Finite Group Databasebeta

← #27135767

Subgroups — #27135767

Order 12352.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4×D193C211yesmaximal, characteristic, complemented
3#20259882C211yesmaximal, characteristic, complemented
4#20259877C212yesmaximal, complemented
5#20259872C212yesmaximal, complemented
6#20259869C211yesmaximal, characteristic, complemented
7#20259867C211yesmaximal, characteristic, complemented
8#20259861C212yesmaximal, complemented
9#20259859C212yesmaximal, complemented
10#20259855C211yesmaximal, characteristic, complemented
11#123×C193C211yesmaximal, characteristic, complemented
12C23×D193C2211yescharacteristic
13#15690554—24no—
14C2×Dic386C2211yescharacteristic, complemented
15C2×D772C2212yes—
16C2×C22⋊D193C2212yes—
17C2×C22⋊D193C2211yescharacteristic
18C2×C22⋊D193C2211yescharacteristic, complemented
19C22×Dic193C2211yescharacteristic
20C22×Dic193C2211yescharacteristic, complemented
21C2×C4×D193—24no—
22C2×C4×D193C2212yes—
23C2×C4×D193C2212yes—
24C2×C4×D193C2211yescharacteristic, complemented
25#15690534C2212yescomplemented
26#15690534C2212yescomplemented
27C22⋊Dic193C2211yescharacteristic
28#15690532C2212yes—
29#15690532C2212yes—
30#15690532C2212yescomplemented
31C4×Dic193C2212yescomplemented
32C2×D4×C193C2211yescharacteristic
33C22×C4×C193C2211yescharacteristic
34C4⋊C4×C193C2212yescomplemented
35C22⋊C4×C193C2211yescharacteristic
36C22⋊C4×C193C2212yescomplemented
37C22×D193—22no—
38C22×D193—24no—
39C22×D193C2311yescharacteristic
40C22×D193C2312yes—
41Dic386—22no—
42D772—24no—
43C22⋊D193—22no—
44C22⋊D193—24no—
45C22⋊D193—22no—
46C2×Dic193—22no—
47C2×Dic193D414yescomplemented
48C2×Dic193C2312yes—
49C2×Dic193C2311yescharacteristic
50C2×Dic193C2311yescharacteristic
51C2×Dic193C2311yescharacteristic
52C4×D193—24no—
53C4×D193—24no—
54C4×D193—22no—
55C23×C193C2311yescharacteristic
56C23×C193C2311yescharacteristic
57D4×C193—22no—
58C2×C4×C193—22no—
59C2×C4×C193C2312yes—
60C2×C4×C193C2312yes—
61C2×C4×C193C2311yescharacteristic
62C22×C193—22no—
63C22×C193—21no—
64C22×C193—22no—
65C22×C193C2×D412yes—
66C22×C193C2411yescharacteristic
67C2×D193—28no—
68C2×D193—22no—
69C2×D193—22no—
70C2×D193—21no—
71C2×D193—22no—
72C2×D193#4114yescomplemented
73C4×C193—22no—
74C4×C193—21no—
75C4×C193—22no—
76Dic193—21no—
77Dic193—22no—
78Dic193C2×D412yes—
79Dic193C2×D412yes—
80C2×C193—22no—
81C2×C193—41no—
82C2×C193C22×D411yescharacteristic
83C2×C193C2×#4112yes—
84D193—24no—
85D193—41no—
86C193#52511yescharacteristic, complemented, Sylow
87#525—1931nomaximal, Sylow
88C2×#41—1931no—
89C23×C4—1931no—
90#123—1931no—
91#123D19311yescharacteristic, complemented
92C22⋊Q8—1932no—
93#121—1932no—
94#120—1931no—
95#120—1931no—
96C4×D4—1932no—
97C4×D4—1932no—
98#117—1931no—
99C24—1931no—
100#41—3864no—
101C2×Q8—1931no—
102C2×D4—1932no—
103C2×D4—1931no—
104C2×D4—1931no—
105C2×D4—1932no—
106C2×D4C2×D19311yescharacteristic
107C22×C4—3864no—
108C22×C4—1931no—
109C22×C4—1932no—
110C22×C4—1932no—
111C22×C4—1931no—
112C22×C4—1931no—
113C22×C4C2×D19311yescharacteristic
114C4⋊C4—1932no—
115C4⋊C4—1932no—
116C4⋊C4C2×D19312yescomplemented
117C22⋊C4—1931no—
118C22⋊C4—1932no—
119C22⋊C4—1932no—
120C22⋊C4—1932no—
121C22⋊C4C2×D19311yescharacteristic
122C22⋊C4C2×D19312yescomplemented
123C42—1932no—
124C23—3864no—
125C23—3862no—
126C23—1931no—
127C23—1932no—
128C23C22×D19311yescharacteristic
129C23C22×D19311yescharacteristic
130Q8—3862no—
131D4—22no—
132D4—3864no—
133D4—3864no—
134D4—3862no—
135D4—3862no—
136C2×C4—22no—
137C2×C4—3864no—
138C2×C4—3864no—
139C2×C4—3862no—
140C2×C4—3862no—
141C2×C4—1932no—
142C2×C4—1931no—
143C2×C4—1934no—
144C2×C4—1931no—
145C2×C4—1931no—
146C2×C4C22×D19312yes—
147C2×C4C22×D19311yescharacteristic
148C2×C4C22×D19312yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—3868no—
153C22—3861no—
154C22—3862no—
155C22—3862no—
156C22—3862no—
157C22—1934no—
158C22D4×D19312yes—
159C22C23×D19311yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—3861no—
164C4—3862no—
165C4—1932no—
166C4—1932no—
167C2—22no—
168C2—41no—
169C2—7721no—
170C2—3864no—
171C2C2×D4×D19311yescharacteristic
172C2#41×D19312yes—
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#123×C193C2
hypercenter#123D193
hypocenterC193#525
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#15690554C2semidirect
2. C22×C4×D193C2semidirect
3. #20259882C2semidirect
4. #20259877C2semidirect
5. #20259872C2semidirect
6. #20259869C2semidirect
7. #20259867C2semidirect
8. #20259861C2semidirect
9. #20259859C2semidirect
10. #20259855C2semidirect
11. #123×C193C2semidirect
14. C2×Dic386C22semidirect
18. C2×C22⋊D193C22semidirect
20. C22×Dic193C22semidirect
24. C2×C4×D193C22semidirect
25. #15690534C22semidirect
26. #15690534C22semidirect
30. #15690532C22semidirect
31. C4×Dic193C22semidirect
34. C4⋊C4×C193C22semidirect
36. C22⋊C4×C193C22semidirect
47. C2×Dic193D4semidirect
72. C2×D193#41semidirect
86. C193#525semidirect
91. #123D193semidirect
116. C4⋊C4C2×D193semidirect
122. 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×C4×D1932
#202598822
#202598772
#202598722
#202598692
#202598672
#202598612
#202598592
#202598552
#123×C1932
#525193