Finite Group Databasebeta

← #27135443

Subgroups — #27135443

Order 12352.

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

normal subgroups only

This group has 142 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#20259946C211yesmaximal, characteristic
2#20259932C211yesmaximal, characteristic, complemented
3#20259929C211yesmaximal, characteristic, complemented
4#20259899C211yesmaximal, characteristic, complemented
5#20259827C211yesmaximal, characteristic, complemented
6#20259817C211yesmaximal, characteristic, complemented
7#104×C193C211yesmaximal, characteristic, complemented
8#15690555—21no—
9#15690555C2211yescharacteristic
10#15690554—22no—
11#15690554—21no—
12C2×Dic386—21no—
13C2×Dic386C2211yescharacteristic
14C2×Dic386C2211yescharacteristic
15Q8×D193—21no—
16Dic772—21no—
17#15690545—21no—
18#15690544—21no—
19#15690543—21no—
20#15690539C2211yescharacteristic
21C4⋊Dic193—22no—
22#41×C193C2211yescharacteristic
23M16×C193C2211yescharacteristic
24C42×C193C2211yescharacteristic
25Dic386—22no—
26Dic386—21no—
27Dic386—21no—
28Dic386—21no—
29Dic386—21no—
30Dic386D411yescharacteristic
31D772D411yescharacteristic
32C22⋊D193—22no—
33C22⋊D193—21no—
34C2×Dic193—22no—
35C2×Dic193—21no—
36C2×Dic193D412yes—
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—21no—
50C4×C193—22no—
51C4×C193—21no—
52C4×C193C2×D411yescharacteristic
53C4×C193C2×D411yescharacteristic
54Dic193—21no—
55Dic193—22no—
56Dic193—41no—
57C2×C193—21no—
58C2×C193—41no—
59C2×C193#12011yescharacteristic
60D193—41no—
61C193#45611yescharacteristic, complemented, Sylow
62#456—1931nomaximal, Sylow
632−1+4—1931no—
64#137—1931no—
65#137—1931no—
66C4⋊Q8—1931no—
67#104—1931no—
68#104D19311yescharacteristic, complemented
69#101—1931no—
70#41—3861no—
71#41—3861no—
72#41—3862no—
73#41—1931no—
74#41C2×D19311yescharacteristic
75C2×Q8—3861no—
76C2×Q8—3861no—
77C2×Q8—1931no—
78C2×Q8—1931no—
79Q16—3861no—
80Q16—3861no—
81SD16—3861no—
82SD16—3861no—
83M16—1931no—
84M16C2×D19311yescharacteristic
85C4⋊C4—3862no—
86C42C2×D19311yescharacteristic
87Q8—3862no—
88Q8—3861no—
89Q8—3861no—
90Q8—3861no—
91Q8—3861no—
92Q8—1931no—
93Q8D77211yescharacteristic
94D4—21no—
95D4—3862no—
96D4—3861no—
97D4—1931no—
98D4D77211yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—3862no—
102C2×C4—3861no—
103C2×C4—3862no—
104C2×C4—3861no—
105C2×C4—1932no—
106C2×C4C22×D19311yescharacteristic
107C8—21no—
108C8—3861no—
109C22—21no—
110C22—3861no—
111C22D4×D19311yescharacteristic
112C4—22no—
113C4—21no—
114C4—7721no—
115C4—3861no—
116C4—3862no—
117C4D4×D19311yescharacteristic
118C4C2×D77211yescharacteristic
119C2—21no—
120C2—41no—
121C2—7721no—
122C2#2025986911yescharacteristic
subgroupG/H
centerC2#20259869
commutator, cosocleC2×C4×C193C23
FrattiniC2×C4C22×D193
Fitting#104×C193C2
socleC2×C193#120
prime radical#20259946C2
hypercenter#104D193
hypocenterC193#456
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. #20259899C2semidirect
5. #20259827C2semidirect
6. #20259817C2semidirect
7. #104×C193C2semidirect
61. C193#456semidirect
68. #104D193semidirect

Maximal subgroups

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

subgroupindex
#202599462
#202599322
#202599292
#202598992
#202598272
#202598172
#104×C1932
#456193