Finite Group Databasebeta

← #20293431

Subgroups — #20293431

Order 6848.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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#16812155C211yesmaximal, characteristic, complemented
2#16812154C211yesmaximal, characteristic, complemented
3#16812137C211yesmaximal, characteristic, complemented
4#16812136C212yesmaximal, complemented
5#16812133C212yesmaximal, complemented
6#16812130C212yesmaximal, complemented
7#16812129C211yesmaximal, characteristic, complemented
8C2×#13133803C211yesmaximal, characteristic, complemented
9D8×D107C212yesmaximal, complemented
10#16812117C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12#13133819C2212yescomplemented
13#13133818—21no—
14#13133818C2211yescharacteristic
15#13133818C2211yescharacteristic, complemented
16#13133817—21no—
17#13133817C2212yescomplemented
18C2×D428—21no—
19C2×D428C2211yescharacteristic, complemented
20C2×C22⋊D107—22no—
21D4×D107—22no—
22D4×D107—21no—
23D4×D107C2212yescomplemented
24D4×D107C2212yescomplemented
25#13133808C2211yescharacteristic, complemented
26#13133807C2212yescomplemented
27#13133806C2211yescharacteristic, complemented
28#13133805C2211yescharacteristic, complemented
29D856C2212yescomplemented
30#13133803C2211yescharacteristic, complemented
31#13133803C2212yescomplemented
32#13133803C2212yescomplemented
33#13133801C2212yescomplemented
34C2×C107⋊C8C2211yescharacteristic, complemented
35C8×D107C2212yescomplemented
36#41×C107C2211yescharacteristic, complemented
37C2×D4×C107C2211yescharacteristic, complemented
38SD16×C107C2212yescomplemented
39D8×C107C2212yescomplemented
40M16×C107C2211yescharacteristic, complemented
41C23×C107—21no—
42C22×D107—21no—
43C22×D107—22no—
44C22×D107—22no—
45Q8×C107C2311yescharacteristic
46D4×C107—21no—
47D4×C107—21no—
48D4×C107C2311yescharacteristic
49D4×C107C2312yes—
50C2×C4×C107—21no—
51C2×C4×C107C2311yescharacteristic
52C22⋊D107—21no—
53C22⋊D107—22no—
54C22⋊D107—22no—
55C22⋊D107—22no—
56C22⋊D107D412yes—
57C2×Dic107—22no—
58D428—22no—
59D428—21no—
60D428D411yescharacteristic
61D428C2312yes—
62D428C2311yescharacteristic
63C4×D107—21no—
64C4×D107—22no—
65C4×D107C2312yes—
66Dic214D411yescharacteristic
67Dic214C2311yescharacteristic
68C8×C107C2312yes—
69C107⋊C8C2311yescharacteristic
70C107⋊C8C2311yescharacteristic
71C22×C107—22no—
72C22×C107—21no—
73C22×C107—81no—
74C22×C107C2×D411yescharacteristic
75C2×D107—22no—
76C2×D107—21no—
77C2×D107—82no—
78C2×D107—82no—
79C2×D107—81no—
80C2×D107C2×D412yes—
81C4×C107—21no—
82C4×C107C2×D411yescharacteristic
83C4×C107C2411yescharacteristic
84Dic107—21no—
85Dic107C2×D412yes—
86C2×C107—21no—
87C2×C107—41no—
88C2×C107—42no—
89C2×C107C22×D411yescharacteristic
90D107—22no—
91D107—42no—
92D107—41no—
93C107#57611yescharacteristic, complemented, Sylow
94#576—1071nomaximal, Sylow
95H(5,𝔽2)—1071no—
96H(5,𝔽2)—1071no—
97AGL(1,ℤ/8)—1072no—
98AGL(1,ℤ/8)—1072no—
99AGL(1,ℤ/8)—1071no—
100AGL(1,ℤ/8)D10711yescharacteristic, complemented
101#135—1071no—
102#135—1072no—
103C2×D8—1071no—
104C2×D8—1072no—
105#131—1071no—
106#41—1071no—
107#41—1072no—
108#41—1072no—
109#41—1071no—
110#41—2141no—
111#41—2141no—
112#41C2×D10711yescharacteristic, complemented
113C2×D4—1072no—
114C2×D4—1072no—
115C2×D4—1071no—
116C2×D4—2141no—
117C2×D4—2142no—
118C2×D4—2141no—
119C2×D4—2142no—
120C2×D4C2×D10711yescharacteristic, complemented
121Q16—1071no—
122SD16—1071no—
123SD16—1071no—
124SD16—1072no—
125SD16C2×D10712yescomplemented
126D8—1072no—
127D8—1071no—
128D8—1072no—
129D8—1072no—
130D8C2×D10712yescomplemented
131M16—1072no—
132M16C2×D10711yescharacteristic, complemented
133C2×C8—1071no—
134C2×C8—1072no—
135C23—21no—
136C23—2142no—
137C23—2142no—
138C23—2141no—
139Q8—1071no—
140Q8—1071no—
141Q8C22×D10711yescharacteristic
142D4—21no—
143D4—21no—
144D4—1072no—
145D4—1071no—
146D4—1072no—
147D4—1071no—
148D4—2142no—
149D4—2142no—
150D4—2142no—
151D4—2142no—
152D4—2141no—
153D4—2141no—
154D4C22×D10712yes—
155D4C22×D10711yescharacteristic
156C2×C4—21no—
157C2×C4—1072no—
158C2×C4—2142no—
159C2×C4—2142no—
160C2×C4—2141no—
161C2×C4C22×D10711yescharacteristic
162C8—1071no—
163C8—1071no—
164C8C22×D10712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—8561no—
169C22—8562no—
170C22—8562no—
171C22—1072no—
172C22—2142no—
173C22—2141no—
174C22D4×D10711yescharacteristic
175C4—21no—
176C4—1072no—
177C4—2141no—
178C4D4×D10711yescharacteristic
179C4C23×D10711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—4282no—
184C2—4281no—
185C2—2142no—
186C2C2×D4×D10711yescharacteristic
subgroupG/H
centerC2C2×D4×D107
commutatorC4×C107C24
FrattiniC4C23×D107
FittingAGL(1,ℤ/8)×C107C2
socleC2×C107C22×D4
hypercenterAGL(1,ℤ/8)D107
hypocenterC107#576
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16812155C2semidirect
2. #16812154C2semidirect
3. #16812137C2semidirect
4. #16812136C2semidirect
5. #16812133C2semidirect
6. #16812130C2semidirect
7. #16812129C2semidirect
8. C2×#13133803C2semidirect
9. D8×D107C2semidirect
10. #16812117C2semidirect
11. AGL(1,ℤ/8)×C107C2semidirect
12. #13133819C22semidirect
15. #13133818C22semidirect
17. #13133817C22semidirect
19. C2×D428C22semidirect
23. D4×D107C22semidirect
24. D4×D107C22semidirect
25. #13133808C22semidirect
26. #13133807C22semidirect
27. #13133806C22semidirect
28. #13133805C22semidirect
29. D856C22semidirect
30. #13133803C22semidirect
31. #13133803C22semidirect
32. #13133803C22semidirect
33. #13133801C22semidirect
34. C2×C107⋊C8C22semidirect
35. C8×D107C22semidirect
36. #41×C107C22semidirect
37. C2×D4×C107C22semidirect
38. SD16×C107C22semidirect
39. D8×C107C22semidirect
40. M16×C107C22semidirect
93. C107#576semidirect
100. AGL(1,ℤ/8)D107semidirect
112. #41C2×D107semidirect
120. C2×D4C2×D107semidirect
125. SD16C2×D107semidirect
130. D8C2×D107semidirect
132. M16C2×D107semidirect