Finite Group Databasebeta

← #27135769

Subgroups — #27135769

Order 12352.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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#20259881C212yesmaximal, complemented
4#20259879C211yesmaximal, characteristic, complemented
5#20259878C211yesmaximal, characteristic, complemented
6#20259870C211yesmaximal, characteristic, complemented
7#20259867C212yesmaximal, complemented
8D4×Dic193C212yesmaximal, complemented
9#20259859C212yesmaximal, complemented
10#20259857C211yesmaximal, characteristic, complemented
11#121×C193C211yesmaximal, characteristic, complemented
12C23×D193C2211yescharacteristic
13#15690554—24no—
14C2×Dic386C2211yescharacteristic, complemented
15C2×C22⋊D193C2212yes—
16C2×C22⋊D193C2212yes—
17C22×Dic193C2211yescharacteristic
18C22×Dic193C2212yescomplemented
19C2×C4×D193—24no—
20C2×C4×D193C2211yescharacteristic
21C2×C4×D193C2211yescharacteristic
22C2×C4×D193C2211yescharacteristic
23C2×C4×D193C2211yescharacteristic
24C4⋊Dic193C2211yescharacteristic, complemented
25C4⋊Dic193C2212yescomplemented
26#15690534C2212yescomplemented
27C22⋊Dic193C2212yes—
28C22⋊Dic193C2212yescomplemented
29#15690532C2212yes—
30#15690532C2212yes—
31C4×Dic193C2212yescomplemented
32C2×D4×C193C2212yes—
33C2×D4×C193C2211yescharacteristic, complemented
34C22×C4×C193C2211yescharacteristic
35C4⋊C4×C193C2211yescharacteristic, complemented
36C22⋊C4×C193C2212yescomplemented
37C22×D193—22no—
38C22×D193—22no—
39C22×D193—22no—
40C22×D193C2311yescharacteristic
41C22×D193C2311yescharacteristic
42Dic386—22no—
43C22⋊D193—24no—
44C22⋊D193—24no—
45C2×Dic193—22no—
46C2×Dic193—24no—
47C2×Dic193C2312yes—
48C2×Dic193C2312yes—
49C2×Dic193C2311yescharacteristic
50C2×Dic193C2311yescharacteristic
51C4×D193—22no—
52C4×D193—22no—
53C4×D193—22no—
54C4×D193D414yescomplemented
55C23×C193C2312yes—
56C23×C193C2311yescharacteristic
57D4×C193—24no—
58D4×C193—22no—
59C2×C4×C193—22no—
60C2×C4×C193C2311yescharacteristic
61C2×C4×C193C2312yes—
62C2×C4×C193C2311yescharacteristic
63C22×C193—21no—
64C22×C193—21no—
65C22×C193—22no—
66C22×C193—22no—
67C22×C193—22no—
68C22×C193#4112yescomplemented
69C22×C193C2411yescharacteristic
70C2×D193—28no—
71C2×D193—21no—
72C2×D193—21no—
73C2×D193—21no—
74C2×D193—21no—
75C2×D193C2×D412yes—
76C2×D193#4112yescomplemented
77C4×C193—21no—
78C4×C193—22no—
79C4×C193C2×D412yes—
80Dic193—21no—
81Dic193—22no—
82Dic193—22no—
83Dic193C2×D412yes—
84C2×C193—22no—
85C2×C193—42no—
86C2×C193C22×D411yescharacteristic
87C2×C193C2×#4111yescharacteristic
88C2×C193C2×#4111yescharacteristic
89D193—22no—
90D193—22no—
91C193#52511yescharacteristic, complemented, Sylow
92#525—1931nomaximal, Sylow
93C2×#41—1931no—
94C23×C4—1931no—
95#123—1932no—
96C22⋊Q8—1931no—
97C22⋊Q8—1931no—
98#121—1931no—
99#121D19311yescharacteristic, complemented
100#120—1932no—
101C4×D4—1932no—
102C4×D4—1932no—
103#117—1931no—
104C24—1931no—
105#41—3864no—
106C2×Q8—1931no—
107C2×D4—1932no—
108C2×D4—1932no—
109C2×D4C2×D19312yes—
110C2×D4C2×D19311yescharacteristic, complemented
111C22×C4—3864no—
112C22×C4—1931no—
113C22×C4—1932no—
114C22×C4—1931no—
115C22×C4—1931no—
116C22×C4—1931no—
117C22×C4—1931no—
118C22×C4C2×D19311yescharacteristic
119C4⋊C4—1932no—
120C4⋊C4—1931no—
121C4⋊C4—1932no—
122C4⋊C4C2×D19311yescharacteristic, complemented
123C22⋊C4—1932no—
124C22⋊C4—1932no—
125C22⋊C4—1932no—
126C22⋊C4—1932no—
127C22⋊C4C2×D19312yescomplemented
128C42—1932no—
129C23—3862no—
130C23—3862no—
131C23—3862no—
132C23—1931no—
133C23—1931no—
134C23C22×D19312yes—
135C23C22×D19311yescharacteristic
136Q8—3862no—
137D4—24no—
138D4—22no—
139D4—3864no—
140D4—3864no—
141C2×C4—22no—
142C2×C4—3862no—
143C2×C4—3862no—
144C2×C4—3862no—
145C2×C4—3864no—
146C2×C4—3862no—
147C2×C4—1932no—
148C2×C4—1931no—
149C2×C4—1932no—
150C2×C4—1931no—
151C2×C4—1934no—
152C2×C4C22×D19312yes—
153C2×C4C22×D19311yescharacteristic
154C2×C4C22×D19311yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3868no—
161C22—3861no—
162C22—3861no—
163C22—3861no—
164C22—3861no—
165C22—1932no—
166C22—1932no—
167C22#1569055412yescomplemented
168C22C23×D19311yescharacteristic
169C4—21no—
170C4—22no—
171C4—3861no—
172C4—3862no—
173C4—3862no—
174C4—1932no—
175C4D4×D19312yes—
176C2—22no—
177C2—42no—
178C2—3862no—
179C2—3862no—
180C2C2×D4×D19311yescharacteristic
181C2#41×D19311yescharacteristic
182C2C2×#1569055411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#121×C193C2
hypercenter#121D193
hypocenterC193#525
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15690554C2semidirect
2. C22×C4×D193C2semidirect
3. #20259881C2semidirect
4. #20259879C2semidirect
5. #20259878C2semidirect
6. #20259870C2semidirect
7. #20259867C2semidirect
8. D4×Dic193C2semidirect
9. #20259859C2semidirect
10. #20259857C2semidirect
11. #121×C193C2semidirect
14. C2×Dic386C22semidirect
18. C22×Dic193C22semidirect
24. C4⋊Dic193C22semidirect
25. C4⋊Dic193C22semidirect
26. #15690534C22semidirect
28. C22⋊Dic193C22semidirect
31. C4×Dic193C22semidirect
33. C2×D4×C193C22semidirect
35. C4⋊C4×C193C22semidirect
36. C22⋊C4×C193C22semidirect
54. C4×D193D4semidirect
68. C22×C193#41semidirect
76. C2×D193#41semidirect
91. C193#525semidirect
99. #121D193semidirect
110. C2×D4C2×D193semidirect
122. C4⋊C4C2×D193semidirect
127. C22⋊C4C2×D193semidirect
167. C22#15690554semidirect

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
#202598812
#202598792
#202598782
#202598702
#202598672
D4×Dic1932
#202598592
#202598572
#121×C1932
#525193