Finite Group Databasebeta

← #20292813

Subgroups — #20292813

Order 6848.

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#16812154C211yesmaximal, characteristic, complemented
2#16812078C212yesmaximal, complemented
3#16812024C211yesmaximal, characteristic, complemented
4#16812023C212yesmaximal, complemented
5#120×C107C211yesmaximal, characteristic, complemented
6#13133818—21no—
7#13133817—22no—
8C2×C22⋊D107—21no—
9C2×C22⋊D107—22no—
10C2×C22⋊D107C2212yes—
11D4×D107—22no—
12C22⋊Dic107—22no—
13C22⋊Dic107C2212yescomplemented
14C24×C107C2211yescharacteristic, complemented
15C2×D4×C107—21no—
16C2×D4×C107C2211yescharacteristic
17C22⋊C4×C107—21no—
18C22⋊C4×C107C2211yescharacteristic, complemented
19C23×C107—21no—
20C23×C107—22no—
21C23×C107—42no—
22C23×C107D411yescharacteristic, complemented
23C23×C107C2311yescharacteristic
24C22×D107—21no—
25C22×D107D412yescomplemented
26D4×C107—22no—
27D4×C107—42no—
28C2×C4×C107—21no—
29C2×C4×C107D411yescharacteristic, complemented
30C22⋊D107—22no—
31C22⋊D107—22no—
32C22⋊D107—22no—
33C22⋊D107—44no—
34C2×Dic107—21no—
35C2×Dic107—22no—
36C2×Dic107D412yescomplemented
37D428—21no—
38C4×D107—22no—
39Dic214—21no—
40C22×C107—24no—
41C22×C107—22no—
42C22×C107—21no—
43C22×C107—41no—
44C22×C107—41no—
45C22×C107—42no—
46C22×C107—41no—
47C22×C107—41no—
48C22×C107C2×D412yes—
49C22×C107C2×D411yescharacteristic
50C2×D107—22no—
51C2×D107—42no—
52C2×D107—81no—
53C4×C107—21no—
54C4×C107—41no—
55Dic107—22no—
56Dic107—42no—
57C2×C107—22no—
58C2×C107—21no—
59C2×C107—41no—
60C2×C107—42no—
61C2×C107#12011yescharacteristic
62D107—42no—
63C107B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1071nomaximal, Sylow
65H(5,𝔽2)—1071no—
66#120—1072no—
67#120D10711yescharacteristic, complemented
68#99—1071no—
69#99—1072no—
70C24C2×D10711yescharacteristic, complemented
71#41—2142no—
72#41—2141no—
73C2×D4—21no—
74C2×D4—1072no—
75C2×D4—2142no—
76C2×D4—2142no—
77C2×D4—2141no—
78C2×D4C2×D10711yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—1072no—
81C22⋊C4—2142no—
82C22⋊C4C2×D10711yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—1072no—
87C23—2141no—
88C23C22⋊D10711yescharacteristic, complemented
89C23C22×D10711yescharacteristic
90Q8—2141no—
91D4—22no—
92D4—42no—
93D4—4284no—
94D4—2141no—
95D4—2142no—
96D4—2142no—
97D4—2142no—
98C2×C4—21no—
99C2×C4—1072no—
100C2×C4—2142no—
101C2×C4—2142no—
102C2×C4—2141no—
103C2×C4C22⋊D10711yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—8561no—
113C22—4282no—
114C22—2142no—
115C22D4×D10712yes—
116C22C2×C22⋊D10711yescharacteristic
117C4—21no—
118C4—41no—
119C4—4282no—
120C4—2142no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—4282no—
126C2#1681207711yescharacteristic
subgroupG/H
centerC2#16812077
commutatorC23×C107C23
FrattiniC23C22×D107
Fitting#120×C107C2
socleC2×C107#120
hypercenter#120D107
hypocenterC107B(4,𝔽2)
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #16812154C2semidirect
2. #16812078C2semidirect
3. #16812024C2semidirect
4. #16812023C2semidirect
5. #120×C107C2semidirect
13. C22⋊Dic107C22semidirect
14. C24×C107C22semidirect
18. C22⋊C4×C107C22semidirect
22. C23×C107D4semidirect
25. C22×D107D4semidirect
29. C2×C4×C107D4semidirect
36. C2×Dic107D4semidirect
63. C107B(4,𝔽2)semidirect
67. #120D107semidirect
70. C24C2×D107semidirect
82. C22⋊C4C2×D107semidirect
88. C23C22⋊D107semidirect
103. C2×C4C22⋊D107semidirect