
C2×#156907782
C2×#120×C972
C2×H(3,𝔽4)97