Finite Group Databasebeta

← #20293436

Subgroups — #20293436

Order 6848.

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

normal subgroups only

This group has 258 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#16812157C211yesmaximal, characteristic, complemented
2#16812155C211yesmaximal, characteristic, complemented
3#16812141C212yesmaximal, complemented
4#16812137C211yesmaximal, characteristic, complemented
5#16812136C212yesmaximal, complemented
6#16812132C212yesmaximal, complemented
7#16812129C211yesmaximal, characteristic, complemented
8C2×#13133805C211yesmaximal, characteristic, complemented
9SD16×D107C212yesmaximal, complemented
10#16812117C211yesmaximal, characteristic, complemented
11#137×C107C211yesmaximal, characteristic, complemented
12#13133819—21no—
13#13133819C2212yes—
14#13133819C2212yescomplemented
15#13133818—22no—
16#13133818—21no—
17#13133818C2211yescharacteristic
18#13133818C2211yescharacteristic, complemented
19Q8×D107—21no—
20Q8×D107C2212yescomplemented
21C2×D428—21no—
22C2×D428C2211yescharacteristic
23D4×D107—22no—
24D4×D107C2212yes—
25#13133808C2211yescharacteristic, complemented
26#13133807C2212yescomplemented
27#13133806C2211yescharacteristic, complemented
28#13133805C2212yescomplemented
29#13133805C2212yescomplemented
30#13133805C2211yescharacteristic, complemented
31D856C2212yescomplemented
32#13133803C2211yescharacteristic, complemented
33#13133801C2212yescomplemented
34C2×C107⋊C8C2211yescharacteristic, complemented
35C8×D107C2212yescomplemented
36#41×C107C2211yescharacteristic
37C2×Q8×C107C2211yescharacteristic, complemented
38Q16×C107C2212yescomplemented
39SD16×C107C2212yescomplemented
40M16×C107C2211yescharacteristic, complemented
41C22×D107—21no—
42C22×D107—22no—
43Q8×C107—21no—
44Q8×C107C2312yes—
45Q8×C107C2311yescharacteristic
46D4×C107—21no—
47D4×C107C2311yescharacteristic
48C2×C4×C107—21no—
49C2×C4×C107—21no—
50C2×C4×C107C2311yescharacteristic
51C22⋊D107—21no—
52C22⋊D107—22no—
53C22⋊D107D412yes—
54D428—22no—
55D428—22no—
56D428—21no—
57D428D411yescharacteristic
58D428C2311yescharacteristic
59D428C2312yes—
60C4×D107—22no—
61C4×D107—21no—
62C4×D107—22no—
63C4×D107—22no—
64C4×D107C2312yes—
65Dic214—22no—
66Dic214D411yescharacteristic
67Dic214C2311yescharacteristic
68C8×C107C2312yes—
69C107⋊C8C2311yescharacteristic
70C107⋊C8C2311yescharacteristic
71C22×C107—21no—
72C22×C107C2×D411yescharacteristic
73C2×D107—21no—
74C2×D107—22no—
75C2×D107—82no—
76C2×D107—81no—
77C2×D107C2×D412yes—
78C4×C107—22no—
79C4×C107—21no—
80C4×C107C2×D411yescharacteristic
81C4×C107C2411yescharacteristic
82Dic107—21no—
83Dic107C2×D412yes—
84C2×C107—21no—
85C2×C107—41no—
86C2×C107C22×D411yescharacteristic
87D107—22no—
88D107—41no—
89D107—42no—
90C107#57711yescharacteristic, complemented, Sylow
91#577—1071nomaximal, Sylow
922−1+4—1071no—
93H(5,𝔽2)—1071no—
94#137—1072no—
95#137D10711yescharacteristic, complemented
96AGL(1,ℤ/8)—1072no—
97AGL(1,ℤ/8)—1071no—
98#135—1071no—
99#135—1072no—
100C2×SD16—1071no—
101C2×SD16—1072no—
102#131—1071no—
103#41—1071no—
104#41—1072no—
105#41—1072no—
106#41—1071no—
107#41—2142no—
108#41—2141no—
109#41—2141no—
110#41C2×D10711yescharacteristic
111C2×Q8—1072no—
112C2×Q8—2141no—
113C2×Q8C2×D10711yescharacteristic, complemented
114C2×D4—1072no—
115C2×D4—1071no—
116C2×D4—2141no—
117C2×D4—2142no—
118Q16—1071no—
119Q16C2×D10712yescomplemented
120SD16—1072no—
121SD16—1072no—
122SD16—1071no—
123SD16—1071no—
124SD16—1072no—
125SD16C2×D10712yescomplemented
126D8—1071no—
127D8—1072no—
128M16—1072no—
129M16C2×D10711yescharacteristic, complemented
130C2×C8—1071no—
131C2×C8—1072no—
132C23—2142no—
133C23—2141no—
134Q8—21no—
135Q8—1071no—
136Q8—1071no—
137Q8—2142no—
138Q8C22×D10712yes—
139Q8C22×D10711yescharacteristic
140D4—21no—
141D4—1072no—
142D4—1071no—
143D4—1071no—
144D4—1072no—
145D4—2142no—
146D4—2142no—
147D4—2142no—
148D4—2141no—
149D4—2141no—
150D4C22×D10711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—1072no—
154C2×C4—2142no—
155C2×C4—2142no—
156C2×C4—2142no—
157C2×C4—2141no—
158C2×C4C22×D10711yescharacteristic
159C8—1071no—
160C8—1071no—
161C8C22×D10712yes—
162C22—21no—
163C22—8561no—
164C22—8562no—
165C22—1072no—
166C22—2141no—
167C22—2142no—
168C22D4×D10711yescharacteristic
169C4—21no—
170C4—22no—
171C4—1072no—
172C4—2141no—
173C4D4×D10711yescharacteristic
174C4C23×D10711yescharacteristic
175C2—21no—
176C2—41no—
177C2—4282no—
178C2—4281no—
179C2—2142no—
180C2C2×D4×D10711yescharacteristic
subgroupG/H
centerC2C2×D4×D107
commutatorC4×C107C24
FrattiniC4C23×D107
Fitting#137×C107C2
socleC2×C107C22×D4
hypercenter#137D107
hypocenterC107#577
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #16812157C2semidirect
2. #16812155C2semidirect
3. #16812141C2semidirect
4. #16812137C2semidirect
5. #16812136C2semidirect
6. #16812132C2semidirect
7. #16812129C2semidirect
8. C2×#13133805C2semidirect
9. SD16×D107C2semidirect
10. #16812117C2semidirect
11. #137×C107C2semidirect
14. #13133819C22semidirect
18. #13133818C22semidirect
20. Q8×D107C22semidirect
25. #13133808C22semidirect
26. #13133807C22semidirect
27. #13133806C22semidirect
28. #13133805C22semidirect
29. #13133805C22semidirect
30. #13133805C22semidirect
31. D856C22semidirect
32. #13133803C22semidirect
33. #13133801C22semidirect
34. C2×C107⋊C8C22semidirect
35. C8×D107C22semidirect
37. C2×Q8×C107C22semidirect
38. Q16×C107C22semidirect
39. SD16×C107C22semidirect
40. M16×C107C22semidirect
90. C107#577semidirect
95. #137D107semidirect
113. C2×Q8C2×D107semidirect
119. Q16C2×D107semidirect
125. SD16C2×D107semidirect
129. M16C2×D107semidirect