Finite Group Databasebeta

← #20293192

Subgroups — #20293192

Order 6848.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C22×C22⋊D107C211yesmaximal, characteristic, complemented
3#16812105C212yesmaximal, complemented
4#16812095C212yesmaximal, complemented
5#16812082C212yesmaximal, complemented
6#16812081C212yesmaximal, complemented
7#16812079C211yesmaximal, characteristic, complemented
8#16812078C211yesmaximal, characteristic, complemented
9#16812077C211yesmaximal, characteristic, complemented
10#16812065C211yesmaximal, characteristic, complemented
11#120×C107C211yesmaximal, characteristic, complemented
12C23×D107C2211yescharacteristic
13#13133817—24no—
14C2×Dic214C2211yescharacteristic, complemented
15C2×D428C2212yescomplemented
16C2×C22⋊D107—24no—
17C2×C22⋊D107C2211yescharacteristic
18C2×C22⋊D107C2212yes—
19C2×C22⋊D107C2212yes—
20C2×C22⋊D107C2211yescharacteristic, complemented
21C2×C22⋊D107C2212yescomplemented
22C2×C22⋊D107C2212yescomplemented
23C22×Dic107C2211yescharacteristic
24C22×Dic107C2211yescharacteristic, complemented
25C2×C4×D107C2211yescharacteristic, complemented
26Dic107⋊C4C2212yescomplemented
27C22⋊Dic107C2211yescharacteristic
28C22⋊Dic107C2212yescomplemented
29(C2×D107)⋊C4C2212yescomplemented
30(C2×D107)⋊C4C2212yescomplemented
31C4×Dic107C2212yescomplemented
32C24×C107C2211yescharacteristic
33C2×D4×C107C2211yescharacteristic
34C2×D4×C107C2212yescomplemented
35C22⋊C4×C107C2211yescharacteristic
36C22⋊C4×C107C2212yescomplemented
37C23×C107—22no—
38C23×C107—42no—
39C23×C107C2312yes—
40C23×C107C2311yescharacteristic
41C23×C107C2311yescharacteristic
42C22×D107—22no—
43C22×D107—42no—
44C22×D107C2312yes—
45C22×D107C2311yescharacteristic
46D4×C107—22no—
47D4×C107—42no—
48C2×C4×C107C2312yes—
49C2×C4×C107C2311yescharacteristic
50C22⋊D107—22no—
51C22⋊D107—24no—
52C22⋊D107—24no—
53C22⋊D107—22no—
54C22⋊D107—42no—
55C22⋊D107—42no—
56C2×Dic107—22no—
57C2×Dic107D414yescomplemented
58C2×Dic107C2312yes—
59C2×Dic107C2311yescharacteristic
60C2×Dic107C2311yescharacteristic
61C2×Dic107C2311yescharacteristic
62D428—42no—
63C4×D107—22no—
64Dic214—22no—
65C22×C107—22no—
66C22×C107—21no—
67C22×C107—22no—
68C22×C107—22no—
69C22×C107—22no—
70C22×C107—22no—
71C22×C107—44no—
72C22×C107C2×D412yes—
73C22×C107C2411yescharacteristic
74C2×D107—21no—
75C2×D107—22no—
76C2×D107—22no—
77C2×D107—22no—
78C2×D107—22no—
79C2×D107—44no—
80C4×C107—21no—
81C4×C107—22no—
82Dic107—21no—
83Dic107—22no—
84Dic107C2×D412yes—
85Dic107C2×D412yes—
86C2×C107—22no—
87C2×C107—41no—
88C2×C107—42no—
89C2×C107C22×D411yescharacteristic
90C2×C107H(5,𝔽2)12yes—
91D107—41no—
92D107—42no—
93C107#53511yescharacteristic, complemented, Sylow
94#535—1071nomaximal, Sylow
95C2×#41—1071no—
96C22×D4—1071no—
97#127—1072no—
98#124—1072no—
99#121—1072no—
100#121—1072no—
101#120—1071no—
102#120—1071no—
103#120—1071no—
104#120D10711yescharacteristic, complemented
105#117—1071no—
106C24—1071no—
107C24C2×D10711yescharacteristic
108#41—2144no—
109C2×Q8—1071no—
110C2×D4—1072no—
111C2×D4—1072no—
112C2×D4—1071no—
113C2×D4—1072no—
114C2×D4—1072no—
115C2×D4—1071no—
116C2×D4—1072no—
117C2×D4—2144no—
118C2×D4C2×D10711yescharacteristic
119C2×D4C2×D10712yescomplemented
120C22×C4—1071no—
121C22×C4—1071no—
122C22×C4—1071no—
123C4⋊C4—1072no—
124C22⋊C4—1071no—
125C22⋊C4—1072no—
126C22⋊C4—1072no—
127C22⋊C4—1072no—
128C22⋊C4C2×D10711yescharacteristic
129C22⋊C4C2×D10712yescomplemented
130C42—1072no—
131C23—22no—
132C23—42no—
133C23—1072no—
134C23—1071no—
135C23—4282no—
136C23—2142no—
137C23C22×D10712yes—
138C23C22×D10711yescharacteristic
139C23C22×D10711yescharacteristic
140Q8—2142no—
141D4—22no—
142D4—42no—
143D4—4282no—
144D4—4282no—
145D4—4282no—
146D4—2144no—
147D4—2144no—
148D4—2142no—
149D4—2142no—
150C2×C4—1074no—
151C2×C4—1071no—
152C2×C4—1071no—
153C2×C4—1072no—
154C2×C4—1071no—
155C2×C4—2142no—
156C2×C4—2142no—
157C2×C4C22×D10712yes—
158C2×C4C22×D10711yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—4284no—
167C22—2141no—
168C22—2142no—
169C22—2142no—
170C22—2142no—
171C22—2142no—
172C22D4×D10712yes—
173C22C23×D10711yescharacteristic
174C4—21no—
175C4—22no—
176C4—1072no—
177C4—1072no—
178C4—2141no—
179C4—2142no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—4282no—
184C2—4281no—
185C2C2×D4×D10711yescharacteristic
186C2#1681215412yes—
subgroupG/H
center, FrattiniC22C23×D107
commutator, socleC22×C107C24
Fitting#120×C107C2
hypercenter#120D107
hypocenterC107#535
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#13133817C2semidirect
2. C22×C22⋊D107C2semidirect
3. #16812105C2semidirect
4. #16812095C2semidirect
5. #16812082C2semidirect
6. #16812081C2semidirect
7. #16812079C2semidirect
8. #16812078C2semidirect
9. #16812077C2semidirect
10. #16812065C2semidirect
11. #120×C107C2semidirect
14. C2×Dic214C22semidirect
15. C2×D428C22semidirect
20. C2×C22⋊D107C22semidirect
21. C2×C22⋊D107C22semidirect
22. C2×C22⋊D107C22semidirect
24. C22×Dic107C22semidirect
25. C2×C4×D107C22semidirect
26. Dic107⋊C4C22semidirect
28. C22⋊Dic107C22semidirect
29. (C2×D107)⋊C4C22semidirect
30. (C2×D107)⋊C4C22semidirect
31. C4×Dic107C22semidirect
34. C2×D4×C107C22semidirect
36. C22⋊C4×C107C22semidirect
57. C2×Dic107D4semidirect
93. C107#535semidirect
104. #120D107semidirect
119. C2×D4C2×D107semidirect
129. C22⋊C4C2×D107semidirect