Finite Group Databasebeta

← #27135438

Subgroups — #27135438

Order 12352.

152 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#20259945C211yesmaximal, characteristic
2#20259932C211yesmaximal, characteristic, complemented
3#20259929C211yesmaximal, characteristic, complemented
4#20259887C211yesmaximal, characteristic, complemented
5#20259827C211yesmaximal, characteristic, complemented
6#20259815C211yesmaximal, characteristic, complemented
7#104×C193C211yesmaximal, characteristic, complemented
8#15690556—21no—
9#15690555—21no—
10#15690555C2211yescharacteristic
11C2×Dic386C2211yescharacteristic, complemented
12C2×D772—21no—
13C2×D772C2211yescharacteristic
14D4×D193—22no—
15D4×D193—21no—
16Dic772—21no—
17#15690545—21no—
18#15690544—21no—
19#15690543—21no—
20#15690539C2211yescharacteristic, complemented
21#15690532—22no—
22#41×C193C2211yescharacteristic
23M16×C193C2211yescharacteristic, complemented
24C42×C193C2211yescharacteristic, complemented
25C22×D193—22no—
26C22×D193D412yes—
27Dic386—21no—
28Dic386—21no—
29Dic386D411yescharacteristic
30D772—22no—
31D772—21no—
32D772—21no—
33D772D411yescharacteristic
34C22⋊D193—22no—
35C22⋊D193—21no—
36C2×Dic193—21no—
37C4×D193—22no—
38C4×D193—21no—
39#13095393—21no—
40Q8×C193D411yescharacteristic
41D4×C193—21no—
42D4×C193D411yescharacteristic
43C2×C4×C193—21no—
44C2×C4×C193—21no—
45C2×C4×C193C2311yescharacteristic
46C8×C193—21no—
47C22×C193—21no—
48C22×C193C2×D411yescharacteristic
49C2×D193—22no—
50C2×D193—21no—
51C2×D193—42no—
52C2×D193—82no—
53C4×C193—21no—
54C4×C193—41no—
55C4×C193C2×D411yescharacteristic
56C4×C193C2×D411yescharacteristic
57Dic193—21no—
58Dic193—41no—
59C2×C193—21no—
60C2×C193—41no—
61C2×C193#12011yescharacteristic
62D193—42no—
63D193—41no—
64C193#45511yescharacteristic, complemented, Sylow
65#455—1931nomaximal, Sylow
66H(5,𝔽2)—1931no—
67#137—1931no—
68#137—1931no—
69#124—1931no—
70#104—1931no—
71#104D19311yescharacteristic, complemented
72#100—1931no—
73#41—3861no—
74#41—3861no—
75#41—1931no—
76#41C2×D19311yescharacteristic
77C2×Q8—1931no—
78C2×D4—3861no—
79C2×D4—3862no—
80C2×D4—3861no—
81C2×D4—1931no—
82Q16—3861no—
83Q16—3861no—
84SD16—3861no—
85SD16—3861no—
86M16—1931no—
87M16C2×D19311yescharacteristic, complemented
88C22⋊C4—3862no—
89C42C2×D19311yescharacteristic, complemented
90C23—3862no—
91C23—1932no—
92Q8—3861no—
93Q8—3861no—
94Q8—1931no—
95Q8D77211yescharacteristic
96D4—21no—
97D4—3862no—
98D4—3862no—
99D4—3861no—
100D4—3861no—
101D4—3861no—
102D4—1931no—
103D4D77211yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—3862no—
107C2×C4—3861no—
108C2×C4—3861no—
109C2×C4C22×D19311yescharacteristic
110C8—21no—
111C8—3861no—
112C22—21no—
113C22—7722no—
114C22—15442no—
115C22—3862no—
116C22—3861no—
117C22D4×D19311yescharacteristic
118C4—21no—
119C4—41no—
120C4—7721no—
121C4—3861no—
122C4D4×D19311yescharacteristic
123C4C2×D77211yescharacteristic
124C2—21no—
125C2—41no—
126C2—7722no—
127C2—7721no—
128C2#2025986911yescharacteristic
subgroupG/H
centerC2#20259869
commutator, cosocleC2×C4×C193C23
FrattiniC2×C4C22×D193
Fitting#104×C193C2
socleC2×C193#120
prime radical#20259945C2
hypercenter#104D193
hypocenterC193#455
LCSG ▷ C2×C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #20259932C2semidirect
3. #20259929C2semidirect
4. #20259887C2semidirect
5. #20259827C2semidirect
6. #20259815C2semidirect
7. #104×C193C2semidirect
11. C2×Dic386C22semidirect
20. #15690539C22semidirect
23. M16×C193C22semidirect
24. C42×C193C22semidirect
64. C193#455semidirect
71. #104D193semidirect
87. M16C2×D193semidirect
89. C42C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599452
#202599322
#202599292
#202598872
#202598272
#202598152
#104×C1932
#455193