Finite Group Databasebeta

← #20292806

Subgroups — #20292806

Order 6848.

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

normal subgroups only

This group has 152 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#16812155C211yesmaximal, characteristic
2#16812142C211yesmaximal, characteristic, complemented
3#16812139C211yesmaximal, characteristic, complemented
4#16812097C211yesmaximal, characteristic, complemented
5#16812037C211yesmaximal, characteristic, complemented
6#16812025C211yesmaximal, characteristic, complemented
7#104×C107C211yesmaximal, characteristic, complemented
8#13133819—21no—
9#13133818—21no—
10#13133818C2211yescharacteristic
11C2×Dic214C2211yescharacteristic, complemented
12C2×D428—21no—
13C2×D428C2211yescharacteristic
14D4×D107—22no—
15D4×D107—21no—
16Dic428—21no—
17#13133808—21no—
18#13133807—21no—
19#13133806—21no—
20#13133802C2211yescharacteristic, complemented
21(C2×D107)⋊C4—22no—
22#41×C107C2211yescharacteristic
23M16×C107C2211yescharacteristic, complemented
24C42×C107C2211yescharacteristic, complemented
25C22×D107—22no—
26C22×D107D412yes—
27Q8×C107D411yescharacteristic
28D4×C107—21no—
29D4×C107D411yescharacteristic
30C2×C4×C107—21no—
31C2×C4×C107—21no—
32C2×C4×C107C2311yescharacteristic
33C22⋊D107—22no—
34C22⋊D107—21no—
35C2×Dic107—21no—
36D428—22no—
37D428—21no—
38D428—21no—
39D428D411yescharacteristic
40C4×D107—22no—
41C4×D107—21no—
42Dic214—21no—
43Dic214—21no—
44Dic214D411yescharacteristic
45C8×C107—21no—
46C107⋊C8—21no—
47C22×C107—21no—
48C22×C107C2×D411yescharacteristic
49C2×D107—22no—
50C2×D107—21no—
51C2×D107—42no—
52C2×D107—82no—
53C4×C107—21no—
54C4×C107—41no—
55C4×C107C2×D411yescharacteristic
56C4×C107C2×D411yescharacteristic
57Dic107—21no—
58Dic107—41no—
59C2×C107—21no—
60C2×C107—41no—
61C2×C107#12011yescharacteristic
62D107—42no—
63D107—41no—
64C107#45511yescharacteristic, complemented, Sylow
65#455—1071nomaximal, Sylow
66H(5,𝔽2)—1071no—
67#137—1071no—
68#137—1071no—
69#124—1071no—
70#104—1071no—
71#104D10711yescharacteristic, complemented
72#100—1071no—
73#41—1071no—
74#41—2141no—
75#41—2141no—
76#41C2×D10711yescharacteristic
77C2×Q8—1071no—
78C2×D4—1071no—
79C2×D4—2141no—
80C2×D4—2142no—
81C2×D4—2141no—
82Q16—2141no—
83Q16—2141no—
84SD16—2141no—
85SD16—2141no—
86M16—1071no—
87M16C2×D10711yescharacteristic, complemented
88C22⋊C4—2142no—
89C42C2×D10711yescharacteristic, complemented
90C23—1072no—
91C23—2142no—
92Q8—1071no—
93Q8—2141no—
94Q8—2141no—
95Q8D42811yescharacteristic
96D4—21no—
97D4—1071no—
98D4—2142no—
99D4—2142no—
100D4—2141no—
101D4—2141no—
102D4—2141no—
103D4D42811yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—2142no—
107C2×C4—2141no—
108C2×C4—2141no—
109C2×C4C22×D10711yescharacteristic
110C8—21no—
111C8—2141no—
112C22—21no—
113C22—8562no—
114C22—4282no—
115C22—2141no—
116C22—2142no—
117C22D4×D10711yescharacteristic
118C4—21no—
119C4—41no—
120C4—4281no—
121C4—2141no—
122C4D4×D10711yescharacteristic
123C4C2×D42811yescharacteristic
124C2—21no—
125C2—41no—
126C2—4281no—
127C2—4282no—
128C2#1681207911yescharacteristic
subgroupG/H
centerC2#16812079
commutatorC2×C4×C107C23
FrattiniC2×C4C22×D107
Fitting#104×C107C2
socleC2×C107#120
hypercenter#104D107
hypocenterC107#455
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #16812142C2semidirect
3. #16812139C2semidirect
4. #16812097C2semidirect
5. #16812037C2semidirect
6. #16812025C2semidirect
7. #104×C107C2semidirect
11. C2×Dic214C22semidirect
20. #13133802C22semidirect
23. M16×C107C22semidirect
24. C42×C107C22semidirect
64. C107#455semidirect
71. #104D107semidirect
87. M16C2×D107semidirect
89. C42C2×D107semidirect