Finite Group Databasebeta

← #20293274

Subgroups — #20293274

Order 6848.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13133818C211yesmaximal, characteristic, complemented
2C2×D4×D107C211yesmaximal, characteristic, complemented
3#16812096C211yesmaximal, characteristic, complemented
4#16812091C212yesmaximal, complemented
5#16812089C211yesmaximal, characteristic, complemented
6#16812080C212yesmaximal, complemented
7#16812077C212yesmaximal, complemented
8#16812071C211yesmaximal, characteristic, complemented
9C4×C22⋊D107C211yesmaximal, characteristic, complemented
10C22⋊C4×D107C212yesmaximal, complemented
11#121×C107C211yesmaximal, characteristic, complemented
12C23×D107C2212yes—
13#13133818—24no—
14C2×Dic214C2211yescharacteristic, complemented
15C2×D428C2211yescharacteristic
16C2×C22⋊D107C2211yescharacteristic
17C2×C22⋊D107C2212yes—
18C2×C22⋊D107C2212yescomplemented
19D4×D107—24no—
20C2×C4×D107C2211yescharacteristic
21C2×C4×D107C2212yescomplemented
22C2×C4×D107C2212yescomplemented
23C4⋊Dic107C2212yescomplemented
24Dic107⋊C4C2211yescharacteristic, complemented
25C22⋊Dic107C2212yescomplemented
26C22⋊Dic107C2211yescharacteristic, complemented
27C22⋊Dic107C2212yescomplemented
28(C2×D107)⋊C4C2211yescharacteristic
29(C2×D107)⋊C4C2212yes—
30(C2×D107)⋊C4C2212yescomplemented
31C4×Dic107C2211yescharacteristic, complemented
32C2×D4×C107C2211yescharacteristic
33C2×D4×C107C2212yescomplemented
34C22×C4×C107C2211yescharacteristic, complemented
35C4⋊C4×C107C2211yescharacteristic, complemented
36C22⋊C4×C107C2212yescomplemented
37C23×C107C2312yes—
38C23×C107C2311yescharacteristic
39C22×D107—24no—
40C22×D107—24no—
41C22×D107—42no—
42C22×D107C2312yes—
43C22×D107C2311yescharacteristic
44D4×C107—22no—
45D4×C107—42no—
46C2×C4×C107—22no—
47C2×C4×C107C2311yescharacteristic
48C2×C4×C107C2312yes—
49C2×C4×C107C2311yescharacteristic
50C22⋊D107—24no—
51C22⋊D107—24no—
52C22⋊D107—22no—
53C2×Dic107C2311yescharacteristic
54C2×Dic107C2311yescharacteristic
55C2×Dic107C2312yes—
56C2×Dic107C2311yescharacteristic
57D428D414yescomplemented
58C4×D107—24no—
59C4×D107—22no—
60C4×D107—42no—
61Dic214—22no—
62C22×C107—21no—
63C22×C107—22no—
64C22×C107—21no—
65C22×C107—22no—
66C22×C107—21no—
67C22×C107—22no—
68C22×C107C2411yescharacteristic
69C2×D107—21no—
70C2×D107—22no—
71C2×D107—22no—
72C2×D107—21no—
73C2×D107—48no—
74C2×D107C2×D414yes—
75C2×D107#4112yescomplemented
76C4×C107—21no—
77C4×C107—22no—
78C4×C107C2×D412yes—
79Dic107—22no—
80Dic107—21no—
81Dic107—21no—
82Dic107—21no—
83C2×C107—42no—
84C2×C107—41no—
85C2×C107C22×D411yescharacteristic
86C2×C107C2×#4111yescharacteristic
87C2×C107H(5,𝔽2)11yescharacteristic
88D107—22no—
89D107—24no—
90C107#54611yescharacteristic, complemented, Sylow
91#546—1071nomaximal, Sylow
92C2×#41—1071no—
93C22×D4—1071no—
94#124—1071no—
95#123—1072no—
96C22⋊Q8—1071no—
97#121—1072no—
98#121D10711yescharacteristic, complemented
99#120—1072no—
100C4×D4—1071no—
101C4×D4—1071no—
102C2×C22⋊C4—1072no—
103C24—1072no—
104#41—2144no—
105C2×Q8—1071no—
106C2×D4—1071no—
107C2×D4—1072no—
108C2×D4—1072no—
109C2×D4—1071no—
110C2×D4—2144no—
111C2×D4C2×D10711yescharacteristic
112C2×D4C2×D10712yescomplemented
113C22×C4—1072no—
114C22×C4—1072no—
115C22×C4—1071no—
116C22×C4C2×D10711yescharacteristic, complemented
117C4⋊C4—1071no—
118C4⋊C4—1072no—
119C4⋊C4C2×D10711yescharacteristic, complemented
120C22⋊C4—1072no—
121C22⋊C4—1071no—
122C22⋊C4—1072no—
123C22⋊C4—1071no—
124C22⋊C4—1072no—
125C22⋊C4—1072no—
126C22⋊C4C2×D10712yescomplemented
127C42—1071no—
128C23—1071no—
129C23—1072no—
130C23—4282no—
131C23—2144no—
132C23—2144no—
133C23C22×D10712yes—
134C23C22×D10711yescharacteristic
135Q8—2142no—
136D4—22no—
137D4—42no—
138D4—1074no—
139D4—2144no—
140D4—2144no—
141D4—2142no—
142C2×C4—22no—
143C2×C4—1071no—
144C2×C4—1071no—
145C2×C4—1072no—
146C2×C4—1071no—
147C2×C4—4282no—
148C2×C4—2144no—
149C2×C4—2142no—
150C2×C4C22×D10712yes—
151C2×C4C22×D10711yescharacteristic
152C2×C4C22×D10711yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—1074no—
160C22—1072no—
161C22—4288no—
162C22—2141no—
163C22—2142no—
164C22—2142no—
165C22—2141no—
166C22C23×D10711yescharacteristic
167C4—21no—
168C4—22no—
169C4—2141no—
170C4—2142no—
171C4—2141no—
172C4—2141no—
173C4D4×D10712yes—
174C2—42no—
175C2—41no—
176C2—2142no—
177C2—2144no—
178C2C2×D4×D10711yescharacteristic
179C2#41×D10711yescharacteristic
180C2#1681215411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#121×C107C2
hypercenter#121D107
hypocenterC107#546
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#13133818C2semidirect
2. C2×D4×D107C2semidirect
3. #16812096C2semidirect
4. #16812091C2semidirect
5. #16812089C2semidirect
6. #16812080C2semidirect
7. #16812077C2semidirect
8. #16812071C2semidirect
9. C4×C22⋊D107C2semidirect
10. C22⋊C4×D107C2semidirect
11. #121×C107C2semidirect
14. C2×Dic214C22semidirect
18. C2×C22⋊D107C22semidirect
21. C2×C4×D107C22semidirect
22. C2×C4×D107C22semidirect
23. C4⋊Dic107C22semidirect
24. Dic107⋊C4C22semidirect
25. C22⋊Dic107C22semidirect
26. C22⋊Dic107C22semidirect
27. C22⋊Dic107C22semidirect
30. (C2×D107)⋊C4C22semidirect
31. C4×Dic107C22semidirect
33. C2×D4×C107C22semidirect
34. C22×C4×C107C22semidirect
35. C4⋊C4×C107C22semidirect
36. C22⋊C4×C107C22semidirect
57. D428D4semidirect
75. C2×D107#41semidirect
90. C107#546semidirect
98. #121D107semidirect
112. C2×D4C2×D107semidirect
116. C22×C4C2×D107semidirect
119. C4⋊C4C2×D107semidirect
126. C22⋊C4C2×D107semidirect