Finite Group Databasebeta

← #27135448

Subgroups — #27135448

Order 12352.

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

normal subgroups only

This group has 160 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#20259944C211yesmaximal, characteristic
2#20259882C212yesmaximal, complemented
3#20259814C211yesmaximal, characteristic, complemented
4#20259813C212yesmaximal, complemented
5#123×C193C211yesmaximal, characteristic, complemented
6#15690555—21no—
7#15690554—22no—
8C2×C22⋊D193—21no—
9C2×C22⋊D193C2212yes—
10D4×D193—22no—
11#15690534—22no—
12C22⋊Dic193C2212yescomplemented
13#15690532—22no—
14C2×D4×C193C2211yescharacteristic
15C22×C4×C193C2211yescharacteristic, complemented
16C4⋊C4×C193—21no—
17C22⋊C4×C193—21no—
18C22⋊C4×C193C2211yescharacteristic, complemented
19C22×D193—21no—
20C22×D193D412yes—
21Dic386—21no—
22D772—21no—
23C22⋊D193—22no—
24C22⋊D193—22no—
25C22⋊D193—22no—
26C2×Dic193—21no—
27C2×Dic193—22no—
28C2×Dic193D412yes—
29C4×D193—22no—
30C23×C193D411yescharacteristic
31C23×C193C2311yescharacteristic
32D4×C193—22no—
33C2×C4×C193—21no—
34C2×C4×C193—21no—
35C2×C4×C193—22no—
36C2×C4×C193D411yescharacteristic
37C22×C193—21no—
38C22×C193—41no—
39C22×C193—41no—
40C22×C193C2×D412yes—
41C22×C193C2×D411yescharacteristic
42C2×D193—22no—
43C2×D193—42no—
44C2×D193—81no—
45C4×C193—21no—
46C4×C193—41no—
47C4×C193—41no—
48Dic193—22no—
49Dic193—42no—
50C2×C193—22no—
51C2×C193—21no—
52C2×C193—41no—
53C2×C193#12011yescharacteristic
54D193—42no—
55C193#45811yescharacteristic, complemented, Sylow
56#458—1931nomaximal, Sylow
57H(5,𝔽2)—1931no—
58#123—1932no—
59#123D19311yescharacteristic, complemented
60#99—1931no—
61#99—1932no—
62#41—3862no—
63#41—3861no—
64C2×D4—3862no—
65C2×D4—3861no—
66C2×D4—1932no—
67C2×D4C2×D19311yescharacteristic
68C22×C4C2×D19311yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—3862no—
71C22⋊C4—21no—
72C22⋊C4—3862no—
73C22⋊C4—1932no—
74C22⋊C4C2×D19311yescharacteristic, complemented
75C23—3861no—
76C23—1932no—
77C23C22⋊D19311yescharacteristic
78C23C22×D19311yescharacteristic
79Q8—3861no—
80D4—22no—
81D4—3861no—
82D4—3862no—
83D4—3862no—
84D4—3862no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3862no—
89C2×C4—3862no—
90C2×C4—3861no—
91C2×C4—1932no—
92C2×C4C22⋊D19311yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—7722no—
97C22—15441no—
98C22—3862no—
99C22D4×D19312yes—
100C22C2×C22⋊D19311yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—7722no—
105C4—3862no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—7722no—
110C2#2025986711yescharacteristic
subgroupG/H
centerC2#20259867
commutator, cosocleC23×C193C23
FrattiniC23C22×D193
Fitting#123×C193C2
socleC2×C193#120
prime radical#20259944C2
hypercenter#123D193
hypocenterC193#458
LCSG ▷ C23×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #20259882C2semidirect
3. #20259814C2semidirect
4. #20259813C2semidirect
5. #123×C193C2semidirect
12. C22⋊Dic193C22semidirect
15. C22×C4×C193C22semidirect
18. C22⋊C4×C193C22semidirect
55. C193#458semidirect
59. #123D193semidirect
68. C22×C4C2×D193semidirect
74. C22⋊C4C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599442
#202598822
#202598142
#202598132
#123×C1932
#458193