Finite Group Databasebeta

← #27135824

Subgroups — #27135824

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×#15690554C211yesmaximal, characteristic, complemented
2C22×C22⋊D193C211yesmaximal, characteristic, complemented
3#20259895C212yesmaximal, complemented
4#20259885C212yesmaximal, complemented
5#20259872C212yesmaximal, complemented
6#20259871C212yesmaximal, complemented
7#20259869C211yesmaximal, characteristic, complemented
8#20259868C211yesmaximal, characteristic, complemented
9#20259867C211yesmaximal, characteristic, complemented
10#20259855C211yesmaximal, characteristic, complemented
11#120×C193C211yesmaximal, characteristic, complemented
12C23×D193C2211yescharacteristic
13#15690554—24no—
14C2×Dic386C2211yescharacteristic, complemented
15C2×D772C2212yescomplemented
16C2×C22⋊D193—24no—
17C2×C22⋊D193C2211yescharacteristic
18C2×C22⋊D193C2212yes—
19C2×C22⋊D193C2212yes—
20C2×C22⋊D193C2211yescharacteristic, complemented
21C2×C22⋊D193C2212yescomplemented
22C2×C22⋊D193C2212yescomplemented
23C22×Dic193C2211yescharacteristic
24C22×Dic193C2211yescharacteristic, complemented
25C2×C4×D193C2211yescharacteristic, complemented
26#15690534C2212yescomplemented
27C22⋊Dic193C2211yescharacteristic
28C22⋊Dic193C2212yescomplemented
29#15690532C2212yescomplemented
30#15690532C2212yescomplemented
31C4×Dic193C2212yescomplemented
32C24×C193C2211yescharacteristic
33C2×D4×C193C2211yescharacteristic
34C2×D4×C193C2212yescomplemented
35C22⋊C4×C193C2211yescharacteristic
36C22⋊C4×C193C2212yescomplemented
37C22×D193—22no—
38C22×D193—42no—
39C22×D193C2312yes—
40C22×D193C2311yescharacteristic
41Dic386—22no—
42D772—42no—
43C22⋊D193—22no—
44C22⋊D193—24no—
45C22⋊D193—24no—
46C22⋊D193—22no—
47C22⋊D193—42no—
48C22⋊D193—42no—
49C2×Dic193—22no—
50C2×Dic193D414yescomplemented
51C2×Dic193C2312yes—
52C2×Dic193C2311yescharacteristic
53C2×Dic193C2311yescharacteristic
54C2×Dic193C2311yescharacteristic
55C4×D193—22no—
56C23×C193—22no—
57C23×C193—42no—
58C23×C193C2312yes—
59C23×C193C2311yescharacteristic
60C23×C193C2311yescharacteristic
61D4×C193—22no—
62D4×C193—42no—
63C2×C4×C193C2312yes—
64C2×C4×C193C2311yescharacteristic
65C22×C193—22no—
66C22×C193—21no—
67C22×C193—22no—
68C22×C193—22no—
69C22×C193—22no—
70C22×C193—22no—
71C22×C193—44no—
72C22×C193C2×D412yes—
73C22×C193C2411yescharacteristic
74C2×D193—21no—
75C2×D193—22no—
76C2×D193—22no—
77C2×D193—22no—
78C2×D193—22no—
79C2×D193—44no—
80C4×C193—21no—
81C4×C193—22no—
82Dic193—21no—
83Dic193—22no—
84Dic193C2×D412yes—
85Dic193C2×D412yes—
86C2×C193—22no—
87C2×C193—41no—
88C2×C193—42no—
89C2×C193C22×D411yescharacteristic
90C2×C193H(5,𝔽2)12yes—
91D193—41no—
92D193—42no—
93C193#53511yescharacteristic, complemented, Sylow
94#535—1931nomaximal, Sylow
95C2×#41—1931no—
96C22×D4—1931no—
97#127—1932no—
98#124—1932no—
99#121—1932no—
100#121—1932no—
101#120—1931no—
102#120—1931no—
103#120—1931no—
104#120D19311yescharacteristic, complemented
105#117—1931no—
106C24—1931no—
107C24C2×D19311yescharacteristic
108#41—3864no—
109C2×Q8—1931no—
110C2×D4—3864no—
111C2×D4—1932no—
112C2×D4—1932no—
113C2×D4—1931no—
114C2×D4—1932no—
115C2×D4—1932no—
116C2×D4—1931no—
117C2×D4—1932no—
118C2×D4C2×D19311yescharacteristic
119C2×D4C2×D19312yescomplemented
120C22×C4—1931no—
121C22×C4—1931no—
122C22×C4—1931no—
123C4⋊C4—1932no—
124C22⋊C4—1931no—
125C22⋊C4—1932no—
126C22⋊C4—1932no—
127C22⋊C4—1932no—
128C22⋊C4C2×D19311yescharacteristic
129C22⋊C4C2×D19312yescomplemented
130C42—1932no—
131C23—22no—
132C23—42no—
133C23—7722no—
134C23—3862no—
135C23—1932no—
136C23—1931no—
137C23C22×D19312yes—
138C23C22×D19311yescharacteristic
139C23C22×D19311yescharacteristic
140Q8—3862no—
141D4—22no—
142D4—42no—
143D4—7722no—
144D4—7722no—
145D4—7722no—
146D4—3862no—
147D4—3864no—
148D4—3864no—
149D4—3862no—
150C2×C4—3862no—
151C2×C4—3862no—
152C2×C4—1932no—
153C2×C4—1931no—
154C2×C4—1934no—
155C2×C4—1931no—
156C2×C4—1931no—
157C2×C4C22×D19312yes—
158C2×C4C22×D19311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—7724no—
167C22—3861no—
168C22—3862no—
169C22—3862no—
170C22—3862no—
171C22—3862no—
172C22D4×D19312yes—
173C22C23×D19311yescharacteristic
174C4—21no—
175C4—22no—
176C4—3861no—
177C4—3862no—
178C4—1932no—
179C4—1932no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—7722no—
184C2—7721no—
185C2C2×D4×D19311yescharacteristic
186C2#2025994412yes—
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#120×C193C2
hypercenter#120D193
hypocenterC193#535
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#15690554C2semidirect
2. C22×C22⋊D193C2semidirect
3. #20259895C2semidirect
4. #20259885C2semidirect
5. #20259872C2semidirect
6. #20259871C2semidirect
7. #20259869C2semidirect
8. #20259868C2semidirect
9. #20259867C2semidirect
10. #20259855C2semidirect
11. #120×C193C2semidirect
14. C2×Dic386C22semidirect
15. C2×D772C22semidirect
20. C2×C22⋊D193C22semidirect
21. C2×C22⋊D193C22semidirect
22. C2×C22⋊D193C22semidirect
24. C22×Dic193C22semidirect
25. C2×C4×D193C22semidirect
26. #15690534C22semidirect
28. C22⋊Dic193C22semidirect
29. #15690532C22semidirect
30. #15690532C22semidirect
31. C4×Dic193C22semidirect
34. C2×D4×C193C22semidirect
36. C22⋊C4×C193C22semidirect
50. C2×Dic193D4semidirect
93. C193#535semidirect
104. #120D193semidirect
119. C2×D4C2×D193semidirect
129. C22⋊C4C2×D193semidirect

Maximal subgroups

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

subgroupindex
C2×#156905542
C22×C22⋊D1932
#202598952
#202598852
#202598722
#202598712
#202598692
#202598682
#202598672
#202598552
#120×C1932
#535193