Finite Group Databasebeta

← #20298786

Subgroups — #20298786

Order 7232.

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#16813667C211yesmaximal, characteristic, complemented
2#16813592C212yesmaximal, complemented
3#16813590C211yesmaximal, characteristic, complemented
4#16813536C212yesmaximal, complemented
5#99×C113C211yesmaximal, characteristic, complemented
6C23×D113C2211yescharacteristic, complemented
7#14268647—21no—
8#14268646—22no—
9C2×D452—22no—
10C2×C22⋊D113—21no—
11C2×C22⋊D113—21no—
12C2×C22⋊D113C2212yes—
13D4×D113—22no—
14C22⋊Dic113C2211yescharacteristic, complemented
15#14268624—21no—
16#14268624—22no—
17C2×D4×C113C2211yescharacteristic
18C22⋊C4×C113C2212yescomplemented
19C23×C113D411yescharacteristic, complemented
20C23×C113C2311yescharacteristic
21C22×D113—21no—
22C22×D113—22no—
23C22×D113—21no—
24C22×D113—42no—
25C22×D113D412yescomplemented
26D4×C113—22no—
27C2×C4×C113—22no—
28C2×C4×C113D411yescharacteristic, complemented
29C22⋊D113—22no—
30C22⋊D113—22no—
31C22⋊D113—22no—
32C22⋊D113—42no—
33C2×Dic113—21no—
34C2×Dic113—21no—
35C2×Dic113D412yescomplemented
36D452—21no—
37D452—44no—
38C4×D113—22no—
39Dic226—21no—
40C22×C113—21no—
41C22×C113—41no—
42C22×C113—41no—
43C22×C113C2×D412yes—
44C22×C113C2×D411yescharacteristic
45C2×D113—24no—
46C2×D113—22no—
47C2×D113—22no—
48C2×D113—41no—
49C2×D113—42no—
50C2×D113—41no—
51C2×D113—42no—
52C2×D113—81no—
53C4×C113—21no—
54C4×C113—42no—
55Dic113—22no—
56Dic113—41no—
57C2×C113—21no—
58C2×C113—22no—
59C2×C113—41no—
60C2×C113#12011yescharacteristic
61D113—42no—
62D113—42no—
63C113B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1131nomaximal, Sylow
65H(5,𝔽2)—1131no—
66#120—1131no—
67#120—1132no—
68#99—1132no—
69#99D11311yescharacteristic, complemented
70C24—1131no—
71#41—2262no—
72#41—2261no—
73C2×D4—1132no—
74C2×D4—2262no—
75C2×D4—2262no—
76C2×D4—2261no—
77C2×D4—2261no—
78C2×D4C2×D11311yescharacteristic
79C22⋊C4—1131no—
80C22⋊C4—2262no—
81C22⋊C4—2261no—
82C22⋊C4C2×D11312yescomplemented
83C23—1132no—
84C23—4522no—
85C23—2261no—
86C23—2262no—
87C23—2261no—
88C23D45211yescharacteristic, complemented
89C23C22×D11311yescharacteristic
90Q8—2261no—
91D4—22no—
92D4—4524no—
93D4—4522no—
94D4—2261no—
95D4—2262no—
96D4—2262no—
97D4—2262no—
98C2×C4—22no—
99C2×C4—1132no—
100C2×C4—2262no—
101C2×C4—2261no—
102C2×C4—2261no—
103C2×C4D45211yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—9041no—
108C22—4522no—
109C22—4522no—
110C22—4521no—
111C22—4521no—
112C22—2264no—
113C22—2262no—
114C22—2262no—
115C22D4×D11312yes—
116C22C2×D45211yescharacteristic
117C4—21no—
118C4—42no—
119C4—4521no—
120C4—2262no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—4522no—
125C2—4522no—
126C2#1681359211yescharacteristic
subgroupG/H
centerC2#16813592
commutatorC23×C113C23
FrattiniC23C22×D113
Fitting#99×C113C2
socleC2×C113#120
hypercenter#99D113
hypocenterC113B(4,𝔽2)
LCSG ▷ C23×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #16813667C2semidirect
2. #16813592C2semidirect
3. #16813590C2semidirect
4. #16813536C2semidirect
5. #99×C113C2semidirect
6. C23×D113C22semidirect
14. C22⋊Dic113C22semidirect
18. C22⋊C4×C113C22semidirect
19. C23×C113D4semidirect
25. C22×D113D4semidirect
28. C2×C4×C113D4semidirect
35. C2×Dic113D4semidirect
63. C113B(4,𝔽2)semidirect
69. #99D113semidirect
82. C22⋊C4C2×D113semidirect
88. C23D452semidirect
103. C2×C4D452semidirect