Finite Group Databasebeta

← C2×#20259923 #27136044

Subgroups — C2×#20259923 #27136044

Order 12352.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×#15690554C211yesmaximal, characteristic, complemented
2C2×D4×D193C211yesmaximal, characteristic, complemented
3#20259923C218yesmaximal, direct product factor
4C2×#15690544C211yesmaximal, characteristic, complemented
5C2×#15690543C211yesmaximal, characteristic, complemented
6C2×#15690540C211yesmaximal, characteristic, complemented
7C2×#15690538C211yesmaximal, characteristic, complemented
8C2×D8×C193C211yesmaximal, characteristic, complemented
9C23×D193—21no—
10#15690554—22no—
11#15690554C2214yescomplemented
12C2×Dic386C2211yescharacteristic, complemented
13C2×D772C2211yescharacteristic
14C2×C22⋊D193—21no—
15C2×C22⋊D193—21no—
16D4×D193—22no—
17D4×D193C2214yescomplemented
18C22×Dic193—21no—
19C2×C4×D193C2211yescharacteristic
20#15690544C2214yescomplemented
21#15690543C2214yescomplemented
22#15690540C2214yescomplemented
23#15690538C2214yescomplemented
24C2×#13095393C2211yescharacteristic, complemented
25C2×D4×C193C2211yescharacteristic
26C2×D4×C193C2211yescharacteristic, complemented
27D8×C193C2214yescomplemented
28C2×C8×C193C2211yescharacteristic, complemented
29C22×D193—21no—
30C22×D193—24no—
31C22×D193—44no—
32C22×D193D411yescharacteristic
33Dic386—21no—
34Dic386C2312yescomplemented
35D772—21no—
36D772C2312yes—
37C22⋊D193—24no—
38C22⋊D193—24no—
39C2×Dic193—21no—
40C2×Dic193—24no—
41C2×Dic193D411yescharacteristic
42C4×D193D412yes—
43C4×D193C2312yes—
44#13095393C2312yescomplemented
45C23×C193—21no—
46C23×C193—21no—
47D4×C193—21no—
48D4×C193—21no—
49D4×C193C2312yes—
50D4×C193C2312yescomplemented
51C2×C4×C193C2311yescharacteristic
52C8×C193C2312yescomplemented
53C22×C193—22no—
54C22×C193—22no—
55C22×C193—42no—
56C22×C193—42no—
57C22×C193C2×D411yescharacteristic
58C2×D193—22no—
59C2×D193—22no—
60C2×D193—48no—
61C2×D193—42no—
62C2×D193C2×D412yes—
63C4×C193C2×D411yescharacteristic
64C4×C193C2411yescharacteristic
65Dic193—22no—
66Dic193C2×D412yes—
67C2×C193—42no—
68C2×C193—42no—
69C2×C193AGL(1,ℤ/8)12yescomplemented
70C2×C193C22×D411yescharacteristic
71D193—22no—
72D193—42no—
73C193C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
74C2×AGL(1,ℤ/8)—1931nomaximal, Sylow
75C2×#41—1931no—
76C22×D4—1931no—
77AGL(1,ℤ/8)—1938no—
78C2×SD16—1931no—
79C2×SD16—1931no—
80C2×D8—1931no—
81C2×D8D19311yescharacteristic, complemented
82C2×M16—1931no—
83C24—3861no—
84#41—3862no—
85#41—1934no—
86C2×Q8—1931no—
87C2×D4—3861no—
88C2×D4—3862no—
89C2×D4—3861no—
90C2×D4—1934no—
91C2×D4—1931no—
92C2×D4C2×D19311yescharacteristic
93C2×D4C2×D19311yescharacteristic, complemented
94C22×C4—3861no—
95C22×C4—1931no—
96SD16—1934no—
97SD16—1934no—
98D8—1934no—
99D8C2×D19314yescomplemented
100M16—1934no—
101C2×C8—1931no—
102C2×C8C2×D19311yescharacteristic, complemented
103C23—21no—
104C23—21no—
105C23—7724no—
106C23—3864no—
107C23—3861no—
108C23—1931no—
109Q8—3861no—
110Q8—1932no—
111D4—21no—
112D4—21no—
113D4—3864no—
114D4—3864no—
115D4—3861no—
116D4—1932no—
117D4C22×D19312yes—
118D4C22×D19312yescomplemented
119C2×C4—3861no—
120C2×C4—3864no—
121C2×C4—1931no—
122C2×C4—1932no—
123C2×C4—1932no—
124C2×C4C22×D19311yescharacteristic
125C8—1932no—
126C8C22×D19312yescomplemented
127C22—22no—
128C22—22no—
129C22—42no—
130C22—42no—
131C22—7728no—
132C22—7722no—
133C22—3862no—
134C22—3862no—
135C22—1932no—
136C22D4×D19311yescharacteristic
137C4—3862no—
138C4—1932no—
139C4D4×D19311yescharacteristic
140C4C23×D19311yescharacteristic
141C2—42no—
142C2—42no—
143C2—7722no—
144C2—3862no—
145C2#2025992312yesdirect product factor
146C2C2×D4×D19311yescharacteristic
subgroupG/H
centerC22D4×D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
FittingC2×D8×C193C2
socleC22×C193C2×D4
hypercenterC2×D8D193
hypocenterC193C2×AGL(1,ℤ/8)
LCSG ▷ C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15690554C2semidirect
2. C2×D4×D193C2semidirect
3. #20259923C2direct (N × H)
4. C2×#15690544C2semidirect
5. C2×#15690543C2semidirect
6. C2×#15690540C2semidirect
7. C2×#15690538C2semidirect
8. C2×D8×C193C2semidirect
11. #15690554C22semidirect
12. C2×Dic386C22semidirect
17. D4×D193C22semidirect
20. #15690544C22semidirect
21. #15690543C22semidirect
22. #15690540C22semidirect
23. #15690538C22semidirect
24. C2×#13095393C22semidirect
26. C2×D4×C193C22semidirect
27. D8×C193C22semidirect
28. C2×C8×C193C22semidirect
34. Dic386C23semidirect
44. #13095393C23semidirect
50. D4×C193C23semidirect
52. C8×C193C23semidirect
69. C2×C193AGL(1,ℤ/8)semidirect
73. C193C2×AGL(1,ℤ/8)semidirect
81. C2×D8D193semidirect
93. C2×D4C2×D193semidirect
99. D8C2×D193semidirect
102. C2×C8C2×D193semidirect
118. D4C22×D193semidirect
126. C8C22×D193semidirect
145. C2#20259923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156905542
C2×D4×D1932
#202599232
C2×#156905442
C2×#156905432
C2×#156905402
C2×#156905382
C2×D8×C1932
C2×AGL(1,ℤ/8)193