Finite Group Databasebeta

← #27135440

Subgroups — #27135440

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#20259930C211yesmaximal, characteristic, complemented
3#20259897C211yesmaximal, characteristic, complemented
4#20259826C211yesmaximal, characteristic, complemented
5#20259825C211yesmaximal, characteristic, complemented
6#20259817C211yesmaximal, characteristic, complemented
7#137×C193C211yesmaximal, characteristic, complemented
8#15690555—21no—
9#15690555C2211yescharacteristic
10#15690554—22no—
11#15690554—21no—
12C2×Dic386—21no—
13C2×Dic386C2211yescharacteristic
14Q8×D193—21no—
15#15690545—21no—
16#15690542—21no—
17#15690539C2211yescharacteristic
18#15690534—22no—
19C4×Dic193C2211yescharacteristic
20#41×C193C2211yescharacteristic
21C2×Q8×C193C2211yescharacteristic
22Q16×C193—21no—
23SD16×C193—21no—
24M16×C193C2211yescharacteristic
25Dic386—22no—
26Dic386—21no—
27Dic386—21no—
28Dic386D411yescharacteristic
29D772D411yescharacteristic
30C22⋊D193—22no—
31C22⋊D193—21no—
32C2×Dic193—22no—
33C2×Dic193—21no—
34C2×Dic193D412yes—
35C4×D193—22no—
36C4×D193—21no—
37#13095393—21no—
38Q8×C193—21no—
39Q8×C193—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—21no—
51C4×C193—41no—
52C4×C193C2×D411yescharacteristic
53C4×C193C2×D411yescharacteristic
54Dic193—22no—
55Dic193—21no—
56Dic193—22no—
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#137D19311yescharacteristic, complemented
66C4⋊Q8—1931no—
67#104—1931no—
68#104—1931no—
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×Q8C2×D19311yescharacteristic
79Q16—21no—
80Q16—3861no—
81SD16—21no—
82SD16—3861no—
83M16—1931no—
84M16C2×D19311yescharacteristic
85C4⋊C4—3862no—
86C42—1931no—
87Q8—21no—
88Q8—21no—
89Q8—3862no—
90Q8—3861no—
91Q8—3861no—
92Q8—1931no—
93Q8C22⋊D19311yescharacteristic
94D4—21no—
95D4—3862no—
96D4—3861no—
97D4—1931no—
98D4C22⋊D19311yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—3862no—
102C2×C4—3862no—
103C2×C4—3861no—
104C2×C4—3861no—
105C2×C4—1932no—
106C2×C4C22×D19311yescharacteristic
107C8—21no—
108C8—3861no—
109C22—21no—
110C22—3861no—
111C22D4×D19311yescharacteristic
112C4—21no—
113C4—41no—
114C4—3862no—
115C4—3861no—
116C4—3862no—
117C4D4×D19311yescharacteristic
118C4C2×C22⋊D19311yescharacteristic
119C2—21no—
120C2—41no—
121C2—7721no—
122C2#2025986711yescharacteristic
subgroupG/H
centerC2#20259867
commutator, cosocleC2×C4×C193C23
FrattiniC2×C4C22×D193
Fitting#137×C193C2
socleC2×C193#120
prime radical#20259946C2
hypercenter#137D193
hypocenterC193#456
LCSG ▷ C2×C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20259930C2semidirect
3. #20259897C2semidirect
4. #20259826C2semidirect
5. #20259825C2semidirect
6. #20259817C2semidirect
7. #137×C193C2semidirect
61. C193#456semidirect
65. #137D193semidirect

Maximal subgroups

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

subgroupindex
#202599462
#202599302
#202598972
#202598262
#202598252
#202598172
#137×C1932
#456193