Finite Group Databasebeta

← #20292816

Subgroups — #20292816

Order 6848.

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

normal subgroups only

This group has 160 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#16812154C211yesmaximal, characteristic
2#16812092C212yesmaximal, complemented
3#16812024C211yesmaximal, characteristic, complemented
4#16812023C212yesmaximal, complemented
5#123×C107C211yesmaximal, characteristic, complemented
6#13133818—21no—
7#13133817—22no—
8C2×C22⋊D107—21no—
9C2×C22⋊D107C2212yes—
10D4×D107—22no—
11Dic107⋊C4—22no—
12C22⋊Dic107C2212yescomplemented
13(C2×D107)⋊C4—22no—
14C2×D4×C107C2211yescharacteristic
15C22×C4×C107C2211yescharacteristic, complemented
16C4⋊C4×C107—21no—
17C22⋊C4×C107—21no—
18C22⋊C4×C107C2211yescharacteristic, complemented
19C23×C107D411yescharacteristic
20C23×C107C2311yescharacteristic
21C22×D107—21no—
22C22×D107D412yes—
23D4×C107—22no—
24C2×C4×C107—21no—
25C2×C4×C107—21no—
26C2×C4×C107—22no—
27C2×C4×C107D411yescharacteristic
28C22⋊D107—22no—
29C22⋊D107—22no—
30C22⋊D107—22no—
31C2×Dic107—21no—
32C2×Dic107—22no—
33C2×Dic107D412yes—
34D428—21no—
35C4×D107—22no—
36Dic214—21no—
37C22×C107—21no—
38C22×C107—41no—
39C22×C107—41no—
40C22×C107C2×D412yes—
41C22×C107C2×D411yescharacteristic
42C2×D107—22no—
43C2×D107—42no—
44C2×D107—81no—
45C4×C107—21no—
46C4×C107—41no—
47C4×C107—41no—
48Dic107—22no—
49Dic107—42no—
50C2×C107—22no—
51C2×C107—21no—
52C2×C107—41no—
53C2×C107#12011yescharacteristic
54D107—42no—
55C107#45811yescharacteristic, complemented, Sylow
56#458—1071nomaximal, Sylow
57H(5,𝔽2)—1071no—
58#123—1072no—
59#123D10711yescharacteristic, complemented
60#99—1071no—
61#99—1072no—
62#41—2142no—
63#41—2141no—
64C2×D4—1072no—
65C2×D4—2142no—
66C2×D4—2141no—
67C2×D4C2×D10711yescharacteristic
68C22×C4C2×D10711yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—2142no—
71C22⋊C4—21no—
72C22⋊C4—1072no—
73C22⋊C4—2142no—
74C22⋊C4C2×D10711yescharacteristic, complemented
75C23—1072no—
76C23—2141no—
77C23C22⋊D10711yescharacteristic
78C23C22×D10711yescharacteristic
79Q8—2141no—
80D4—22no—
81D4—2141no—
82D4—2142no—
83D4—2142no—
84D4—2142no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—1072no—
89C2×C4—2142no—
90C2×C4—2142no—
91C2×C4—2141no—
92C2×C4C22⋊D10711yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—8561no—
97C22—4282no—
98C22—2142no—
99C22D4×D10712yes—
100C22C2×C22⋊D10711yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—4282no—
105C4—2142no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—4282no—
110C2#1681207711yescharacteristic
subgroupG/H
centerC2#16812077
commutatorC23×C107C23
FrattiniC23C22×D107
Fitting#123×C107C2
socleC2×C107#120
hypercenter#123D107
hypocenterC107#458
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #16812092C2semidirect
3. #16812024C2semidirect
4. #16812023C2semidirect
5. #123×C107C2semidirect
12. C22⋊Dic107C22semidirect
15. C22×C4×C107C22semidirect
18. C22⋊C4×C107C22semidirect
55. C107#458semidirect
59. #123D107semidirect
68. C22×C4C2×D107semidirect
74. C22⋊C4C2×D107semidirect