Finite Group Databasebeta

← #20293433

Subgroups — #20293433

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#16812154C211yesmaximal, characteristic, complemented
3#16812140C211yesmaximal, characteristic, complemented
4#16812138C212yesmaximal, complemented
5#16812136C212yesmaximal, complemented
6#16812135C211yesmaximal, characteristic, complemented
7#16812131C211yesmaximal, characteristic, complemented
8#16812130C212yesmaximal, complemented
9SD16×D107C212yesmaximal, complemented
10#16812115C211yesmaximal, characteristic, complemented
11C2×SD16×C107C211yesmaximal, characteristic, complemented
12#13133819—21no—
13#13133819C2212yescomplemented
14#13133818—22no—
15#13133818C2211yescharacteristic
16#13133818C2211yescharacteristic
17#13133818C2211yescharacteristic, complemented
18#13133817—21no—
19#13133817C2212yes—
20Q8×D107—21no—
21Q8×D107C2212yescomplemented
22C2×C22⋊D107—22no—
23D4×D107—21no—
24D4×D107C2212yes—
25Dic428C2211yescharacteristic, complemented
26#13133808C2212yescomplemented
27#13133807C2211yescharacteristic, complemented
28#13133807C2211yescharacteristic, complemented
29#13133806C2212yescomplemented
30#13133805C2212yescomplemented
31D856C2211yescharacteristic, complemented
32#13133803C2212yescomplemented
33#13133802C2211yescharacteristic, complemented
34#13133801C2212yescomplemented
35C8×D107C2212yescomplemented
36C2×Q8×C107C2211yescharacteristic, complemented
37C2×D4×C107C2211yescharacteristic
38SD16×C107C2212yescomplemented
39SD16×C107C2212yescomplemented
40C2×C8×C107C2211yescharacteristic, complemented
41C23×C107—21no—
42C22×D107—22no—
43Q8×C107—21no—
44Q8×C107C2312yes—
45D4×C107—21no—
46D4×C107C2312yes—
47C2×C4×C107—21no—
48C2×C4×C107C2311yescharacteristic
49C22⋊D107—21no—
50C22⋊D107—21no—
51C22⋊D107—22no—
52C22⋊D107—22no—
53C22⋊D107D412yes—
54C2×Dic107—22no—
55D428—22no—
56D428D411yescharacteristic
57D428C2311yescharacteristic
58D428C2311yescharacteristic
59C4×D107—22no—
60C4×D107—21no—
61C4×D107—21no—
62C4×D107—22no—
63C4×D107C2312yes—
64Dic214—22no—
65Dic214D411yescharacteristic
66Dic214C2311yescharacteristic
67Dic214C2311yescharacteristic
68C8×C107C2311yescharacteristic
69C8×C107C2311yescharacteristic
70C107⋊C8C2312yes—
71C22×C107—22no—
72C22×C107—81no—
73C22×C107C2×D411yescharacteristic
74C2×D107—21no—
75C2×D107—21no—
76C2×D107—82no—
77C2×D107C2×D412yes—
78C4×C107—22no—
79C4×C107C2×D411yescharacteristic
80C4×C107C2411yescharacteristic
81Dic107—21no—
82Dic107—21no—
83Dic107C2×D412yes—
84C2×C107—21no—
85C2×C107—42no—
86C2×C107C22×D411yescharacteristic
87D107—22no—
88D107—41no—
89D107—41no—
90C107#57711yescharacteristic, complemented, Sylow
91#577—1071nomaximal, Sylow
922−1+4—1071no—
93H(5,𝔽2)—1071no—
94#137—1071no—
95#137—1072no—
96AGL(1,ℤ/8)—1071no—
97AGL(1,ℤ/8)—1072no—
98#135—1072no—
99#135—1071no—
100C2×SD16—1072no—
101C2×SD16D10711yescharacteristic, complemented
102#131—1071no—
103#41—1071no—
104#41—1072no—
105#41—1072no—
106#41—1071no—
107#41—1071no—
108#41—2142no—
109#41—2141no—
110#41—2141no—
111C2×Q8—1072no—
112C2×Q8—2141no—
113C2×Q8C2×D10711yescharacteristic, complemented
114C2×D4—1072no—
115C2×D4—2141no—
116C2×D4—2142no—
117C2×D4C2×D10711yescharacteristic
118Q16—1072no—
119Q16—1071no—
120SD16—1072no—
121SD16—1072no—
122SD16—1071no—
123SD16—1071no—
124SD16C2×D10712yescomplemented
125SD16C2×D10712yescomplemented
126D8—1072no—
127D8—1071no—
128M16—1071no—
129M16—1072no—
130C2×C8—1072no—
131C2×C8C2×D10711yescharacteristic, complemented
132C23—21no—
133C23—2142no—
134Q8—21no—
135Q8—1071no—
136Q8—1071no—
137Q8—1071no—
138Q8—2142no—
139Q8C22×D10712yes—
140D4—21no—
141D4—1072no—
142D4—1071no—
143D4—1071no—
144D4—1071no—
145D4—2142no—
146D4—2142no—
147D4—2142no—
148D4—2141no—
149D4—2141no—
150D4C22×D10712yes—
151C2×C4—21no—
152C2×C4—1072no—
153C2×C4—2142no—
154C2×C4—2142no—
155C2×C4—2142no—
156C2×C4—2141no—
157C2×C4—2141no—
158C2×C4C22×D10711yescharacteristic
159C8—1072no—
160C8C22×D10711yescharacteristic
161C8C22×D10711yescharacteristic
162C22—22no—
163C22—81no—
164C22—8562no—
165C22—1072no—
166C22—2141no—
167C22—2141no—
168C22D4×D10711yescharacteristic
169C4—22no—
170C4—1072no—
171C4—2141no—
172C4—2141no—
173C4D4×D10711yescharacteristic
174C4C23×D10711yescharacteristic
175C2—21no—
176C2—42no—
177C2—4281no—
178C2—4281no—
179C2—2142no—
180C2C2×D4×D10711yescharacteristic
subgroupG/H
centerC2C2×D4×D107
commutatorC4×C107C24
FrattiniC4C23×D107
FittingC2×SD16×C107C2
socleC2×C107C22×D4
hypercenterC2×SD16D107
hypocenterC107#577
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #16812157C2semidirect
2. #16812154C2semidirect
3. #16812140C2semidirect
4. #16812138C2semidirect
5. #16812136C2semidirect
6. #16812135C2semidirect
7. #16812131C2semidirect
8. #16812130C2semidirect
9. SD16×D107C2semidirect
10. #16812115C2semidirect
11. C2×SD16×C107C2semidirect
13. #13133819C22semidirect
17. #13133818C22semidirect
21. Q8×D107C22semidirect
25. Dic428C22semidirect
26. #13133808C22semidirect
27. #13133807C22semidirect
28. #13133807C22semidirect
29. #13133806C22semidirect
30. #13133805C22semidirect
31. D856C22semidirect
32. #13133803C22semidirect
33. #13133802C22semidirect
34. #13133801C22semidirect
35. C8×D107C22semidirect
36. C2×Q8×C107C22semidirect
38. SD16×C107C22semidirect
39. SD16×C107C22semidirect
40. C2×C8×C107C22semidirect
90. C107#577semidirect
101. C2×SD16D107semidirect
113. C2×Q8C2×D107semidirect
124. SD16C2×D107semidirect
125. SD16C2×D107semidirect
131. C2×C8C2×D107semidirect