Finite Group Databasebeta

← #135×D193 #27136037

Subgroups — #135×D193 #27136037

Order 12352.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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#20259922C211yesmaximal, characteristic, complemented
3#20259921C211yesmaximal, characteristic, complemented
4#20259920C212yesmaximal, complemented
5#20259919C212yesmaximal, complemented
6#20259918C211yesmaximal, characteristic, complemented
7Q16×D193C211yesmaximal, characteristic, complemented
8SD16×D193C212yesmaximal, complemented
9D8×D193C211yesmaximal, characteristic, complemented
10C2×C8×D193C211yesmaximal, characteristic, complemented
11#135×C193C211yesmaximal, characteristic, complemented
12#15690556C2212yescomplemented
13#15690555—22no—
14#15690555C2212yescomplemented
15#15690554—22no—
16#15690554C2212yescomplemented
17Q8×D193C2212yes—
18D4×D193—22no—
19D4×D193C2212yes—
20C2×C4×D193—22no—
21C2×C4×D193C2211yescharacteristic
22Dic772C2211yescharacteristic, complemented
23#15690545C2212yescomplemented
24#15690544C2212yescomplemented
25#15690543C2212yescomplemented
26#15690542C2212yescomplemented
27D1544C2211yescharacteristic, complemented
28#15690540C2212yescomplemented
29C2×#13095393C2211yescharacteristic, complemented
30C8×D193C2211yescharacteristic
31C8×D193C2211yescharacteristic
32C8×D193C2211yescharacteristic, complemented
33C8×D193C2211yescharacteristic, complemented
34#41×C193C2212yescomplemented
35Q16×C193C2211yescharacteristic, complemented
36SD16×C193C2212yescomplemented
37D8×C193C2211yescharacteristic, complemented
38C2×C8×C193C2211yescharacteristic, complemented
39C22×D193—22no—
40C22×D193D411yescharacteristic
41Dic386—22no—
42Dic386C2312yes—
43D772—22no—
44D772C2312yes—
45C22⋊D193—22no—
46C22⋊D193—22no—
47C22⋊D193—22no—
48C2×Dic193—22no—
49C2×Dic193D411yescharacteristic
50C4×D193—22no—
51C4×D193—22no—
52C4×D193—22no—
53C4×D193D411yescharacteristic
54C4×D193D411yescharacteristic
55C4×D193C2311yescharacteristic
56C4×D193C2311yescharacteristic
57#13095393C2311yescharacteristic
58#13095393C2311yescharacteristic
59Q8×C193C2312yes—
60D4×C193—22no—
61D4×C193C2312yes—
62C2×C4×C193—22no—
63C2×C4×C193C2311yescharacteristic
64C8×C193C2311yescharacteristic
65C8×C193C2311yescharacteristic
66C22×C193—22no—
67C22×C193C2×D411yescharacteristic
68C2×D193—22no—
69C2×D193—22no—
70C2×D193—44no—
71C2×D193C2×D411yescharacteristic
72C2×D193C2×D411yescharacteristic
73C4×C193—22no—
74C4×C193C2×D411yescharacteristic
75C4×C193C2411yescharacteristic
76Dic193—22no—
77Dic193C2×D411yescharacteristic
78Dic193C2×D411yescharacteristic
79C2×C193—21no—
80C2×C193—42no—
81C2×C193C22×D411yescharacteristic
82D193—21no—
83D193—42no—
84D193#13512yesdirect product factor
85C193C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—1931nomaximal, Sylow
87C2×#41—1932no—
88#135—1932no—
89#135—1931no—
90#135—1932no—
91#135—1931no—
92#135—1931no—
93#135D19311yescharacteristic, direct product factor
94C2×Q16—1931no—
95C2×SD16—1932no—
96C2×D8—1931no—
97C22×C8—1931no—
98#41—3862no—
99#41—3862no—
100#41—1932no—
101#41—1932no—
102#41—1932no—
103#41C2×D19312yescomplemented
104C2×Q8—1932no—
105C2×D4—3862no—
106C2×D4—1932no—
107C22×C4—3862no—
108C22×C4—1931no—
109Q16—1932no—
110Q16—1931no—
111Q16C2×D19311yescharacteristic, complemented
112SD16—1932no—
113SD16—1932no—
114SD16—1932no—
115SD16C2×D19312yescomplemented
116D8—1932no—
117D8—1931no—
118D8C2×D19311yescharacteristic, complemented
119C2×C8—1931no—
120C2×C8—1931no—
121C2×C8—1931no—
122C2×C8—1931no—
123C2×C8—1931no—
124C2×C8C2×D19311yescharacteristic, complemented
125C23—3862no—
126C23—1931no—
127Q8—3862no—
128Q8—1932no—
129Q8C22×D19312yes—
130D4—22no—
131D4—3862no—
132D4—3862no—
133D4—3862no—
134D4—3862no—
135D4—1932no—
136D4C22×D19312yes—
137C2×C4—22no—
138C2×C4—3862no—
139C2×C4—3862no—
140C2×C4—3862no—
141C2×C4—3862no—
142C2×C4—1931no—
143C2×C4—1931no—
144C2×C4—1931no—
145C2×C4—1931no—
146C2×C4—1931no—
147C2×C4C22×D19311yescharacteristic
148C8—1931no—
149C8—1931no—
150C8C22×D19311yescharacteristic
151C8C22×D19311yescharacteristic
152C22—22no—
153C22—7724no—
154C22—3862no—
155C22—3862no—
156C22—1931no—
157C22—1931no—
158C22D4×D19311yescharacteristic
159C4—22no—
160C4—3862no—
161C4—1931no—
162C4—1931no—
163C4D4×D19311yescharacteristic
164C4C23×D19311yescharacteristic
165C2—21no—
166C2—42no—
167C2—7722no—
168C2—3861no—
169C2—1932no—
170C2C2×D4×D19311yescharacteristic
subgroupG/H
centerC4D4×D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
Fitting#135×C193C2
socleC2×C193C22×D4
hypercenter#135D193
hypocenterC193C2×#135
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. #20259922C2semidirect
3. #20259921C2semidirect
4. #20259920C2semidirect
5. #20259919C2semidirect
6. #20259918C2semidirect
7. Q16×D193C2semidirect
8. SD16×D193C2semidirect
9. D8×D193C2semidirect
10. C2×C8×D193C2semidirect
11. #135×C193C2semidirect
12. #15690556C22semidirect
14. #15690555C22semidirect
16. #15690554C22semidirect
22. Dic772C22semidirect
23. #15690545C22semidirect
24. #15690544C22semidirect
25. #15690543C22semidirect
26. #15690542C22semidirect
27. D1544C22semidirect
28. #15690540C22semidirect
29. C2×#13095393C22semidirect
32. C8×D193C22semidirect
33. C8×D193C22semidirect
34. #41×C193C22semidirect
35. Q16×C193C22semidirect
36. SD16×C193C22semidirect
37. D8×C193C22semidirect
38. C2×C8×C193C22semidirect
84. D193#135direct (N × H)
85. C193C2×#135semidirect
93. #135D193direct (N × H)
103. #41C2×D193semidirect
111. Q16C2×D193semidirect
115. SD16C2×D193semidirect
118. D8C2×D193semidirect
124. 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
#202599222
#202599212
#202599202
#202599192
#202599182
Q16×D1932
SD16×D1932
D8×D1932
C2×C8×D1932
#135×C1932
C2×#135193