Finite Group Databasebeta

← #27135817

Subgroups — #27135817

Order 12352.

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

normal subgroups only

This group has 346 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×D4×D193C212yesmaximal, complemented
2#20259884C212yesmaximal, complemented
3#20259881C212yesmaximal, complemented
4#20259873C211yesmaximal, characteristic, complemented
5#20259871C212yesmaximal, complemented
6#20259869C212yesmaximal, complemented
7#20259867C212yesmaximal, complemented
8C2×#15690532C211yesmaximal, characteristic, complemented
9#121×C193C211yesmaximal, characteristic, complemented
10C23×D193C2211yescharacteristic
11C23×D193C2212yescomplemented
12C2×D772C2212yes—
13C2×D772C2211yescharacteristic, complemented
14C2×C22⋊D193C2212yes—
15C2×C22⋊D193C2212yescomplemented
16C2×C22⋊D193C2212yescomplemented
17D4×D193—28no—
18C22×Dic193C2211yescharacteristic, complemented
19C2×C4×D193C2212yescomplemented
20C4⋊Dic193C2211yescharacteristic, complemented
21#15690534C2212yescomplemented
22C22⋊Dic193C2212yescomplemented
23#15690532C2212yescomplemented
24#15690532C2212yescomplemented
25#15690532C2212yescomplemented
26#15690532C2212yescomplemented
27C2×D4×C193C2212yescomplemented
28C2×D4×C193C2211yescharacteristic, complemented
29C22×C4×C193C2211yescharacteristic, complemented
30C4⋊C4×C193C2211yescharacteristic, complemented
31C22⋊C4×C193C2212yescomplemented
32C22×D193—24no—
33C22×D193—41no—
34C22×D193—41no—
35C22×D193—42no—
36C22×D193—42no—
37C22×D193D414yescomplemented
38C22×D193C2312yes—
39C22×D193C2312yes—
40D772—24no—
41D772—41no—
42C22⋊D193—24no—
43C22⋊D193—24no—
44C22⋊D193—42no—
45C2×Dic193—41no—
46C2×Dic193C2312yes—
47C2×Dic193C2312yes—
48C4×D193—24no—
49C23×C193C2312yes—
50C23×C193C2311yescharacteristic
51D4×C193—24no—
52D4×C193—41no—
53C2×C4×C193—41no—
54C2×C4×C193C2311yescharacteristic
55C2×C4×C193C2312yes—
56C2×C4×C193C2311yescharacteristic
57C22×C193—21no—
58C22×C193—21no—
59C22×C193—22no—
60C22×C193—22no—
61C22×C193—22no—
62C22×C193C2×D412yes—
63C22×C193C2411yescharacteristic
64C2×D193—22no—
65C2×D193—22no—
66C2×D193—22no—
67C2×D193—22no—
68C2×D193—22no—
69C2×D193—44no—
70C2×D193—48no—
71C2×D193C2×D414yes—
72C4×C193—22no—
73C4×C193—21no—
74C4×C193—21no—
75Dic193—22no—
76Dic193—22no—
77C2×C193—22no—
78C2×C193—42no—
79C2×C193C22×D411yescharacteristic
80C2×C193H(5,𝔽2)11yescharacteristic
81C2×C193H(5,𝔽2)11yescharacteristic
82D193—24no—
83D193—42no—
84C193#53411yescharacteristic, complemented, Sylow
85#534—1931nomaximal, Sylow
86C22×D4—1932no—
87#123—1932no—
88#123—1932no—
89#121—1932no—
90#121—1931no—
91#121D19311yescharacteristic, complemented
92#120—1932no—
93#120—1932no—
94C2×C22⋊C4—1931no—
95C24—1932no—
96C24—1931no—
97C2×D4—3868no—
98C2×D4—1932no—
99C2×D4—1932no—
100C2×D4—1932no—
101C2×D4—1931no—
102C2×D4—1932no—
103C2×D4C2×D19312yescomplemented
104C2×D4C2×D19311yescharacteristic, complemented
105C22×C4—1931no—
106C22×C4—1932no—
107C22×C4C2×D19311yescharacteristic, complemented
108C4⋊C4—1932no—
109C4⋊C4—1931no—
110C4⋊C4C2×D19311yescharacteristic, complemented
111C22⋊C4—1932no—
112C22⋊C4—1932no—
113C22⋊C4—1932no—
114C22⋊C4—1932no—
115C22⋊C4—1932no—
116C22⋊C4C2×D19312yescomplemented
117C23—7721no—
118C23—7721no—
119C23—7722no—
120C23—7722no—
121C23—3864no—
122C23—1934no—
123C23—1932no—
124C23—1932no—
125C23C22×D19312yes—
126C23C22×D19311yescharacteristic
127D4—24no—
128D4—41no—
129D4—7721no—
130D4—7722no—
131D4—3864no—
132D4—3864no—
133D4—3864no—
134C2×C4—41no—
135C2×C4—7721no—
136C2×C4—3864no—
137C2×C4—1932no—
138C2×C4—1932no—
139C2×C4C22×D19311yescharacteristic
140C2×C4C22×D19311yescharacteristic
141C2×C4C22×D19312yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—7724no—
148C22—7728no—
149C22—3862no—
150C22—3862no—
151C22—3862no—
152C22—3862no—
153C22—3862no—
154C22—1934no—
155C22D4×D19312yes—
156C22C23×D19311yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—3862no—
161C4—3862no—
162C2—22no—
163C2—42no—
164C2—7722no—
165C2—3864no—
166C2C2×D4×D19311yescharacteristic
167C2#2025994411yescharacteristic
168C2#2025994511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#121×C193C2
hypercenter#121D193
hypocenterC193#534
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D193C2semidirect
2. #20259884C2semidirect
3. #20259881C2semidirect
4. #20259873C2semidirect
5. #20259871C2semidirect
6. #20259869C2semidirect
7. #20259867C2semidirect
8. C2×#15690532C2semidirect
9. #121×C193C2semidirect
11. C23×D193C22semidirect
13. C2×D772C22semidirect
15. C2×C22⋊D193C22semidirect
16. C2×C22⋊D193C22semidirect
18. C22×Dic193C22semidirect
19. C2×C4×D193C22semidirect
20. C4⋊Dic193C22semidirect
21. #15690534C22semidirect
22. C22⋊Dic193C22semidirect
23. #15690532C22semidirect
24. #15690532C22semidirect
25. #15690532C22semidirect
26. #15690532C22semidirect
27. C2×D4×C193C22semidirect
28. C2×D4×C193C22semidirect
29. C22×C4×C193C22semidirect
30. C4⋊C4×C193C22semidirect
31. C22⋊C4×C193C22semidirect
37. C22×D193D4semidirect
84. C193#534semidirect
91. #121D193semidirect
103. C2×D4C2×D193semidirect
104. C2×D4C2×D193semidirect
107. C22×C4C2×D193semidirect
110. C4⋊C4C2×D193semidirect
116. C22⋊C4C2×D193semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D1932
#202598842
#202598812
#202598732
#202598712
#202598692
#202598672
C2×#156905322
#121×C1932
#534193