Finite Group Databasebeta

← #27135425

Subgroups — #27135425

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#20259925C211yesmaximal, characteristic, complemented
3#20259895C211yesmaximal, characteristic, complemented
4#20259826C211yesmaximal, characteristic, complemented
5#20259825C211yesmaximal, characteristic, complemented
6#20259815C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C193C211yesmaximal, characteristic, complemented
8#15690556—21no—
9#15690555—21no—
10#15690555C2211yescharacteristic
11C2×D772—21no—
12C2×D772C2211yescharacteristic
13C2×C22⋊D193—22no—
14D4×D193—22no—
15D4×D193—21no—
16#15690543—21no—
17#15690540—21no—
18#15690539C2211yescharacteristic, complemented
19C4×Dic193C2211yescharacteristic, complemented
20#41×C193C2211yescharacteristic
21C2×D4×C193C2211yescharacteristic, complemented
22SD16×C193—21no—
23D8×C193—21no—
24M16×C193C2211yescharacteristic, complemented
25C22×D193—22no—
26C22×D193D412yes—
27Dic386D411yescharacteristic, complemented
28D772—22no—
29D772—21no—
30D772—21no—
31D772D411yescharacteristic, complemented
32C22⋊D193—22no—
33C22⋊D193—21no—
34C22⋊D193—44no—
35C2×Dic193—21no—
36C4×D193—22no—
37C4×D193—21no—
38#13095393—21no—
39C23×C193—21no—
40Q8×C193D411yescharacteristic, complemented
41D4×C193—21no—
42D4×C193—21no—
43D4×C193—21no—
44D4×C193D411yescharacteristic, complemented
45C2×C4×C193—21no—
46C2×C4×C193C2311yescharacteristic
47C8×C193—21no—
48C22×C193—21no—
49C22×C193—42no—
50C22×C193—41no—
51C22×C193C2×D411yescharacteristic
52C2×D193—22no—
53C2×D193—21no—
54C2×D193—42no—
55C2×D193—82no—
56C4×C193—21no—
57C4×C193C2×D411yescharacteristic
58C4×C193C2×D411yescharacteristic
59Dic193—22no—
60Dic193—21no—
61C2×C193—21no—
62C2×C193—41no—
63C2×C193—81no—
64C2×C193#12011yescharacteristic
65D193—42no—
66D193—41no—
67C193#45311yescharacteristic, complemented, Sylow
68#453—1931nomaximal, Sylow
69H(5,𝔽2)—1931no—
70AGL(1,ℤ/8)—1931no—
71AGL(1,ℤ/8)D19311yescharacteristic, complemented
72#127—1931no—
73#104—1931no—
74#104—1931no—
75#100—1931no—
76#41—3861no—
77#41—3861no—
78#41—1931no—
79#41C2×D19311yescharacteristic
80C2×D4—3861no—
81C2×D4—3862no—
82C2×D4—3862no—
83C2×D4—3861no—
84C2×D4—1931no—
85C2×D4C2×D19311yescharacteristic, complemented
86SD16—21no—
87SD16—3861no—
88D8—21no—
89D8—3861no—
90M16—1931no—
91M16C2×D19311yescharacteristic, complemented
92C42—1931no—
93C23—21no—
94C23—3862no—
95C23—1932no—
96Q8—1931no—
97Q8C22⋊D19311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—7724no—
102D4—3862no—
103D4—3862no—
104D4—3861no—
105D4—3861no—
106D4—3861no—
107D4—1931no—
108D4C22⋊D19311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—3862no—
111C2×C4—3861no—
112C2×C4—3861no—
113C2×C4C22×D19311yescharacteristic
114C8—21no—
115C8—3861no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—7722no—
120C22—15442no—
121C22—3862no—
122C22—3861no—
123C22D4×D19311yescharacteristic
124C4—21no—
125C4—3862no—
126C4—3861no—
127C4D4×D19311yescharacteristic
128C4C2×C22⋊D19311yescharacteristic
129C2—21no—
130C2—41no—
131C2—7722no—
132C2—7721no—
133C2—81no—
134C2#2025986711yescharacteristic
subgroupG/H
centerC2#20259867
commutator, cosocleC2×C4×C193C23
FrattiniC2×C4C22×D193
FittingAGL(1,ℤ/8)×C193C2
socleC2×C193#120
hypercenterAGL(1,ℤ/8)D193
hypocenterC193#453
LCSG ▷ C2×C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20259945C2semidirect
2. #20259925C2semidirect
3. #20259895C2semidirect
4. #20259826C2semidirect
5. #20259825C2semidirect
6. #20259815C2semidirect
7. AGL(1,ℤ/8)×C193C2semidirect
18. #15690539C22semidirect
19. C4×Dic193C22semidirect
21. C2×D4×C193C22semidirect
24. M16×C193C22semidirect
27. Dic386D4semidirect
31. D772D4semidirect
40. Q8×C193D4semidirect
44. D4×C193D4semidirect
67. C193#453semidirect
71. AGL(1,ℤ/8)D193semidirect
85. C2×D4C2×D193semidirect
91. M16C2×D193semidirect
97. Q8C22⋊D193semidirect
108. D4C22⋊D193semidirect

Maximal subgroups

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

subgroupindex
#202599452
#202599252
#202598952
#202598262
#202598252
#202598152
AGL(1,ℤ/8)×C1932
#453193