Finite Group Databasebeta

← #27135428

Subgroups — #27135428

Order 12352.

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

normal subgroups only

This group has 168 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, complemented
2#20259927C211yesmaximal, characteristic, complemented
3#20259924C211yesmaximal, characteristic, complemented
4#20259896C211yesmaximal, characteristic, complemented
5#20259827C211yesmaximal, characteristic, complemented
6#20259815C211yesmaximal, characteristic, complemented
7#104×C193C211yesmaximal, characteristic, complemented
8#15690556—21no—
9#15690555—21no—
10#15690555C2211yescharacteristic
11C2×D772—21no—
12C2×D772—22no—
13C2×D772C2211yescharacteristic
14C2×D772C2211yescharacteristic, complemented
15D4×D193—22no—
16D4×D193—21no—
17#15690544—21no—
18#15690542—21no—
19D1544—21no—
20#15690540—21no—
21#15690539C2211yescharacteristic, complemented
22#41×C193C2211yescharacteristic
23M16×C193C2211yescharacteristic, complemented
24C42×C193C2211yescharacteristic, complemented
25C22×D193—22no—
26C22×D193—21no—
27C22×D193D412yes—
28Dic386D411yescharacteristic, complemented
29D772—22no—
30D772—21no—
31D772—21no—
32D772—21no—
33D772—21no—
34D772—44no—
35D772D411yescharacteristic, complemented
36C22⋊D193—22no—
37C22⋊D193—21no—
38C4×D193—22no—
39C4×D193—21no—
40#13095393—21no—
41Q8×C193D411yescharacteristic, complemented
42D4×C193—21no—
43D4×C193D411yescharacteristic, complemented
44C2×C4×C193—21no—
45C2×C4×C193—21no—
46C2×C4×C193C2311yescharacteristic
47C8×C193—21no—
48C22×C193—21no—
49C22×C193C2×D411yescharacteristic
50C2×D193—22no—
51C2×D193—21no—
52C2×D193—42no—
53C2×D193—42no—
54C2×D193—41no—
55C2×D193—82no—
56C4×C193—22no—
57C4×C193—21no—
58C4×C193C2×D411yescharacteristic
59C4×C193C2×D411yescharacteristic
60Dic193—21no—
61C2×C193—21no—
62C2×C193—41no—
63C2×C193#12011yescharacteristic
64D193—42no—
65D193—41no—
66D193—81no—
67C193#45311yescharacteristic, complemented, Sylow
68#453—1931nomaximal, Sylow
69H(5,𝔽2)—1931no—
70AGL(1,ℤ/8)—1931no—
71AGL(1,ℤ/8)—1931no—
72#127—1931no—
73#104—1931no—
74#104D19311yescharacteristic, complemented
75#100—1931no—
76#41—3861no—
77#41—3861no—
78#41—1931no—
79#41C2×D19311yescharacteristic
80C2×D4—3862no—
81C2×D4—3861no—
82C2×D4—3862no—
83C2×D4—3861no—
84C2×D4—1931no—
85C2×D4—1931no—
86SD16—3861no—
87SD16—3861no—
88D8—3861no—
89D8—3861no—
90M16—1931no—
91M16C2×D19311yescharacteristic, complemented
92C42C2×D19311yescharacteristic, complemented
93C23—3862no—
94C23—3861no—
95C23—1932no—
96Q8—1931no—
97Q8D77211yescharacteristic, complemented
98D4—21no—
99D4—7724no—
100D4—3862no—
101D4—3862no—
102D4—3861no—
103D4—3861no—
104D4—3861no—
105D4—3861no—
106D4—3861no—
107D4—1931no—
108D4D77211yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—3862no—
112C2×C4—3861no—
113C2×C4C22×D19311yescharacteristic
114C8—21no—
115C8—3861no—
116C22—21no—
117C22—7722no—
118C22—7722no—
119C22—7721no—
120C22—15442no—
121C22—3862no—
122C22—3861no—
123C22D4×D19311yescharacteristic
124C4—22no—
125C4—21no—
126C4—3861no—
127C4D4×D19311yescharacteristic
128C4C2×D77211yescharacteristic
129C2—21no—
130C2—41no—
131C2—7722no—
132C2—7721no—
133C2—15441no—
134C2#2025986911yescharacteristic
subgroupG/H
centerC2#20259869
commutator, cosocleC2×C4×C193C23
FrattiniC2×C4C22×D193
Fitting#104×C193C2
socleC2×C193#120
hypercenter#104D193
hypocenterC193#453
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
1. #20259945C2semidirect
2. #20259927C2semidirect
3. #20259924C2semidirect
4. #20259896C2semidirect
5. #20259827C2semidirect
6. #20259815C2semidirect
7. #104×C193C2semidirect
14. C2×D772C22semidirect
21. #15690539C22semidirect
23. M16×C193C22semidirect
24. C42×C193C22semidirect
28. Dic386D4semidirect
35. D772D4semidirect
41. Q8×C193D4semidirect
43. D4×C193D4semidirect
67. C193#453semidirect
74. #104D193semidirect
91. M16C2×D193semidirect
92. C42C2×D193semidirect
97. Q8D772semidirect
108. D4D772semidirect

Maximal subgroups

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

subgroupindex
#202599452
#202599272
#202599242
#202598962
#202598272
#202598152
#104×C1932
#453193