Finite Group Databasebeta

← #20293271

Subgroups — #20293271

Order 6848.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#13133817C211yesmaximal, characteristic, complemented
2#16812096C211yesmaximal, characteristic, complemented
3#16812090C212yesmaximal, complemented
4#16812088C211yesmaximal, characteristic, complemented
5#16812081C212yesmaximal, complemented
6#16812080C211yesmaximal, characteristic, complemented
7#16812078C212yesmaximal, complemented
8D4×Dic107C211yesmaximal, characteristic, complemented
9C4×C22⋊D107C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic107C212yesmaximal, complemented
11C22×D4×C107C211yesmaximal, characteristic, complemented
12#13133817—24no—
13C2×Dic214C2211yescharacteristic, complemented
14C2×C22⋊D107C2211yescharacteristic
15C2×C22⋊D107C2212yescomplemented
16C2×C22⋊D107C2212yescomplemented
17C22×Dic107C2212yescomplemented
18C22×Dic107C2212yescomplemented
19C2×C4×D107C2211yescharacteristic, complemented
20C4⋊Dic107C2211yescharacteristic, complemented
21Dic107⋊C4C2211yescharacteristic, complemented
22Dic107⋊C4C2212yescomplemented
23C22⋊Dic107C2211yescharacteristic
24C22⋊Dic107C2212yes—
25C22⋊Dic107C2212yescomplemented
26C22⋊Dic107C2212yescomplemented
27C22⋊Dic107C2212yescomplemented
28C22⋊Dic107C2212yescomplemented
29(C2×D107)⋊C4C2211yescharacteristic, complemented
30C4×Dic107C2211yescharacteristic, complemented
31C24×C107C2212yes—
32C2×D4×C107—24no—
33C2×D4×C107C2211yescharacteristic
34C2×D4×C107C2212yes—
35C2×D4×C107C2211yescharacteristic
36C22×C4×C107C2211yescharacteristic
37C23×C107—24no—
38C23×C107—24no—
39C23×C107—42no—
40C23×C107C2312yes—
41C23×C107C2312yes—
42C23×C107C2311yescharacteristic
43C22×D107C2311yescharacteristic
44D4×C107—24no—
45D4×C107—22no—
46D4×C107D414yescomplemented
47C2×C4×C107—22no—
48C2×C4×C107C2311yescharacteristic
49C2×C4×C107C2311yescharacteristic
50C22⋊D107—24no—
51C22⋊D107—22no—
52C22⋊D107—42no—
53C2×Dic107—24no—
54C2×Dic107—42no—
55C2×Dic107C2311yescharacteristic
56C2×Dic107C2311yescharacteristic
57C2×Dic107C2312yes—
58C2×Dic107C2311yescharacteristic
59C2×Dic107C2312yes—
60C4×D107—22no—
61Dic214—22no—
62C22×C107—21no—
63C22×C107—22no—
64C22×C107—22no—
65C22×C107—21no—
66C22×C107—22no—
67C22×C107—22no—
68C22×C107—22no—
69C22×C107—48no—
70C22×C107C2×D414yes—
71C22×C107#4112yescomplemented
72C22×C107C2411yescharacteristic
73C2×D107—21no—
74C2×D107—21no—
75C2×D107—21no—
76C4×C107—21no—
77C4×C107C2×D412yes—
78Dic107—22no—
79Dic107—21no—
80Dic107—21no—
81Dic107—21no—
82Dic107—22no—
83C2×C107—22no—
84C2×C107—24no—
85C2×C107—42no—
86C2×C107C22×D411yescharacteristic
87C2×C107C2×#4111yescharacteristic
88C2×C107H(5,𝔽2)11yescharacteristic
89D107—41no—
90C107#54611yescharacteristic, complemented, Sylow
91#546—1071nomaximal, Sylow
92C2×#41—1071no—
93C22×D4D10711yescharacteristic, complemented
94#124—1071no—
95#123—1072no—
96C22⋊Q8—1071no—
97#121—1071no—
98#121—1072no—
99#120—1072no—
100C4×D4—1071no—
101C4×D4—1071no—
102C2×C22⋊C4—1072no—
103C24C2×D10712yes—
104#41—2144no—
105C2×Q8—1071no—
106C2×D4—24no—
107C2×D4—1072no—
108C2×D4—1071no—
109C2×D4—1072no—
110C2×D4C2×D10712yes—
111C2×D4C2×D10711yescharacteristic
112C2×D4C2×D10711yescharacteristic
113C22×C4—1072no—
114C22×C4—1072no—
115C22×C4—1071no—
116C22×C4C2×D10711yescharacteristic
117C4⋊C4—1071no—
118C4⋊C4—1071no—
119C4⋊C4—1072no—
120C22⋊C4—1071no—
121C22⋊C4—1072no—
122C22⋊C4—1072no—
123C22⋊C4—1072no—
124C22⋊C4—1072no—
125C22⋊C4—1072no—
126C22⋊C4—1071no—
127C42—1071no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—1071no—
132C23C22×D10712yes—
133C23C22×D10712yes—
134C23C22×D10711yescharacteristic
135Q8—2142no—
136D4—24no—
137D4—22no—
138D4—4282no—
139D4—2144no—
140D4—2142no—
141D4C22⋊D10714yescomplemented
142C2×C4—22no—
143C2×C4—1071no—
144C2×C4—1071no—
145C2×C4—1072no—
146C2×C4—1071no—
147C2×C4—1072no—
148C2×C4—4282no—
149C2×C4—2144no—
150C2×C4—2142no—
151C2×C4C22×D10711yescharacteristic
152C2×C4C22×D10711yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—2141no—
162C22—2141no—
163C22—2141no—
164C22C2×C22⋊D10714yes—
165C22#1313381712yescomplemented
166C22C23×D10711yescharacteristic
167C4—21no—
168C4—2141no—
169C4—2142no—
170C4—2141no—
171C4—2141no—
172C4—2142no—
173C4C2×C22⋊D10712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—4281no—
178C2C22×C22⋊D10711yescharacteristic
179C2C2×#1313381711yescharacteristic
180C2#1681215411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
FittingC22×D4×C107C2
hypercenterC22×D4D107
hypocenterC107#546
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133817C2semidirect
2. #16812096C2semidirect
3. #16812090C2semidirect
4. #16812088C2semidirect
5. #16812081C2semidirect
6. #16812080C2semidirect
7. #16812078C2semidirect
8. D4×Dic107C2semidirect
9. C4×C22⋊D107C2semidirect
10. C2×C22⋊Dic107C2semidirect
11. C22×D4×C107C2semidirect
13. C2×Dic214C22semidirect
15. C2×C22⋊D107C22semidirect
16. C2×C22⋊D107C22semidirect
17. C22×Dic107C22semidirect
18. C22×Dic107C22semidirect
19. C2×C4×D107C22semidirect
20. C4⋊Dic107C22semidirect
21. Dic107⋊C4C22semidirect
22. Dic107⋊C4C22semidirect
25. C22⋊Dic107C22semidirect
26. C22⋊Dic107C22semidirect
27. C22⋊Dic107C22semidirect
28. C22⋊Dic107C22semidirect
29. (C2×D107)⋊C4C22semidirect
30. C4×Dic107C22semidirect
46. D4×C107D4semidirect
71. C22×C107#41semidirect
90. C107#546semidirect
93. C22×D4D107semidirect
141. D4C22⋊D107semidirect
165. C22#13133817semidirect