Finite Group Databasebeta

← #27136056

Subgroups — #27136056

Order 12352.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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#41×D193C212yesmaximal, complemented
2#20259931C212yesmaximal, complemented
3#20259928C212yesmaximal, complemented
4#20259926C212yesmaximal, complemented
5#20259923C212yesmaximal, complemented
6#20259921C211yesmaximal, characteristic, complemented
7#20259919C212yesmaximal, complemented
8C2×#15690538C211yesmaximal, characteristic, complemented
9#135×C193C211yesmaximal, characteristic, complemented
10#15690556C2212yes—
11#15690555—22no—
12#15690555C2212yescomplemented
13#15690554—22no—
14#15690554C2212yes—
15Q8×D193C2212yescomplemented
16D4×D193—22no—
17D4×D193C2212yescomplemented
18C2×C4×D193—22no—
19C2×C4×D193C2211yescharacteristic
20Dic772C2211yescharacteristic, complemented
21#15690545C2212yescomplemented
22#15690544C2212yescomplemented
23#15690543C2212yescomplemented
24#15690542C2212yescomplemented
25D1544C2211yescharacteristic, complemented
26#15690540C2212yescomplemented
27#15690538C2212yescomplemented
28#15690538C2212yescomplemented
29C2×#13095393C2211yescharacteristic, complemented
30#41×C193C2212yescomplemented
31Q16×C193C2211yescharacteristic, complemented
32SD16×C193C2212yescomplemented
33D8×C193C2211yescharacteristic, complemented
34C2×C8×C193C2211yescharacteristic, complemented
35C22×D193—22no—
36C22×D193D411yescharacteristic
37Dic386—22no—
38Dic386C2312yes—
39D772—22no—
40D772C2312yes—
41C22⋊D193—22no—
42C22⋊D193—22no—
43C22⋊D193—22no—
44C2×Dic193—22no—
45C2×Dic193D411yescharacteristic
46C4×D193—22no—
47C4×D193—22no—
48C4×D193—22no—
49C4×D193D412yes—
50C4×D193C2312yes—
51#13095393C2312yes—
52Q8×C193C2312yes—
53D4×C193—22no—
54D4×C193C2312yes—
55C2×C4×C193—22no—
56C2×C4×C193C2311yescharacteristic
57C8×C193C2311yescharacteristic
58C8×C193C2311yescharacteristic
59C22×C193—22no—
60C22×C193C2×D411yescharacteristic
61C2×D193—22no—
62C2×D193—41no—
63C2×D193—44no—
64C2×D193C2×D412yes—
65C4×C193—22no—
66C4×C193C2×D411yescharacteristic
67C4×C193C2411yescharacteristic
68Dic193—22no—
69Dic193C2×D412yes—
70C2×C193—21no—
71C2×C193—42no—
72C2×C193C22×D411yescharacteristic
73D193—22no—
74D193—42no—
75C193#57511yescharacteristic, complemented, Sylow
76#575—1931nomaximal, Sylow
77C2×#41—1932no—
78#137—1932no—
79#137—1932no—
80AGL(1,ℤ/8)—1932no—
81AGL(1,ℤ/8)—1932no—
82#135—1932no—
83#135—1931no—
84#135D19311yescharacteristic, complemented
85C2×M16—1931no—
86#41—3862no—
87#41—3862no—
88#41—1932no—
89#41—1932no—
90#41—1932no—
91#41C2×D19312yescomplemented
92C2×Q8—1932no—
93C2×D4—3862no—
94C2×D4—1932no—
95C22×C4—3862no—
96C22×C4—1931no—
97Q16—1932no—
98Q16—1931no—
99Q16C2×D19311yescharacteristic, complemented
100SD16—1932no—
101SD16—1932no—
102SD16—1932no—
103SD16C2×D19312yescomplemented
104D8—1932no—
105D8—1931no—
106D8C2×D19311yescharacteristic, complemented
107M16—1932no—
108M16—1932no—
109C2×C8—1931no—
110C2×C8C2×D19311yescharacteristic, complemented
111C23—3862no—
112C23—1931no—
113Q8—3862no—
114Q8—1932no—
115Q8C22×D19312yes—
116D4—22no—
117D4—3862no—
118D4—3862no—
119D4—3862no—
120D4—3862no—
121D4—1932no—
122D4C22×D19312yes—
123C2×C4—22no—
124C2×C4—3862no—
125C2×C4—3862no—
126C2×C4—3862no—
127C2×C4—3862no—
128C2×C4—1931no—
129C2×C4—1932no—
130C2×C4—1932no—
131C2×C4C22×D19311yescharacteristic
132C8—1932no—
133C8C22×D19311yescharacteristic
134C8C22×D19311yescharacteristic
135C22—22no—
136C22—7721no—
137C22—7724no—
138C22—3862no—
139C22—1932no—
140C22D4×D19311yescharacteristic
141C4—22no—
142C4—3862no—
143C4—1932no—
144C4D4×D19311yescharacteristic
145C4C23×D19311yescharacteristic
146C2—21no—
147C2—42no—
148C2—7722no—
149C2—3862no—
150C2C2×D4×D19311yescharacteristic
subgroupG/H
centerC4D4×D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
Fitting#135×C193C2
socleC2×C193C22×D4
hypercenter#135D193
hypocenterC193#575
LCSG ▷ C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D193C2semidirect
2. #20259931C2semidirect
3. #20259928C2semidirect
4. #20259926C2semidirect
5. #20259923C2semidirect
6. #20259921C2semidirect
7. #20259919C2semidirect
8. C2×#15690538C2semidirect
9. #135×C193C2semidirect
12. #15690555C22semidirect
15. Q8×D193C22semidirect
17. D4×D193C22semidirect
20. Dic772C22semidirect
21. #15690545C22semidirect
22. #15690544C22semidirect
23. #15690543C22semidirect
24. #15690542C22semidirect
25. D1544C22semidirect
26. #15690540C22semidirect
27. #15690538C22semidirect
28. #15690538C22semidirect
29. C2×#13095393C22semidirect
30. #41×C193C22semidirect
31. Q16×C193C22semidirect
32. SD16×C193C22semidirect
33. D8×C193C22semidirect
34. C2×C8×C193C22semidirect
75. C193#575semidirect
84. #135D193semidirect
91. #41C2×D193semidirect
99. Q16C2×D193semidirect
103. SD16C2×D193semidirect
106. D8C2×D193semidirect
110. C2×C8C2×D193semidirect

Maximal subgroups

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

subgroupindex
#41×D1932
#202599312
#202599282
#202599262
#202599232
#202599212
#202599192
C2×#156905382
#135×C1932
#575193