Finite Group Databasebeta

← #22553500

Subgroups — #22553500

Order 8384.

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#16862291C211yesmaximal, characteristic, complemented
2#16862216C212yesmaximal, complemented
3#16862214C211yesmaximal, characteristic, complemented
4#16862160C212yesmaximal, complemented
5#99×C131C211yesmaximal, characteristic, complemented
6C23×D131C2211yescharacteristic, complemented
7#14523513—21no—
8#14523512—22no—
9C2×D524—22no—
10C2×C22⋊D131—21no—
11C2×C22⋊D131—21no—
12C2×C22⋊D131C2212yes—
13D4×D131—22no—
14C22⋊Dic131C2211yescharacteristic, complemented
15(C2×D131)⋊C4—22no—
16(C2×D131)⋊C4—21no—
17C2×D4×C131C2211yescharacteristic
18C22⋊C4×C131C2212yescomplemented
19C22×D131—21no—
20C22×D131—22no—
21C22×D131—21no—
22C22×D131—42no—
23C22×D131D412yescomplemented
24Dic262—21no—
25D524—21no—
26D524—44no—
27C22⋊D131—22no—
28C22⋊D131—22no—
29C22⋊D131—22no—
30C22⋊D131—42no—
31C2×Dic131—21no—
32C2×Dic131—21no—
33C2×Dic131D412yescomplemented
34C4×D131—22no—
35C23×C131D411yescharacteristic, complemented
36C23×C131C2311yescharacteristic
37D4×C131—22no—
38C2×C4×C131—22no—
39C2×C4×C131D411yescharacteristic, complemented
40C22×C131—21no—
41C22×C131—41no—
42C22×C131—41no—
43C22×C131C2×D412yes—
44C22×C131C2×D411yescharacteristic
45C2×D131—22no—
46C2×D131—24no—
47C2×D131—22no—
48C2×D131—41no—
49C2×D131—42no—
50C2×D131—41no—
51C2×D131—42no—
52C2×D131—81no—
53C4×C131—21no—
54C4×C131—42no—
55Dic131—22no—
56Dic131—41no—
57C2×C131—21no—
58C2×C131—22no—
59C2×C131—41no—
60C2×C131#12011yescharacteristic
61D131—42no—
62D131—42no—
63C131B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1311nomaximal, Sylow
65H(5,𝔽2)—1311no—
66#120—1311no—
67#120—1312no—
68#99—1312no—
69#99D13111yescharacteristic, complemented
70C24—1311no—
71#41—2622no—
72#41—2621no—
73C2×D4—2622no—
74C2×D4—2622no—
75C2×D4—2621no—
76C2×D4—2621no—
77C2×D4—1312no—
78C2×D4C2×D13111yescharacteristic
79C22⋊C4—2622no—
80C22⋊C4—2621no—
81C22⋊C4—1311no—
82C22⋊C4C2×D13112yescomplemented
83C23—2621no—
84C23—2622no—
85C23—2621no—
86C23—5242no—
87C23—1312no—
88C23D52411yescharacteristic, complemented
89C23C22×D13111yescharacteristic
90Q8—2621no—
91D4—22no—
92D4—2621no—
93D4—2622no—
94D4—2622no—
95D4—2622no—
96D4—5244no—
97D4—5242no—
98C2×C4—22no—
99C2×C4—2622no—
100C2×C4—2621no—
101C2×C4—2621no—
102C2×C4—1312no—
103C2×C4D52411yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—2624no—
108C22—2622no—
109C22—2622no—
110C22—5242no—
111C22—5242no—
112C22—5241no—
113C22—5241no—
114C22—10481no—
115C22D4×D13112yes—
116C22C2×D52411yescharacteristic
117C4—21no—
118C4—42no—
119C4—2622no—
120C4—5241no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—5242no—
125C2—5242no—
126C2#1686221611yescharacteristic
subgroupG/H
centerC2#16862216
commutatorC23×C131C23
FrattiniC23C22×D131
Fitting#99×C131C2
socleC2×C131#120
hypercenter#99D131
hypocenterC131B(4,𝔽2)
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #16862291C2semidirect
2. #16862216C2semidirect
3. #16862214C2semidirect
4. #16862160C2semidirect
5. #99×C131C2semidirect
6. C23×D131C22semidirect
14. C22⋊Dic131C22semidirect
18. C22⋊C4×C131C22semidirect
23. C22×D131D4semidirect
33. C2×Dic131D4semidirect
35. C23×C131D4semidirect
39. C2×C4×C131D4semidirect
63. C131B(4,𝔽2)semidirect
69. #99D131semidirect
82. C22⋊C4C2×D131semidirect
88. C23D524semidirect
103. C2×C4D524semidirect