Finite Group Databasebeta

← #20298787

Subgroups — #20298787

Order 7232.

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#16813667C211yesmaximal, characteristic
2#16813605C212yesmaximal, complemented
3#16813537C211yesmaximal, characteristic, complemented
4#16813536C212yesmaximal, complemented
5#123×C113C211yesmaximal, characteristic, complemented
6#14268647—21no—
7#14268646—22no—
8C2×C22⋊D113—21no—
9C2×C22⋊D113C2212yes—
10D4×D113—22no—
11#14268626—22no—
12C22⋊Dic113C2212yescomplemented
13#14268624—22no—
14C2×D4×C113C2211yescharacteristic
15C22×C4×C113C2211yescharacteristic, complemented
16C4⋊C4×C113—21no—
17C22⋊C4×C113—21no—
18C22⋊C4×C113C2211yescharacteristic, complemented
19C23×C113D411yescharacteristic
20C23×C113C2311yescharacteristic
21C22×D113—21no—
22C22×D113D412yes—
23D4×C113—22no—
24C2×C4×C113—21no—
25C2×C4×C113—21no—
26C2×C4×C113—22no—
27C2×C4×C113D411yescharacteristic
28C22⋊D113—22no—
29C22⋊D113—22no—
30C22⋊D113—22no—
31C2×Dic113—21no—
32C2×Dic113—22no—
33C2×Dic113D412yes—
34D452—21no—
35C4×D113—22no—
36Dic226—21no—
37C22×C113—21no—
38C22×C113—41no—
39C22×C113—41no—
40C22×C113C2×D412yes—
41C22×C113C2×D411yescharacteristic
42C2×D113—22no—
43C2×D113—42no—
44C2×D113—81no—
45C4×C113—21no—
46C4×C113—41no—
47C4×C113—41no—
48Dic113—22no—
49Dic113—42no—
50C2×C113—22no—
51C2×C113—21no—
52C2×C113—41no—
53C2×C113#12011yescharacteristic
54D113—42no—
55C113#45811yescharacteristic, complemented, Sylow
56#458—1131nomaximal, Sylow
57H(5,𝔽2)—1131no—
58#123—1132no—
59#123D11311yescharacteristic, complemented
60#99—1131no—
61#99—1132no—
62#41—2262no—
63#41—2261no—
64C2×D4—1132no—
65C2×D4—2262no—
66C2×D4—2261no—
67C2×D4C2×D11311yescharacteristic
68C22×C4C2×D11311yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—2262no—
71C22⋊C4—21no—
72C22⋊C4—1132no—
73C22⋊C4—2262no—
74C22⋊C4C2×D11311yescharacteristic, complemented
75C23—1132no—
76C23—2261no—
77C23C22⋊D11311yescharacteristic
78C23C22×D11311yescharacteristic
79Q8—2261no—
80D4—22no—
81D4—2261no—
82D4—2262no—
83D4—2262no—
84D4—2262no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—1132no—
89C2×C4—2262no—
90C2×C4—2262no—
91C2×C4—2261no—
92C2×C4C22⋊D11311yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—9041no—
97C22—4522no—
98C22—2262no—
99C22D4×D11312yes—
100C22C2×C22⋊D11311yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—4522no—
105C4—2262no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—4522no—
110C2#1681359011yescharacteristic
subgroupG/H
centerC2#16813590
commutatorC23×C113C23
FrattiniC23C22×D113
Fitting#123×C113C2
socleC2×C113#120
hypercenter#123D113
hypocenterC113#458
LCSG ▷ C23×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #16813605C2semidirect
3. #16813537C2semidirect
4. #16813536C2semidirect
5. #123×C113C2semidirect
12. C22⋊Dic113C22semidirect
15. C22×C4×C113C22semidirect
18. C22⋊C4×C113C22semidirect
55. C113#458semidirect
59. #123D113semidirect
68. C22×C4C2×D113semidirect
74. C22⋊C4C2×D113semidirect