Finite Group Databasebeta

← #27164204

Subgroups — #27164204

Order 12736.

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

normal subgroups only

This group has 198 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#20264000C211yesmaximal, characteristic, complemented
2#20263924C212yesmaximal, complemented
3#20263870C211yesmaximal, characteristic, complemented
4#20263869C212yesmaximal, complemented
5#120×C199C211yesmaximal, characteristic, complemented
6#15691886—21no—
7#15691885—22no—
8C2×C22⋊D199—22no—
9C2×C22⋊D199—21no—
10C2×C22⋊D199C2212yes—
11D4×D199—22no—
12C22⋊Dic199—22no—
13C22⋊Dic199C2212yescomplemented
14C24×C199C2211yescharacteristic, complemented
15C2×D4×C199—21no—
16C2×D4×C199C2211yescharacteristic
17C22⋊C4×C199—21no—
18C22⋊C4×C199C2211yescharacteristic, complemented
19C22×D199—21no—
20C22×D199D412yescomplemented
21Dic398—21no—
22D796—21no—
23C22⋊D199—22no—
24C22⋊D199—22no—
25C22⋊D199—22no—
26C22⋊D199—44no—
27C2×Dic199—21no—
28C2×Dic199—22no—
29C2×Dic199D412yescomplemented
30C4×D199—22no—
31C23×C199—21no—
32C23×C199—22no—
33C23×C199—42no—
34C23×C199D411yescharacteristic, complemented
35C23×C199C2311yescharacteristic
36D4×C199—22no—
37D4×C199—42no—
38C2×C4×C199—21no—
39C2×C4×C199D411yescharacteristic, complemented
40C22×C199—22no—
41C22×C199—24no—
42C22×C199—21no—
43C22×C199—41no—
44C22×C199—41no—
45C22×C199—42no—
46C22×C199—41no—
47C22×C199—41no—
48C22×C199C2×D412yes—
49C22×C199C2×D411yescharacteristic
50C2×D199—22no—
51C2×D199—42no—
52C2×D199—81no—
53C4×C199—21no—
54C4×C199—41no—
55Dic199—22no—
56Dic199—42no—
57C2×C199—22no—
58C2×C199—21no—
59C2×C199—41no—
60C2×C199—42no—
61C2×C199#12011yescharacteristic
62D199—42no—
63C199B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1991nomaximal, Sylow
65H(5,𝔽2)—1991no—
66#120—1992no—
67#120D19911yescharacteristic, complemented
68#99—1991no—
69#99—1992no—
70C24C2×D19911yescharacteristic, complemented
71#41—3982no—
72#41—3981no—
73C2×D4—21no—
74C2×D4—3982no—
75C2×D4—3982no—
76C2×D4—3981no—
77C2×D4—1992no—
78C2×D4C2×D19911yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—3982no—
81C22⋊C4—1992no—
82C22⋊C4C2×D19911yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—3981no—
87C23—1992no—
88C23C22⋊D19911yescharacteristic, complemented
89C23C22×D19911yescharacteristic
90Q8—3981no—
91D4—22no—
92D4—42no—
93D4—7964no—
94D4—3981no—
95D4—3982no—
96D4—3982no—
97D4—3982no—
98C2×C4—21no—
99C2×C4—3982no—
100C2×C4—3982no—
101C2×C4—3981no—
102C2×C4—1992no—
103C2×C4C22⋊D19911yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—7962no—
113C22—15921no—
114C22—3982no—
115C22D4×D19912yes—
116C22C2×C22⋊D19911yescharacteristic
117C4—21no—
118C4—41no—
119C4—7962no—
120C4—3982no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—7962no—
126C2#2026392311yescharacteristic
subgroupG/H
centerC2#20263923
commutator, cosocleC23×C199C23
FrattiniC23C22×D199
Fitting#120×C199C2
socleC2×C199#120
hypercenter#120D199
hypocenterC199B(4,𝔽2)
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #20264000C2semidirect
2. #20263924C2semidirect
3. #20263870C2semidirect
4. #20263869C2semidirect
5. #120×C199C2semidirect
13. C22⋊Dic199C22semidirect
14. C24×C199C22semidirect
18. C22⋊C4×C199C22semidirect
20. C22×D199D4semidirect
29. C2×Dic199D4semidirect
34. C23×C199D4semidirect
39. C2×C4×C199D4semidirect
63. C199B(4,𝔽2)semidirect
67. #120D199semidirect
70. C24C2×D199semidirect
82. C22⋊C4C2×D199semidirect
88. C23C22⋊D199semidirect
103. C2×C4C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#202640002
#202639242
#202638702
#202638692
#120×C1992
B(4,𝔽2)199