Finite Group Databasebeta

← #27135825

Subgroups — #27135825

Order 12352.

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

normal subgroups only

This group has 334 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×#15690555C211yesmaximal, characteristic, complemented
2C2×D4×D193C211yesmaximal, characteristic, complemented
3#20259895C212yesmaximal, complemented
4#20259885C212yesmaximal, complemented
5#20259874C211yesmaximal, characteristic, complemented
6#20259873C211yesmaximal, characteristic, complemented
7#20259870C211yesmaximal, characteristic, complemented
8#20259869C212yesmaximal, complemented
9#20259867C212yesmaximal, complemented
10#20259857C211yesmaximal, characteristic, complemented
11#121×C193C211yesmaximal, characteristic, complemented
12C23×D193C2212yes—
13#15690555—24no—
14C2×Dic386C2211yescharacteristic, complemented
15C2×D772C2211yescharacteristic
16C2×D772C2211yescharacteristic
17C2×D772C2212yescomplemented
18C2×C22⋊D193C2211yescharacteristic
19C2×C22⋊D193C2212yes—
20C2×C22⋊D193C2211yescharacteristic, complemented
21C2×C22⋊D193C2212yescomplemented
22C2×C22⋊D193C2212yescomplemented
23D4×D193—24no—
24C2×C4×D193C2211yescharacteristic
25C2×C4×D193C2211yescharacteristic, complemented
26C4⋊Dic193C2211yescharacteristic, complemented
27C22⋊Dic193C2212yescomplemented
28#15690532C2212yes—
29#15690532C2212yescomplemented
30#15690532C2212yescomplemented
31C4×Dic193C2212yescomplemented
32C2×D4×C193C2211yescharacteristic
33C2×D4×C193C2212yescomplemented
34C22×C4×C193C2211yescharacteristic, complemented
35C4⋊C4×C193C2211yescharacteristic, complemented
36C22⋊C4×C193C2212yescomplemented
37C22×D193—24no—
38C22×D193—42no—
39C22×D193—42no—
40C22×D193C2312yes—
41C22×D193C2311yescharacteristic
42C22×D193C2311yescharacteristic
43Dic386—22no—
44D772—22no—
45D772—22no—
46D772—42no—
47C22⋊D193—22no—
48C22⋊D193—24no—
49C22⋊D193—22no—
50C22⋊D193—42no—
51C22⋊D193—42no—
52C2×Dic193C2312yes—
53C2×Dic193C2311yescharacteristic
54C2×Dic193C2311yescharacteristic
55C4×D193—22no—
56C4×D193D414yescomplemented
57C23×C193C2312yes—
58C23×C193C2311yescharacteristic
59D4×C193—22no—
60D4×C193—42no—
61C2×C4×C193—22no—
62C2×C4×C193C2311yescharacteristic
63C2×C4×C193C2312yes—
64C2×C4×C193C2311yescharacteristic
65C22×C193—21no—
66C22×C193—22no—
67C22×C193—21no—
68C22×C193—22no—
69C22×C193—21no—
70C22×C193—22no—
71C22×C193C2411yescharacteristic
72C2×D193—21no—
73C2×D193—21no—
74C2×D193—21no—
75C2×D193—22no—
76C2×D193—21no—
77C2×D193—22no—
78C2×D193—21no—
79C2×D193—22no—
80C2×D193—48no—
81C2×D193C2×D412yes—
82C4×C193—21no—
83C4×C193—22no—
84C4×C193C2×D412yes—
85Dic193—21no—
86Dic193—22no—
87Dic193C2×D412yes—
88C2×C193—42no—
89C2×C193—41no—
90C2×C193C22×D411yescharacteristic
91C2×C193H(5,𝔽2)11yescharacteristic
92C2×C193H(5,𝔽2)11yescharacteristic
93D193—22no—
94D193—41no—
95D193—42no—
96C193#53511yescharacteristic, complemented, Sylow
97#535—1931nomaximal, Sylow
98C2×#41—1931no—
99C22×D4—1931no—
100#127—1932no—
101#124—1932no—
102#121—1931no—
103#121—1931no—
104#121—1931no—
105#121D19311yescharacteristic, complemented
106#120—1932no—
107#120—1932no—
108#117—1931no—
109C24—1932no—
110#41—3864no—
111C2×Q8—1931no—
112C2×D4—3864no—
113C2×D4—1932no—
114C2×D4—1932no—
115C2×D4—1931no—
116C2×D4—1932no—
117C2×D4—1931no—
118C2×D4—1932no—
119C2×D4—1931no—
120C2×D4—1931no—
121C2×D4C2×D19311yescharacteristic
122C2×D4C2×D19312yescomplemented
123C22×C4—1931no—
124C22×C4—1931no—
125C22×C4C2×D19311yescharacteristic, complemented
126C4⋊C4—1931no—
127C4⋊C4C2×D19311yescharacteristic, complemented
128C22⋊C4—1932no—
129C22⋊C4—1932no—
130C22⋊C4—1932no—
131C22⋊C4—1932no—
132C22⋊C4C2×D19312yescomplemented
133C42—1932no—
134C23—7722no—
135C23—7722no—
136C23—3864no—
137C23—1932no—
138C23—1931no—
139C23—1931no—
140C23C22×D19312yes—
141C23C22×D19311yescharacteristic
142Q8—3862no—
143D4—22no—
144D4—42no—
145D4—7722no—
146D4—7722no—
147D4—7722no—
148D4—3862no—
149D4—3862no—
150D4—3864no—
151D4—3862no—
152D4—3862no—
153C2×C4—22no—
154C2×C4—3862no—
155C2×C4—1932no—
156C2×C4—1931no—
157C2×C4—1931no—
158C2×C4—1934no—
159C2×C4C22×D19312yes—
160C2×C4C22×D19311yescharacteristic
161C2×C4C22×D19311yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—7728no—
169C22—3861no—
170C22—3862no—
171C22—3861no—
172C22—3862no—
173C22—3861no—
174C22—3862no—
175C22—3861no—
176C22—3861no—
177C22—1932no—
178C22C23×D19311yescharacteristic
179C4—21no—
180C4—22no—
181C4—3861no—
182C4—3862no—
183C4—1932no—
184C4D4×D19312yes—
185C2—42no—
186C2—41no—
187C2—7722no—
188C2—7721no—
189C2—3862no—
190C2C2×D4×D19311yescharacteristic
191C2#2025994411yescharacteristic
192C2#2025994511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#121×C193C2
hypercenter#121D193
hypocenterC193#535
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15690555C2semidirect
2. C2×D4×D193C2semidirect
3. #20259895C2semidirect
4. #20259885C2semidirect
5. #20259874C2semidirect
6. #20259873C2semidirect
7. #20259870C2semidirect
8. #20259869C2semidirect
9. #20259867C2semidirect
10. #20259857C2semidirect
11. #121×C193C2semidirect
14. C2×Dic386C22semidirect
17. C2×D772C22semidirect
20. C2×C22⋊D193C22semidirect
21. C2×C22⋊D193C22semidirect
22. C2×C22⋊D193C22semidirect
25. C2×C4×D193C22semidirect
26. C4⋊Dic193C22semidirect
27. C22⋊Dic193C22semidirect
29. #15690532C22semidirect
30. #15690532C22semidirect
31. C4×Dic193C22semidirect
33. C2×D4×C193C22semidirect
34. C22×C4×C193C22semidirect
35. C4⋊C4×C193C22semidirect
36. C22⋊C4×C193C22semidirect
56. C4×D193D4semidirect
96. C193#535semidirect
105. #121D193semidirect
122. C2×D4C2×D193semidirect
125. C22×C4C2×D193semidirect
127. C4⋊C4C2×D193semidirect
132. C22⋊C4C2×D193semidirect

Maximal subgroups

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

subgroupindex
C2×#156905552
C2×D4×D1932
#202598952
#202598852
#202598742
#202598732
#202598702
#202598692
#202598672
#202598572
#121×C1932
#535193