Finite Group Databasebeta

← #20292793

Subgroups — #20292793

Order 6848.

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

normal subgroups only

This group has 168 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, complemented
2#16812135C211yesmaximal, characteristic, complemented
3#16812105C211yesmaximal, characteristic, complemented
4#16812036C211yesmaximal, characteristic, complemented
5#16812035C211yesmaximal, characteristic, complemented
6#16812025C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
8#13133819—21no—
9#13133818—21no—
10#13133818C2211yescharacteristic
11C2×D428—21no—
12C2×D428C2211yescharacteristic
13C2×C22⋊D107—22no—
14D4×D107—22no—
15D4×D107—21no—
16#13133806—21no—
17#13133803—21no—
18#13133802C2211yescharacteristic, complemented
19C4×Dic107C2211yescharacteristic, complemented
20#41×C107C2211yescharacteristic
21C2×D4×C107C2211yescharacteristic, complemented
22SD16×C107—21no—
23D8×C107—21no—
24M16×C107C2211yescharacteristic, complemented
25C23×C107—21no—
26C22×D107—22no—
27C22×D107D412yes—
28Q8×C107D411yescharacteristic, complemented
29D4×C107—21no—
30D4×C107—21no—
31D4×C107—21no—
32D4×C107D411yescharacteristic, complemented
33C2×C4×C107—21no—
34C2×C4×C107C2311yescharacteristic
35C22⋊D107—22no—
36C22⋊D107—21no—
37C22⋊D107—44no—
38C2×Dic107—21no—
39D428—22no—
40D428—21no—
41D428—21no—
42D428D411yescharacteristic, complemented
43C4×D107—22no—
44C4×D107—21no—
45Dic214D411yescharacteristic, complemented
46C8×C107—21no—
47C107⋊C8—21no—
48C22×C107—21no—
49C22×C107—42no—
50C22×C107—41no—
51C22×C107C2×D411yescharacteristic
52C2×D107—22no—
53C2×D107—21no—
54C2×D107—42no—
55C2×D107—82no—
56C4×C107—21no—
57C4×C107C2×D411yescharacteristic
58C4×C107C2×D411yescharacteristic
59Dic107—22no—
60Dic107—21no—
61C2×C107—21no—
62C2×C107—41no—
63C2×C107—81no—
64C2×C107#12011yescharacteristic
65D107—42no—
66D107—41no—
67C107#45311yescharacteristic, complemented, Sylow
68#453—1071nomaximal, Sylow
69H(5,𝔽2)—1071no—
70AGL(1,ℤ/8)—1071no—
71AGL(1,ℤ/8)D10711yescharacteristic, complemented
72#127—1071no—
73#104—1071no—
74#104—1071no—
75#100—1071no—
76#41—1071no—
77#41—2141no—
78#41—2141no—
79#41C2×D10711yescharacteristic
80C2×D4—1071no—
81C2×D4—2141no—
82C2×D4—2142no—
83C2×D4—2142no—
84C2×D4—2141no—
85C2×D4C2×D10711yescharacteristic, complemented
86SD16—21no—
87SD16—2141no—
88D8—21no—
89D8—2141no—
90M16—1071no—
91M16C2×D10711yescharacteristic, complemented
92C42—1071no—
93C23—21no—
94C23—1072no—
95C23—2142no—
96Q8—1071no—
97Q8C22⋊D10711yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—1071no—
102D4—4284no—
103D4—2142no—
104D4—2142no—
105D4—2141no—
106D4—2141no—
107D4—2141no—
108D4C22⋊D10711yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—2142no—
111C2×C4—2141no—
112C2×C4—2141no—
113C2×C4C22×D10711yescharacteristic
114C8—21no—
115C8—2141no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—8562no—
120C22—4282no—
121C22—2141no—
122C22—2142no—
123C22D4×D10711yescharacteristic
124C4—21no—
125C4—2142no—
126C4—2141no—
127C4D4×D10711yescharacteristic
128C4C2×C22⋊D10711yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—4281no—
133C2—4282no—
134C2#1681207711yescharacteristic
subgroupG/H
centerC2#16812077
commutatorC2×C4×C107C23
FrattiniC2×C4C22×D107
FittingAGL(1,ℤ/8)×C107C2
socleC2×C107#120
hypercenterAGL(1,ℤ/8)D107
hypocenterC107#453
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16812155C2semidirect
2. #16812135C2semidirect
3. #16812105C2semidirect
4. #16812036C2semidirect
5. #16812035C2semidirect
6. #16812025C2semidirect
7. AGL(1,ℤ/8)×C107C2semidirect
18. #13133802C22semidirect
19. C4×Dic107C22semidirect
21. C2×D4×C107C22semidirect
24. M16×C107C22semidirect
28. Q8×C107D4semidirect
32. D4×C107D4semidirect
42. D428D4semidirect
45. Dic214D4semidirect
67. C107#453semidirect
71. AGL(1,ℤ/8)D107semidirect
85. C2×D4C2×D107semidirect
91. M16C2×D107semidirect
97. Q8C22⋊D107semidirect
108. D4C22⋊D107semidirect