Finite Group Databasebeta

← #27136058

Subgroups — #27136058

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
1C2×#15690556C211yesmaximal, characteristic, complemented
2#41×D193C211yesmaximal, characteristic, complemented
3#20259931C212yesmaximal, complemented
4#20259930C211yesmaximal, characteristic, complemented
5#20259927C211yesmaximal, characteristic, complemented
6#20259926C212yesmaximal, complemented
7#20259924C211yesmaximal, characteristic, complemented
8#20259922C212yesmaximal, complemented
9#20259920C212yesmaximal, complemented
10M16×D193C211yesmaximal, characteristic, complemented
11#137×C193C211yesmaximal, characteristic, complemented
12#15690556—21no—
13#15690556—21no—
14#15690556C2212yes—
15#15690556C2212yescomplemented
16#15690556C2211yescharacteristic, complemented
17#15690555—21no—
18#15690555C2211yescharacteristic
19#15690554—21no—
20#15690554C2211yescharacteristic, complemented
21Q8×D193C2211yescharacteristic, complemented
22C2×D772—21no—
23C2×D772C2211yescharacteristic, complemented
24D4×D193—21no—
25D4×D193C2211yescharacteristic, complemented
26C2×C4×D193—21no—
27C2×C4×D193—21no—
28C2×C4×D193C2211yescharacteristic
29#15690545C2212yescomplemented
30#15690544C2212yescomplemented
31#15690542C2212yescomplemented
32#15690542C2212yescomplemented
33D1544C2212yescomplemented
34#15690540C2212yescomplemented
35#15690539C2211yescharacteristic, complemented
36#15690538C2212yescomplemented
37C8×D193C2212yescomplemented
38#41×C193C2211yescharacteristic
39C2×Q8×C193C2211yescharacteristic, complemented
40Q16×C193C2212yescomplemented
41SD16×C193C2212yescomplemented
42M16×C193C2211yescharacteristic, complemented
43C22×D193—21no—
44C22×D193—21no—
45C22×D193D411yescharacteristic
46Dic386—21no—
47Dic386C2311yescharacteristic
48D772—21no—
49D772—22no—
50D772—22no—
51D772—21no—
52D772C2312yes—
53D772C2311yescharacteristic
54C22⋊D193—21no—
55C22⋊D193—21no—
56C22⋊D193—21no—
57C2×Dic193—21no—
58C2×Dic193D411yescharacteristic
59C4×D193—21no—
60C4×D193—21no—
61C4×D193—21no—
62C4×D193—22no—
63C4×D193—22no—
64C4×D193D411yescharacteristic
65C4×D193D411yescharacteristic
66C4×D193C2311yescharacteristic
67C4×D193C2311yescharacteristic
68#13095393C2312yes—
69Q8×C193—21no—
70Q8×C193C2312yes—
71Q8×C193C2311yescharacteristic
72D4×C193—21no—
73D4×C193C2311yescharacteristic
74C2×C4×C193—21no—
75C2×C4×C193—21no—
76C2×C4×C193C2311yescharacteristic
77C8×C193C2312yes—
78C22×C193—21no—
79C22×C193C2×D411yescharacteristic
80C2×D193—21no—
81C2×D193—22no—
82C2×D193—41no—
83C2×D193—42no—
84C2×D193—42no—
85C2×D193C2×D411yescharacteristic
86C2×D193C2×D411yescharacteristic
87C4×C193—21no—
88C4×C193—22no—
89C4×C193C2×D411yescharacteristic
90C4×C193C2411yescharacteristic
91Dic193—21no—
92Dic193C2×D411yescharacteristic
93Dic193C2×D411yescharacteristic
94C2×C193—21no—
95C2×C193—41no—
96C2×C193C22×D411yescharacteristic
97D193—21no—
98D193—21no—
99D193—41no—
100D193—42no—
101C193#57511yescharacteristic, complemented, Sylow
102#575—1931nomaximal, Sylow
103C2×#41—1931no—
104C2×#41—1931no—
105#137—1931no—
106#137—1932no—
107#137D19311yescharacteristic, complemented
108AGL(1,ℤ/8)—1931no—
109AGL(1,ℤ/8)—1932no—
110AGL(1,ℤ/8)—1931no—
111#135—1932no—
112#135—1932no—
113C2×M16—1931no—
114#41—3861no—
115#41—3861no—
116#41—3861no—
117#41—3861no—
118#41—1932no—
119#41—1931no—
120#41—1932no—
121#41—1931no—
122#41—1931no—
123#41C2×D19311yescharacteristic
124C2×Q8—1931no—
125C2×Q8C2×D19311yescharacteristic, complemented
126C2×D4—3861no—
127C2×D4—3861no—
128C2×D4—1931no—
129C2×D4—1931no—
130C22×C4—3861no—
131C22×C4—3861no—
132C22×C4—1931no—
133Q16—1932no—
134Q16C2×D19312yescomplemented
135SD16—1932no—
136SD16—1932no—
137SD16—1932no—
138SD16C2×D19312yescomplemented
139D8—1932no—
140D8—1932no—
141M16—1931no—
142M16—1932no—
143M16C2×D19311yescharacteristic, complemented
144C2×C8—1932no—
145C23—3861no—
146C23—3861no—
147C23—1931no—
148Q8—21no—
149Q8—3861no—
150Q8—1931no—
151Q8C22×D19311yescharacteristic
152Q8C22×D19312yes—
153D4—21no—
154D4—3861no—
155D4—3862no—
156D4—3862no—
157D4—3861no—
158D4—3861no—
159D4—3861no—
160D4—3861no—
161D4—1932no—
162D4—1931no—
163D4C22×D19311yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3861no—
167C2×C4—3861no—
168C2×C4—3862no—
169C2×C4—3862no—
170C2×C4—3861no—
171C2×C4—3861no—
172C2×C4—1931no—
173C2×C4—1931no—
174C2×C4—1931no—
175C2×C4—1931no—
176C2×C4—1931no—
177C2×C4C22×D19311yescharacteristic
178C8—1932no—
179C8C22×D19312yes—
180C22—21no—
181C22—7722no—
182C22—7721no—
183C22—7722no—
184C22—3862no—
185C22—3861no—
186C22—1931no—
187C22—1931no—
188C22D4×D19311yescharacteristic
189C4—21no—
190C4—22no—
191C4—3861no—
192C4—1931no—
193C4—1931no—
194C4D4×D19311yescharacteristic
195C4C23×D19311yescharacteristic
196C2—21no—
197C2—41no—
198C2—7722no—
199C2—7721no—
200C2—3861no—
201C2—3861no—
202C2C2×D4×D19311yescharacteristic
subgroupG/H
centerC2C2×D4×D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
Fitting#137×C193C2
socleC2×C193C22×D4
hypercenter#137D193
hypocenterC193#575
LCSG ▷ C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#15690556C2semidirect
2. #41×D193C2semidirect
3. #20259931C2semidirect
4. #20259930C2semidirect
5. #20259927C2semidirect
6. #20259926C2semidirect
7. #20259924C2semidirect
8. #20259922C2semidirect
9. #20259920C2semidirect
10. M16×D193C2semidirect
11. #137×C193C2semidirect
15. #15690556C22semidirect
16. #15690556C22semidirect
20. #15690554C22semidirect
21. Q8×D193C22semidirect
23. C2×D772C22semidirect
25. D4×D193C22semidirect
29. #15690545C22semidirect
30. #15690544C22semidirect
31. #15690542C22semidirect
32. #15690542C22semidirect
33. D1544C22semidirect
34. #15690540C22semidirect
35. #15690539C22semidirect
36. #15690538C22semidirect
37. C8×D193C22semidirect
39. C2×Q8×C193C22semidirect
40. Q16×C193C22semidirect
41. SD16×C193C22semidirect
42. M16×C193C22semidirect
101. C193#575semidirect
107. #137D193semidirect
125. C2×Q8C2×D193semidirect
134. Q16C2×D193semidirect
138. SD16C2×D193semidirect
143. M16C2×D193semidirect

Maximal subgroups

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

subgroupindex
C2×#156905562
#41×D1932
#202599312
#202599302
#202599272
#202599262
#202599242
#202599222
#202599202
M16×D1932
#137×C1932
#575193