Finite Group Databasebeta

← #28411669

Subgroups — #28411669

Order 14656.

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#20301033C211yesmaximal, characteristic, complemented
2#20300958C212yesmaximal, complemented
3#20300956C211yesmaximal, characteristic, complemented
4#20300902C212yesmaximal, complemented
5#99×C229C211yesmaximal, characteristic, complemented
6C23×D229C2211yescharacteristic, complemented
7#16814666—21no—
8#16814665—22no—
9C2×D916—22no—
10C2×C22⋊D229—21no—
11C2×C22⋊D229—21no—
12C2×C22⋊D229C2212yes—
13D4×D229—22no—
14C22⋊Dic229C2211yescharacteristic, complemented
15#16814643—22no—
16#16814643—21no—
17C2×D4×C229C2211yescharacteristic
18C22⋊C4×C229C2212yescomplemented
19C22×D229—21no—
20C22×D229—22no—
21C22×D229—21no—
22C22×D229—42no—
23C22×D229D412yescomplemented
24Dic458—21no—
25D916—21no—
26D916—44no—
27C22⋊D229—22no—
28C22⋊D229—22no—
29C22⋊D229—22no—
30C22⋊D229—42no—
31C2×Dic229—21no—
32C2×Dic229—21no—
33C2×Dic229D412yescomplemented
34C4×D229—22no—
35C23×C229D411yescharacteristic, complemented
36C23×C229C2311yescharacteristic
37D4×C229—22no—
38C2×C4×C229—22no—
39C2×C4×C229D411yescharacteristic, complemented
40C22×C229—21no—
41C22×C229—41no—
42C22×C229—41no—
43C22×C229C2×D412yes—
44C22×C229C2×D411yescharacteristic
45C2×D229—22no—
46C2×D229—24no—
47C2×D229—22no—
48C2×D229—41no—
49C2×D229—42no—
50C2×D229—41no—
51C2×D229—42no—
52C2×D229—81no—
53C4×C229—21no—
54C4×C229—42no—
55Dic229—22no—
56Dic229—41no—
57C2×C229—21no—
58C2×C229—22no—
59C2×C229—41no—
60C2×C229#12011yescharacteristic
61D229—42no—
62D229—42no—
63C229B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—2291nomaximal, Sylow
65H(5,𝔽2)—2291no—
66#120—2291no—
67#120—2292no—
68#99—2292no—
69#99D22911yescharacteristic, complemented
70C24—2291no—
71#41—4582no—
72#41—4581no—
73C2×D4—4582no—
74C2×D4—4582no—
75C2×D4—4581no—
76C2×D4—4581no—
77C2×D4—2292no—
78C2×D4C2×D22911yescharacteristic
79C22⋊C4—4582no—
80C22⋊C4—4581no—
81C22⋊C4—2291no—
82C22⋊C4C2×D22912yescomplemented
83C23—9162no—
84C23—4581no—
85C23—4582no—
86C23—4581no—
87C23—2292no—
88C23D91611yescharacteristic, complemented
89C23C22×D22911yescharacteristic
90Q8—4581no—
91D4—22no—
92D4—9164no—
93D4—9162no—
94D4—4581no—
95D4—4582no—
96D4—4582no—
97D4—4582no—
98C2×C4—22no—
99C2×C4—4582no—
100C2×C4—4581no—
101C2×C4—4581no—
102C2×C4—2292no—
103C2×C4D91611yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—18321no—
108C22—9162no—
109C22—9162no—
110C22—9161no—
111C22—9161no—
112C22—4584no—
113C22—4582no—
114C22—4582no—
115C22D4×D22912yes—
116C22C2×D91611yescharacteristic
117C4—21no—
118C4—42no—
119C4—9161no—
120C4—4582no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—9162no—
125C2—9162no—
126C2#2030095811yescharacteristic
subgroupG/H
centerC2#20300958
commutator, cosocleC23×C229C23
FrattiniC23C22×D229
Fitting#99×C229C2
socleC2×C229#120
hypercenter#99D229
hypocenterC229B(4,𝔽2)
LCSG ▷ C23×C229 ▷ C2×C229 ▷ C229
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #20301033C2semidirect
2. #20300958C2semidirect
3. #20300956C2semidirect
4. #20300902C2semidirect
5. #99×C229C2semidirect
6. C23×D229C22semidirect
14. C22⋊Dic229C22semidirect
18. C22⋊C4×C229C22semidirect
23. C22×D229D4semidirect
33. C2×Dic229D4semidirect
35. C23×C229D4semidirect
39. C2×C4×C229D4semidirect
63. C229B(4,𝔽2)semidirect
69. #99D229semidirect
82. C22⋊C4C2×D229semidirect
88. C23D916semidirect
103. C2×C4D916semidirect

Maximal subgroups

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

subgroupindex
#203010332
#203009582
#203009562
#203009022
#99×C2292
B(4,𝔽2)229