Finite Group Databasebeta

← #27136066

Subgroups — #27136066

Order 12352.

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

normal subgroups only

This group has 258 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#20259946C211yesmaximal, characteristic, complemented
2#20259945C211yesmaximal, characteristic, complemented
3#20259931C212yesmaximal, complemented
4#20259929C211yesmaximal, characteristic, complemented
5#20259924C211yesmaximal, characteristic, complemented
6#20259923C212yesmaximal, complemented
7#20259920C212yesmaximal, complemented
8C2×#15690544C211yesmaximal, characteristic, complemented
9SD16×D193C212yesmaximal, complemented
10#20259905C211yesmaximal, characteristic, complemented
11#135×C193C211yesmaximal, characteristic, complemented
12#15690556C2212yes—
13#15690555—21no—
14#15690555—21no—
15#15690555C2211yescharacteristic
16#15690554—22no—
17#15690554C2212yescomplemented
18C2×Dic386—21no—
19C2×Dic386C2211yescharacteristic, complemented
20Q8×D193C2212yescomplemented
21C2×D772—21no—
22C2×D772C2211yescharacteristic
23D4×D193—22no—
24D4×D193C2212yes—
25#15690545C2212yescomplemented
26#15690544C2212yescomplemented
27#15690544C2212yescomplemented
28#15690543C2212yescomplemented
29#15690542C2212yescomplemented
30#15690540C2212yescomplemented
31#15690539C2211yescharacteristic, complemented
32#15690538C2212yescomplemented
33C8×D193C2212yescomplemented
34#41×C193C2211yescharacteristic
35#41×C193C2211yescharacteristic, complemented
36Q16×C193C2211yescharacteristic, complemented
37SD16×C193C2211yescharacteristic, complemented
38SD16×C193C2211yescharacteristic, complemented
39D8×C193C2211yescharacteristic, complemented
40C2×C8×C193C2211yescharacteristic, complemented
41C22×D193—21no—
42C22×D193—22no—
43Dic386—21no—
44Dic386—22no—
45Dic386D411yescharacteristic
46Dic386C2312yes—
47D772—22no—
48D772—21no—
49D772D411yescharacteristic
50D772C2312yes—
51C22⋊D193—22no—
52C22⋊D193—22no—
53C22⋊D193D412yes—
54C2×Dic193—21no—
55C2×Dic193—22no—
56C4×D193—22no—
57C4×D193—22no—
58C4×D193C2312yes—
59#13095393C2312yes—
60Q8×C193C2311yescharacteristic
61Q8×C193C2311yescharacteristic
62D4×C193—21no—
63D4×C193—21no—
64D4×C193C2311yescharacteristic
65D4×C193C2311yescharacteristic
66C2×C4×C193—21no—
67C2×C4×C193—21no—
68C2×C4×C193C2311yescharacteristic
69C8×C193C2311yescharacteristic
70C8×C193C2311yescharacteristic
71C22×C193—21no—
72C22×C193—21no—
73C22×C193C2×D411yescharacteristic
74C2×D193—22no—
75C2×D193—82no—
76C2×D193—81no—
77C2×D193C2×D412yes—
78C4×C193—21no—
79C4×C193—21no—
80C4×C193C2×D411yescharacteristic
81C4×C193C2411yescharacteristic
82Dic193—22no—
83Dic193C2×D412yes—
84C2×C193—21no—
85C2×C193—41no—
86C2×C193—41no—
87C2×C193C22×D411yescharacteristic
88D193—22no—
89D193—42no—
90C193#57711yescharacteristic, complemented, Sylow
91#577—1931nomaximal, Sylow
922−1+4—1931no—
93H(5,𝔽2)—1931no—
94#137—1931no—
95#137—1932no—
96AGL(1,ℤ/8)—1931no—
97AGL(1,ℤ/8)—1932no—
98#135—1932no—
99#135D19311yescharacteristic, complemented
100C2×SD16—1932no—
101C2×SD16—1931no—
102#131—1931no—
103#41—3861no—
104#41—3861no—
105#41—3862no—
106#41—1932no—
107#41—1932no—
108#41—1931no—
109#41C2×D19311yescharacteristic
110#41C2×D19311yescharacteristic, complemented
111C2×Q8—3861no—
112C2×Q8—1932no—
113C2×Q8—1931no—
114C2×D4—3861no—
115C2×D4—3862no—
116C2×D4—1932no—
117C2×D4—1931no—
118Q16—1932no—
119Q16C2×D19311yescharacteristic, complemented
120SD16—1932no—
121SD16—1932no—
122SD16—1932no—
123SD16—1932no—
124SD16C2×D19311yescharacteristic, complemented
125SD16C2×D19311yescharacteristic, complemented
126D8—1932no—
127D8C2×D19311yescharacteristic, complemented
128M16—1931no—
129M16—1932no—
130C2×C8—1932no—
131C2×C8C2×D19311yescharacteristic, complemented
132C23—3862no—
133C23—3861no—
134Q8—3862no—
135Q8—3861no—
136Q8—1932no—
137Q8—1931no—
138Q8C22×D19311yescharacteristic
139Q8C22×D19311yescharacteristic
140D4—21no—
141D4—21no—
142D4—3862no—
143D4—3862no—
144D4—3862no—
145D4—3861no—
146D4—1932no—
147D4—1931no—
148D4—1932no—
149D4C22×D19311yescharacteristic
150D4C22×D19311yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3862no—
154C2×C4—3862no—
155C2×C4—3861no—
156C2×C4—3862no—
157C2×C4—1932no—
158C2×C4C22×D19311yescharacteristic
159C8—1932no—
160C8C22×D19311yescharacteristic
161C8C22×D19311yescharacteristic
162C22—21no—
163C22—21no—
164C22—15441no—
165C22—15442no—
166C22—3862no—
167C22—1932no—
168C22D4×D19311yescharacteristic
169C4—21no—
170C4—21no—
171C4—3862no—
172C4—1932no—
173C4D4×D19311yescharacteristic
174C4C23×D19311yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—7722no—
179C2—3862no—
180C2C2×D4×D19311yescharacteristic
subgroupG/H
centerC2C2×D4×D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
Fitting#135×C193C2
socleC2×C193C22×D4
hypercenter#135D193
hypocenterC193#577
LCSG ▷ C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #20259946C2semidirect
2. #20259945C2semidirect
3. #20259931C2semidirect
4. #20259929C2semidirect
5. #20259924C2semidirect
6. #20259923C2semidirect
7. #20259920C2semidirect
8. C2×#15690544C2semidirect
9. SD16×D193C2semidirect
10. #20259905C2semidirect
11. #135×C193C2semidirect
17. #15690554C22semidirect
19. C2×Dic386C22semidirect
20. Q8×D193C22semidirect
25. #15690545C22semidirect
26. #15690544C22semidirect
27. #15690544C22semidirect
28. #15690543C22semidirect
29. #15690542C22semidirect
30. #15690540C22semidirect
31. #15690539C22semidirect
32. #15690538C22semidirect
33. C8×D193C22semidirect
35. #41×C193C22semidirect
36. Q16×C193C22semidirect
37. SD16×C193C22semidirect
38. SD16×C193C22semidirect
39. D8×C193C22semidirect
40. C2×C8×C193C22semidirect
90. C193#577semidirect
99. #135D193semidirect
110. #41C2×D193semidirect
119. Q16C2×D193semidirect
124. SD16C2×D193semidirect
125. SD16C2×D193semidirect
127. D8C2×D193semidirect
131. C2×C8C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599462
#202599452
#202599312
#202599292
#202599242
#202599232
#202599202
C2×#156905442
SD16×D1932
#202599052
#135×C1932
#577193