Finite Group Databasebeta

← #27135435

Subgroups — #27135435

Order 12352.

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

normal subgroups only

This group has 152 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
2#20259930C211yesmaximal, characteristic, complemented
3#20259888C211yesmaximal, characteristic, complemented
4#20259826C211yesmaximal, characteristic, complemented
5#20259825C211yesmaximal, characteristic, complemented
6#20259815C211yesmaximal, characteristic, complemented
7#137×C193C211yesmaximal, characteristic, complemented
8#15690556—21no—
9#15690555—21no—
10#15690555C2211yescharacteristic
11C2×D772—21no—
12C2×D772C2211yescharacteristic
13D4×D193—22no—
14D4×D193—21no—
15#15690545—21no—
16#15690542—21no—
17#15690539C2211yescharacteristic, complemented
18#15690532—22no—
19C4×Dic193C2211yescharacteristic, complemented
20#41×C193C2211yescharacteristic
21C2×Q8×C193C2211yescharacteristic, complemented
22Q16×C193—21no—
23SD16×C193—21no—
24M16×C193C2211yescharacteristic, complemented
25C22×D193—22no—
26C22×D193D412yes—
27Dic386D411yescharacteristic
28D772—22no—
29D772—21no—
30D772—21no—
31D772D411yescharacteristic
32C22⋊D193—22no—
33C22⋊D193—21no—
34C2×Dic193—21no—
35C4×D193—22no—
36C4×D193—21no—
37#13095393—21no—
38Q8×C193—21no—
39Q8×C193—21no—
40Q8×C193D411yescharacteristic
41D4×C193—21no—
42D4×C193D411yescharacteristic
43C2×C4×C193—21no—
44C2×C4×C193—21no—
45C2×C4×C193C2311yescharacteristic
46C8×C193—21no—
47C22×C193—21no—
48C22×C193C2×D411yescharacteristic
49C2×D193—22no—
50C2×D193—21no—
51C2×D193—42no—
52C2×D193—82no—
53C4×C193—21no—
54C4×C193—41no—
55C4×C193C2×D411yescharacteristic
56C4×C193C2×D411yescharacteristic
57Dic193—21no—
58Dic193—41no—
59C2×C193—21no—
60C2×C193—41no—
61C2×C193#12011yescharacteristic
62D193—42no—
63D193—41no—
64C193#45511yescharacteristic, complemented, Sylow
65#455—1931nomaximal, Sylow
66H(5,𝔽2)—1931no—
67#137—1931no—
68#137D19311yescharacteristic, complemented
69#124—1931no—
70#104—1931no—
71#104—1931no—
72#100—1931no—
73#41—3861no—
74#41—3861no—
75#41—1931no—
76#41C2×D19311yescharacteristic
77C2×Q8C2×D19311yescharacteristic, complemented
78C2×D4—3861no—
79C2×D4—3862no—
80C2×D4—3861no—
81C2×D4—1931no—
82Q16—21no—
83Q16—3861no—
84SD16—21no—
85SD16—3861no—
86M16—1931no—
87M16C2×D19311yescharacteristic, complemented
88C22⋊C4—3862no—
89C42—1931no—
90C23—3862no—
91C23—1932no—
92Q8—21no—
93Q8—21no—
94Q8—1931no—
95Q8C22⋊D19311yescharacteristic
96D4—21no—
97D4—3862no—
98D4—3862no—
99D4—3861no—
100D4—3861no—
101D4—3861no—
102D4—1931no—
103D4C22⋊D19311yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—3862no—
107C2×C4—3861no—
108C2×C4—3861no—
109C2×C4C22×D19311yescharacteristic
110C8—21no—
111C8—3861no—
112C22—21no—
113C22—7722no—
114C22—15442no—
115C22—3862no—
116C22—3861no—
117C22D4×D19311yescharacteristic
118C4—21no—
119C4—41no—
120C4—7721no—
121C4—3861no—
122C4D4×D19311yescharacteristic
123C4C2×C22⋊D19311yescharacteristic
124C2—21no—
125C2—41no—
126C2—7722no—
127C2—7721no—
128C2#2025986711yescharacteristic
subgroupG/H
centerC2#20259867
commutator, cosocleC2×C4×C193C23
FrattiniC2×C4C22×D193
Fitting#137×C193C2
socleC2×C193#120
prime radical#20259945C2
hypercenter#137D193
hypocenterC193#455
LCSG ▷ C2×C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20259930C2semidirect
3. #20259888C2semidirect
4. #20259826C2semidirect
5. #20259825C2semidirect
6. #20259815C2semidirect
7. #137×C193C2semidirect
17. #15690539C22semidirect
19. C4×Dic193C22semidirect
21. C2×Q8×C193C22semidirect
24. M16×C193C22semidirect
64. C193#455semidirect
68. #137D193semidirect
77. C2×Q8C2×D193semidirect
87. M16C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599452
#202599302
#202598882
#202598262
#202598252
#202598152
#137×C1932
#455193