Finite Group Databasebeta

← #22553501

Subgroups — #22553501

Order 8384.

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

normal subgroups only

This group has 160 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
2#16862229C212yesmaximal, complemented
3#16862161C211yesmaximal, characteristic, complemented
4#16862160C212yesmaximal, complemented
5#123×C131C211yesmaximal, characteristic, complemented
6#14523513—21no—
7#14523512—22no—
8C2×C22⋊D131—21no—
9C2×C22⋊D131C2212yes—
10D4×D131—22no—
11Dic131⋊C4—22no—
12C22⋊Dic131C2212yescomplemented
13(C2×D131)⋊C4—22no—
14C2×D4×C131C2211yescharacteristic
15C22×C4×C131C2211yescharacteristic, complemented
16C4⋊C4×C131—21no—
17C22⋊C4×C131—21no—
18C22⋊C4×C131C2211yescharacteristic, complemented
19C22×D131—21no—
20C22×D131D412yes—
21Dic262—21no—
22D524—21no—
23C22⋊D131—22no—
24C22⋊D131—22no—
25C22⋊D131—22no—
26C2×Dic131—21no—
27C2×Dic131—22no—
28C2×Dic131D412yes—
29C4×D131—22no—
30C23×C131D411yescharacteristic
31C23×C131C2311yescharacteristic
32D4×C131—22no—
33C2×C4×C131—21no—
34C2×C4×C131—21no—
35C2×C4×C131—22no—
36C2×C4×C131D411yescharacteristic
37C22×C131—21no—
38C22×C131—41no—
39C22×C131—41no—
40C22×C131C2×D412yes—
41C22×C131C2×D411yescharacteristic
42C2×D131—22no—
43C2×D131—42no—
44C2×D131—81no—
45C4×C131—21no—
46C4×C131—41no—
47C4×C131—41no—
48Dic131—22no—
49Dic131—42no—
50C2×C131—22no—
51C2×C131—21no—
52C2×C131—41no—
53C2×C131#12011yescharacteristic
54D131—42no—
55C131#45811yescharacteristic, complemented, Sylow
56#458—1311nomaximal, Sylow
57H(5,𝔽2)—1311no—
58#123—1312no—
59#123D13111yescharacteristic, complemented
60#99—1311no—
61#99—1312no—
62#41—2622no—
63#41—2621no—
64C2×D4—2622no—
65C2×D4—2621no—
66C2×D4—1312no—
67C2×D4C2×D13111yescharacteristic
68C22×C4C2×D13111yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—2622no—
71C22⋊C4—21no—
72C22⋊C4—2622no—
73C22⋊C4—1312no—
74C22⋊C4C2×D13111yescharacteristic, complemented
75C23—2621no—
76C23—1312no—
77C23C22⋊D13111yescharacteristic
78C23C22×D13111yescharacteristic
79Q8—2621no—
80D4—22no—
81D4—2621no—
82D4—2622no—
83D4—2622no—
84D4—2622no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—2622no—
89C2×C4—2621no—
90C2×C4—2622no—
91C2×C4—1312no—
92C2×C4C22⋊D13111yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—2622no—
97C22—5242no—
98C22—10481no—
99C22D4×D13112yes—
100C22C2×C22⋊D13111yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—2622no—
105C4—5242no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—5242no—
110C2#1686221411yescharacteristic
subgroupG/H
centerC2#16862214
commutatorC23×C131C23
FrattiniC23C22×D131
Fitting#123×C131C2
socleC2×C131#120
hypercenter#123D131
hypocenterC131#458
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #16862229C2semidirect
3. #16862161C2semidirect
4. #16862160C2semidirect
5. #123×C131C2semidirect
12. C22⋊Dic131C22semidirect
15. C22×C4×C131C22semidirect
18. C22⋊C4×C131C22semidirect
55. C131#458semidirect
59. #123D131semidirect
68. C22×C4C2×D131semidirect
74. C22⋊C4C2×D131semidirect