Finite Group Databasebeta

← #20292808

Subgroups — #20292808

Order 6848.

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

normal subgroups only

This group has 142 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#16812156C211yesmaximal, characteristic
2#16812140C211yesmaximal, characteristic, complemented
3#16812107C211yesmaximal, characteristic, complemented
4#16812036C211yesmaximal, characteristic, complemented
5#16812035C211yesmaximal, characteristic, complemented
6#16812027C211yesmaximal, characteristic, complemented
7#137×C107C211yesmaximal, characteristic, complemented
8#13133818—21no—
9#13133818C2211yescharacteristic
10#13133817—21no—
11#13133817—22no—
12C2×Dic214—21no—
13C2×Dic214C2211yescharacteristic
14Q8×D107—21no—
15#13133808—21no—
16#13133805—21no—
17#13133802C2211yescharacteristic
18Dic107⋊C4—22no—
19C4×Dic107C2211yescharacteristic
20#41×C107C2211yescharacteristic
21C2×Q8×C107C2211yescharacteristic
22Q16×C107—21no—
23SD16×C107—21no—
24M16×C107C2211yescharacteristic
25Q8×C107—21no—
26Q8×C107—21no—
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—22no—
36C2×Dic107—21no—
37C2×Dic107D412yes—
38D428D411yescharacteristic
39C4×D107—22no—
40C4×D107—21no—
41Dic214—22no—
42Dic214—21no—
43Dic214—21no—
44Dic214D411yescharacteristic
45C8×C107—21no—
46C107⋊C8—21no—
47C22×C107—21no—
48C22×C107C2×D411yescharacteristic
49C2×D107—21no—
50C4×C107—21no—
51C4×C107—41no—
52C4×C107C2×D411yescharacteristic
53C4×C107C2×D411yescharacteristic
54Dic107—22no—
55Dic107—22no—
56Dic107—21no—
57C2×C107—21no—
58C2×C107—41no—
59C2×C107#12011yescharacteristic
60D107—41no—
61C107#45611yescharacteristic, complemented, Sylow
62#456—1071nomaximal, Sylow
632−1+4—1071no—
64#137—1071no—
65#137D10711yescharacteristic, complemented
66C4⋊Q8—1071no—
67#104—1071no—
68#104—1071no—
69#101—1071no—
70#41—1071no—
71#41—2141no—
72#41—2142no—
73#41—2141no—
74#41C2×D10711yescharacteristic
75C2×Q8—1071no—
76C2×Q8—2141no—
77C2×Q8—2141no—
78C2×Q8C2×D10711yescharacteristic
79Q16—21no—
80Q16—2141no—
81SD16—21no—
82SD16—2141no—
83M16—1071no—
84M16C2×D10711yescharacteristic
85C4⋊C4—2142no—
86C42—1071no—
87Q8—21no—
88Q8—21no—
89Q8—1071no—
90Q8—2142no—
91Q8—2141no—
92Q8—2141no—
93Q8C22⋊D10711yescharacteristic
94D4—21no—
95D4—1071no—
96D4—2142no—
97D4—2141no—
98D4C22⋊D10711yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—1072no—
102C2×C4—2142no—
103C2×C4—2142no—
104C2×C4—2141no—
105C2×C4—2141no—
106C2×C4C22×D10711yescharacteristic
107C8—21no—
108C8—2141no—
109C22—21no—
110C22—2141no—
111C22D4×D10711yescharacteristic
112C4—21no—
113C4—41no—
114C4—2142no—
115C4—2142no—
116C4—2141no—
117C4D4×D10711yescharacteristic
118C4C2×C22⋊D10711yescharacteristic
119C2—21no—
120C2—41no—
121C2—4281no—
122C2#1681207711yescharacteristic
subgroupG/H
centerC2#16812077
commutatorC2×C4×C107C23
FrattiniC2×C4C22×D107
Fitting#137×C107C2
socleC2×C107#120
hypercenter#137D107
hypocenterC107#456
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #16812140C2semidirect
3. #16812107C2semidirect
4. #16812036C2semidirect
5. #16812035C2semidirect
6. #16812027C2semidirect
7. #137×C107C2semidirect
61. C107#456semidirect
65. #137D107semidirect