Finite Group Databasebeta

← #20293305

Subgroups — #20293305

Order 6848.

248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 95 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#16812102C212yesmaximal, complemented
2#16812096C211yesmaximal, characteristic, complemented
3#16812091C212yesmaximal, complemented
4#16812087C212yesmaximal, complemented
5#16812077C211yesmaximal, characteristic
6C4×C22⋊D107C212yesmaximal, complemented
7#16812067C212yesmaximal, complemented
8C22⋊C4×D107C212yesmaximal, complemented
9#123×C107C211yesmaximal, characteristic, complemented
10C23×D107C2211yescharacteristic
11C2×Dic214C2211yescharacteristic, complemented
12C2×C22⋊D107C2212yes—
13C2×C4×D107C2212yes—
14C2×C4×D107C2212yes—
15C4⋊Dic107C2212yescomplemented
16Dic107⋊C4C2212yescomplemented
17Dic107⋊C4C2212yescomplemented
18C22⋊Dic107C2212yes—
19C22⋊Dic107C2211yescharacteristic
20C22⋊Dic107C2212yescomplemented
21(C2×D107)⋊C4C2212yes—
22(C2×D107)⋊C4C2212yes—
23(C2×D107)⋊C4C2212yes—
24C4×Dic107C2211yescharacteristic, complemented
25C4×Dic107C2212yescomplemented
26C2×D4×C107C2211yescharacteristic
27C22×C4×C107C2211yescharacteristic, complemented
28C4⋊C4×C107C2212yescomplemented
29C22⋊C4×C107C2212yes—
30C22⋊C4×C107C2211yescharacteristic, complemented
31C23×C107C2311yescharacteristic
32C23×C107C2311yescharacteristic
33C22×D107—24no—
34C22×D107—41no—
35C22×D107C2312yes—
36D4×C107—41no—
37C2×C4×C107—41no—
38C2×C4×C107C2312yes—
39C2×C4×C107C2312yes—
40C2×C4×C107C2311yescharacteristic
41C22⋊D107—24no—
42C2×Dic107C2312yes—
43C2×Dic107C2312yes—
44C2×Dic107C2312yes—
45C4×D107—24no—
46C4×D107—42no—
47Dic214—41no—
48C22×C107—22no—
49C22×C107—22no—
50C22×C107—21no—
51C22×C107—21no—
52C22×C107C2411yescharacteristic
53C2×D107—22no—
54C2×D107—22no—
55C2×D107—44no—
56C2×D107#4114yes—
57C4×C107—22no—
58C4×C107—21no—
59C4×C107—22no—
60Dic107—22no—
61Dic107—22no—
62Dic107—22no—
63C2×C107—41no—
64C2×C107—41no—
65C2×C107C2×#4112yes—
66C2×C107H(5,𝔽2)11yescharacteristic
67D107—24no—
68C107#55111yescharacteristic, complemented, Sylow
69#551—1071nomaximal, Sylow
70#126—1072no—
71#124—1071no—
72#123—1072no—
73#123D10711yescharacteristic, complemented
74C22⋊Q8—1072no—
75#120—1071no—
76C4×D4—1072no—
77#117—1072no—
78C2×C22⋊C4—1072no—
79C24—1071no—
80C2×Q8—1071no—
81C2×D4—1072no—
82C2×D4C2×D10711yescharacteristic
83C22×C4—1072no—
84C22×C4—1072no—
85C22×C4C2×D10711yescharacteristic, complemented
86C4⋊C4—1072no—
87C4⋊C4—1072no—
88C4⋊C4—1072no—
89C4⋊C4C2×D10712yescomplemented
90C22⋊C4—1072no—
91C22⋊C4—1072no—
92C22⋊C4—1071no—
93C22⋊C4—1072no—
94C22⋊C4—1072no—
95C22⋊C4—1072no—
96C22⋊C4C2×D10712yes—
97C22⋊C4C2×D10711yescharacteristic, complemented
98C42—1071no—
99C42—1072no—
100C23—1072no—
101C23—4281no—
102C23—2144no—
103C23C22×D10711yescharacteristic
104C23C22×D10711yescharacteristic
105Q8—4281no—
106D4—41no—
107D4—2144no—
108C2×C4—41no—
109C2×C4—1072no—
110C2×C4—1072no—
111C2×C4—1072no—
112C2×C4—4282no—
113C2×C4—2144no—
114C2×C4C22×D10711yescharacteristic
115C2×C4C22×D10712yes—
116C2×C4C22×D10712yes—
117C22—22no—
118C22—22no—
119C22—21no—
120C22—21no—
121C22—1074no—
122C22—4284no—
123C22—2142no—
124C22—2142no—
125C22C23×D10711yescharacteristic
126C4—22no—
127C4—22no—
128C4—21no—
129C4—2142no—
130C4—2142no—
131C4—2142no—
132C2—41no—
133C2—41no—
134C2—2144no—
135C2#41×D10712yes—
136C2#1681215411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#123×C107C2
hypercenter#123D107
hypocenterC107#551
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #16812102C2semidirect
2. #16812096C2semidirect
3. #16812091C2semidirect
4. #16812087C2semidirect
6. C4×C22⋊D107C2semidirect
7. #16812067C2semidirect
8. C22⋊C4×D107C2semidirect
9. #123×C107C2semidirect
11. C2×Dic214C22semidirect
15. C4⋊Dic107C22semidirect
16. Dic107⋊C4C22semidirect
17. Dic107⋊C4C22semidirect
20. C22⋊Dic107C22semidirect
24. C4×Dic107C22semidirect
25. C4×Dic107C22semidirect
27. C22×C4×C107C22semidirect
28. C4⋊C4×C107C22semidirect
30. C22⋊C4×C107C22semidirect
68. C107#551semidirect
73. #123D107semidirect
85. C22×C4C2×D107semidirect
89. C4⋊C4C2×D107semidirect
97. C22⋊C4C2×D107semidirect