Finite Group Databasebeta

← #20292803

Subgroups — #20292803

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#16812140C211yesmaximal, characteristic, complemented
3#16812098C211yesmaximal, characteristic, complemented
4#16812036C211yesmaximal, characteristic, complemented
5#16812035C211yesmaximal, characteristic, complemented
6#16812025C211yesmaximal, characteristic, complemented
7#137×C107C211yesmaximal, characteristic, complemented
8#13133819—21no—
9#13133818—21no—
10#13133818C2211yescharacteristic
11C2×D428—21no—
12C2×D428C2211yescharacteristic
13D4×D107—22no—
14D4×D107—21no—
15#13133808—21no—
16#13133805—21no—
17#13133802C2211yescharacteristic, complemented
18(C2×D107)⋊C4—22no—
19C4×Dic107C2211yescharacteristic, complemented
20#41×C107C2211yescharacteristic
21C2×Q8×C107C2211yescharacteristic, complemented
22Q16×C107—21no—
23SD16×C107—21no—
24M16×C107C2211yescharacteristic, complemented
25C22×D107—22no—
26C22×D107D412yes—
27Q8×C107—21no—
28Q8×C107—21no—
29Q8×C107D411yescharacteristic
30D4×C107—21no—
31D4×C107D411yescharacteristic
32C2×C4×C107—21no—
33C2×C4×C107—21no—
34C2×C4×C107C2311yescharacteristic
35C22⋊D107—22no—
36C22⋊D107—21no—
37C2×Dic107—21no—
38D428—22no—
39D428—21no—
40D428—21no—
41D428D411yescharacteristic
42C4×D107—22no—
43C4×D107—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#137D10711yescharacteristic, complemented
69#124—1071no—
70#104—1071no—
71#104—1071no—
72#100—1071no—
73#41—1071no—
74#41—2141no—
75#41—2141no—
76#41C2×D10711yescharacteristic
77C2×Q8C2×D10711yescharacteristic, complemented
78C2×D4—1071no—
79C2×D4—2141no—
80C2×D4—2142no—
81C2×D4—2141no—
82Q16—21no—
83Q16—2141no—
84SD16—21no—
85SD16—2141no—
86M16—1071no—
87M16C2×D10711yescharacteristic, complemented
88C22⋊C4—2142no—
89C42—1071no—
90C23—1072no—
91C23—2142no—
92Q8—21no—
93Q8—21no—
94Q8—1071no—
95Q8C22⋊D10711yescharacteristic
96D4—21no—
97D4—1071no—
98D4—2142no—
99D4—2142no—
100D4—2141no—
101D4—2141no—
102D4—2141no—
103D4C22⋊D10711yescharacteristic
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×C22⋊D10711yescharacteristic
124C2—21no—
125C2—41no—
126C2—4281no—
127C2—4282no—
128C2#1681207711yescharacteristic
subgroupG/H
centerC2#16812077
commutatorC2×C4×C107C23
FrattiniC2×C4C22×D107
Fitting#137×C107C2
socleC2×C107#120
hypercenter#137D107
hypocenterC107#455
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. #16812098C2semidirect
4. #16812036C2semidirect
5. #16812035C2semidirect
6. #16812025C2semidirect
7. #137×C107C2semidirect
17. #13133802C22semidirect
19. C4×Dic107C22semidirect
21. C2×Q8×C107C22semidirect
24. M16×C107C22semidirect
64. C107#455semidirect
68. #137D107semidirect
77. C2×Q8C2×D107semidirect
87. M16C2×D107semidirect