Finite Group Databasebeta

← #27135450

Subgroups — #27135450

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#20259883C212yesmaximal, complemented
3#20259880C211yesmaximal, characteristic, complemented
4#20259813C212yesmaximal, complemented
5#99×C193C211yesmaximal, characteristic, complemented
6#15690555—21no—
7#15690554—22no—
8C2×C22⋊D193—21no—
9C2×C22⋊D193C2212yes—
10D4×D193—22no—
11C22×Dic193C2211yescharacteristic, complemented
12C4⋊Dic193—22no—
13#15690534—21no—
14C22⋊Dic193—21no—
15C22⋊Dic193C2211yescharacteristic, complemented
16#15690532—22no—
17C2×D4×C193C2211yescharacteristic
18C22⋊C4×C193C2212yescomplemented
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×Dic193—21no—
29C2×Dic193—21no—
30C2×Dic193D412yes—
31C4×D193—22no—
32C23×C193D411yescharacteristic
33C23×C193C2311yescharacteristic
34D4×C193—22no—
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—42no—
47Dic193—22no—
48Dic193—41no—
49Dic193—41no—
50C2×C193—21no—
51C2×C193—22no—
52C2×C193—41no—
53C2×C193#12011yescharacteristic
54D193—42no—
55C193#45811yescharacteristic, complemented, Sylow
56#458—1931nomaximal, Sylow
57H(5,𝔽2)—1931no—
58#123—1931no—
59#123—1932no—
60#99—1932no—
61#99D19311yescharacteristic, complemented
62#41—3862no—
63#41—3861no—
64C2×D4—3862no—
65C2×D4—3861no—
66C2×D4—1932no—
67C2×D4C2×D19311yescharacteristic
68C22×C4—1931no—
69C4⋊C4—3862no—
70C4⋊C4—3861no—
71C22⋊C4—3862no—
72C22⋊C4—3861no—
73C22⋊C4—1931no—
74C22⋊C4C2×D19312yescomplemented
75C23—3861no—
76C23—1932no—
77C23D77211yescharacteristic
78C23C22×D19311yescharacteristic
79Q8—3861no—
80D4—22no—
81D4—3861no—
82D4—3862no—
83D4—3862no—
84D4—3862no—
85C2×C4—22no—
86C2×C4—3862no—
87C2×C4—3862no—
88C2×C4—3861no—
89C2×C4—3861no—
90C2×C4—3861no—
91C2×C4—1932no—
92C2×C4D77211yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—7722no—
97C22—15441no—
98C22—3862no—
99C22D4×D19312yes—
100C22C2×D77211yescharacteristic
101C4—21no—
102C4—42no—
103C4—7721no—
104C4—7721no—
105C4—3862no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—7722no—
110C2#2025986911yescharacteristic
subgroupG/H
centerC2#20259869
commutator, cosocleC23×C193C23
FrattiniC23C22×D193
Fitting#99×C193C2
socleC2×C193#120
prime radical#20259944C2
hypercenter#99D193
hypocenterC193#458
LCSG ▷ C23×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #20259883C2semidirect
3. #20259880C2semidirect
4. #20259813C2semidirect
5. #99×C193C2semidirect
11. C22×Dic193C22semidirect
15. C22⋊Dic193C22semidirect
18. C22⋊C4×C193C22semidirect
55. C193#458semidirect
61. #99D193semidirect
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
#202598832
#202598802
#202598132
#99×C1932
#458193