Finite Group Databasebeta

← #20293366

Subgroups — #20293366

Order 6848.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#16812105C211yesmaximal, characteristic, complemented
2#16812102C212yesmaximal, complemented
3#16812091C212yesmaximal, complemented
4#16812090C211yesmaximal, characteristic, complemented
5#16812082C212yesmaximal, complemented
6#16812081C212yesmaximal, complemented
7#16812080C212yesmaximal, complemented
8#16812078C212yesmaximal, complemented
9#120×C107C211yesmaximal, characteristic, complemented
10C2×D428C2211yescharacteristic, complemented
11C2×C22⋊D107C2212yes—
12C2×C22⋊D107C2212yes—
13C2×C22⋊D107C2212yes—
14C2×C22⋊D107C2212yescomplemented
15C22×Dic107C2211yescharacteristic, complemented
16C2×C4×D107C2212yescomplemented
17C4⋊Dic107C2212yescomplemented
18Dic107⋊C4C2212yescomplemented
19Dic107⋊C4C2212yescomplemented
20C22⋊Dic107C2212yes—
21C22⋊Dic107C2212yes—
22C22⋊Dic107C2212yes—
23C22⋊Dic107C2211yescharacteristic, complemented
24(C2×D107)⋊C4C2212yescomplemented
25C4×Dic107C2211yescharacteristic, complemented
26C24×C107C2211yescharacteristic
27C2×D4×C107C2212yes—
28C2×D4×C107C2211yescharacteristic
29C22⋊C4×C107C2212yes—
30C22⋊C4×C107C2211yescharacteristic
31C23×C107—42no—
32C23×C107—41no—
33C23×C107C2312yes—
34C23×C107C2311yescharacteristic
35C23×C107C2311yescharacteristic
36C22×D107C2312yes—
37D4×C107—42no—
38D4×C107—41no—
39C2×C4×C107C2311yescharacteristic
40C2×C4×C107C2312yes—
41C22⋊D107—42no—
42C22⋊D107—42no—
43C22⋊D107—42no—
44C22⋊D107—42no—
45C2×Dic107—41no—
46C2×Dic107C2312yes—
47C2×Dic107C2312yes—
48C2×Dic107C2312yes—
49D428—41no—
50C4×D107—42no—
51C22×C107—21no—
52C22×C107—22no—
53C22×C107—22no—
54C22×C107—22no—
55C22×C107—22no—
56C22×C107—22no—
57C22×C107—21no—
58C22×C107—44no—
59C22×C107C2411yescharacteristic
60C2×D107—22no—
61C2×D107—22no—
62C2×D107—22no—
63C4×C107—22no—
64C4×C107—21no—
65Dic107—22no—
66Dic107—22no—
67Dic107—22no—
68C2×C107—41no—
69C2×C107—42no—
70C2×C107—41no—
71C2×C107H(5,𝔽2)12yes—
72C2×C107H(5,𝔽2)11yescharacteristic
73D107—42no—
74C107#56011yescharacteristic, complemented, Sylow
75#560—1071nomaximal, Sylow
76#127—1071no—
77#126—1072no—
78#123—1071no—
79#123—1072no—
80#121—1072no—
81#121—1072no—
82#121—1072no—
83#120—1072no—
84#120D10711yescharacteristic, complemented
85C24C2×D10711yescharacteristic
86C2×D4—1072no—
87C2×D4—1072no—
88C2×D4—1072no—
89C2×D4—1072no—
90C2×D4—1071no—
91C2×D4C2×D10712yes—
92C2×D4C2×D10711yescharacteristic
93C22×C4—1071no—
94C22×C4—1072no—
95C4⋊C4—1072no—
96C4⋊C4—1072no—
97C4⋊C4—1072no—
98C22⋊C4—1072no—
99C22⋊C4—1072no—
100C22⋊C4—1072no—
101C22⋊C4—1071no—
102C22⋊C4—1072no—
103C22⋊C4C2×D10712yes—
104C22⋊C4C2×D10711yescharacteristic
105C42—1071no—
106C23—42no—
107C23—41no—
108C23—1072no—
109C23C22×D10712yes—
110C23C22×D10711yescharacteristic
111C23C22×D10711yescharacteristic
112D4—42no—
113D4—41no—
114D4—4281no—
115D4—4282no—
116D4—4282no—
117D4—4282no—
118D4—4282no—
119C2×C4—1072no—
120C2×C4—1072no—
121C2×C4—1072no—
122C2×C4—4282no—
123C2×C4—4281no—
124C2×C4C22×D10711yescharacteristic
125C2×C4C22×D10712yes—
126C22—21no—
127C22—22no—
128C22—22no—
129C22—21no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—44no—
134C22—2142no—
135C22—2142no—
136C22—2142no—
137C22C23×D10711yescharacteristic
138C4—22no—
139C4—21no—
140C4—2142no—
141C4—2142no—
142C4—2142no—
143C2—42no—
144C2—41no—
145C2—41no—
146C2—4282no—
147C2#1681215412yes—
148C2#1681215411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#120×C107C2
hypercenter#120D107
hypocenterC107#560
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #16812105C2semidirect
2. #16812102C2semidirect
3. #16812091C2semidirect
4. #16812090C2semidirect
5. #16812082C2semidirect
6. #16812081C2semidirect
7. #16812080C2semidirect
8. #16812078C2semidirect
9. #120×C107C2semidirect
10. C2×D428C22semidirect
14. C2×C22⋊D107C22semidirect
15. C22×Dic107C22semidirect
16. C2×C4×D107C22semidirect
17. C4⋊Dic107C22semidirect
18. Dic107⋊C4C22semidirect
19. Dic107⋊C4C22semidirect
23. C22⋊Dic107C22semidirect
24. (C2×D107)⋊C4C22semidirect
25. C4×Dic107C22semidirect
74. C107#560semidirect
84. #120D107semidirect