Finite Group Databasebeta

← #27135430

Subgroups — #27135430

Order 12352.

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

normal subgroups only

This group has 146 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, complemented
2#20259925C211yesmaximal, characteristic, complemented
3#20259886C211yesmaximal, characteristic, complemented
4#20259826C211yesmaximal, characteristic, complemented
5#20259825C211yesmaximal, characteristic, complemented
6#20259817C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C193C211yesmaximal, characteristic, complemented
8#15690555—21no—
9#15690555C2211yescharacteristic
10#15690554—22no—
11#15690554—21no—
12C2×Dic386—21no—
13C2×Dic386C2211yescharacteristic
14Q8×D193—21no—
15#15690543—21no—
16#15690540—21no—
17#15690539C2211yescharacteristic
18C22⋊Dic193—22no—
19C4×Dic193C2211yescharacteristic
20#41×C193C2211yescharacteristic
21C2×D4×C193C2211yescharacteristic
22SD16×C193—21no—
23D8×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—
38C23×C193—21no—
39Q8×C193D411yescharacteristic
40D4×C193—21no—
41D4×C193—21no—
42D4×C193—21no—
43D4×C193D411yescharacteristic
44C2×C4×C193—21no—
45C2×C4×C193C2311yescharacteristic
46C8×C193—21no—
47C22×C193—21no—
48C22×C193—42no—
49C22×C193—41no—
50C22×C193C2×D411yescharacteristic
51C2×D193—21no—
52C4×C193—21no—
53C4×C193C2×D411yescharacteristic
54C4×C193C2×D411yescharacteristic
55Dic193—21no—
56Dic193—22no—
57Dic193—41no—
58C2×C193—21no—
59C2×C193—41no—
60C2×C193—81no—
61C2×C193#12011yescharacteristic
62D193—41no—
63C193#45411yescharacteristic, complemented, Sylow
64#454—1931nomaximal, Sylow
652−1+4—1931no—
66AGL(1,ℤ/8)—1931no—
67AGL(1,ℤ/8)D19311yescharacteristic, complemented
68#124—1931no—
69#104—1931no—
70#104—1931no—
71#101—1931no—
72#41—3861no—
73#41—3861no—
74#41—3862no—
75#41—1931no—
76#41C2×D19311yescharacteristic
77C2×Q8—3861no—
78C2×Q8—3861no—
79C2×Q8—1931no—
80C2×D4C2×D19311yescharacteristic
81SD16—21no—
82SD16—3861no—
83D8—21no—
84D8—3861no—
85M16—1931no—
86M16C2×D19311yescharacteristic
87C22⋊C4—3862no—
88C42—1931no—
89C23—21no—
90Q8—3862no—
91Q8—3861no—
92Q8—3861no—
93Q8—1931no—
94Q8C22⋊D19311yescharacteristic
95D4—21no—
96D4—21no—
97D4—21no—
98D4—3862no—
99D4—3861no—
100D4—1931no—
101D4C22⋊D19311yescharacteristic
102C2×C4—21no—
103C2×C4—3862no—
104C2×C4—3861no—
105C2×C4—3862no—
106C2×C4—3861no—
107C2×C4—1932no—
108C2×C4C22×D19311yescharacteristic
109C8—21no—
110C8—3861no—
111C22—21no—
112C22—42no—
113C22—41no—
114C22—3861no—
115C22D4×D19311yescharacteristic
116C4—21no—
117C4—7721no—
118C4—3861no—
119C4—3862no—
120C4D4×D19311yescharacteristic
121C4C2×C22⋊D19311yescharacteristic
122C2—21no—
123C2—41no—
124C2—7721no—
125C2—81no—
126C2#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#454
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. #20259946C2semidirect
2. #20259925C2semidirect
3. #20259886C2semidirect
4. #20259826C2semidirect
5. #20259825C2semidirect
6. #20259817C2semidirect
7. AGL(1,ℤ/8)×C193C2semidirect
63. C193#454semidirect
67. AGL(1,ℤ/8)D193semidirect

Maximal subgroups

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

subgroupindex
#202599462
#202599252
#202598862
#202598262
#202598252
#202598172
AGL(1,ℤ/8)×C1932
#454193