Finite Group Databasebeta

← #28348261

Subgroups — #28348261

Order 13696.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293432C212yesmaximal, complemented
2#20293429C212yesmaximal, complemented
3#20293004C211yesmaximal, characteristic, complemented
4#20292898C212yesmaximal, complemented
5#20292897C212yesmaximal, complemented
6#20292796C214yesmaximal, complemented
7#20292745C211yesmaximal, characteristic, complemented
8#443×C107C211yesmaximal, characteristic, complemented
9#16812155C2214yescomplemented
10#16812137—24no—
11#16812137C2214yescomplemented
12#16812136—24no—
13#16812134C2214yescomplemented
14#16812132—22no—
15#16812131—22no—
16#16812131C2211yescharacteristic
17C2×D856—22no—
18C2×D856C2212yescomplemented
19C2×D856C2212yescomplemented
20D8×D107—22no—
21#16812115C2212yescomplemented
22#16812106C2212yescomplemented
23#16812045C2211yescharacteristic, complemented
24#16812037C2212yescomplemented
25#16812025C2214yescomplemented
26#135×C107C2211yescharacteristic
27#131×C107C2212yescomplemented
28#108×C107C2211yescharacteristic, complemented
29#104×C107C2212yescomplemented
30C4×C8×C107C2211yescharacteristic, complemented
31#13133819—24no—
32#13133818—24no—
33#13133818C2312yes—
34C2×D428—24no—
35C2×D428—42no—
36C2×D428C2314yes—
37D4×D107—24no—
38D4×D107—44no—
39Dic428D411yescharacteristic, complemented
40#13133807—24no—
41#13133807D412yescomplemented
42#13133805—24no—
43D856—24no—
44D856—22no—
45D856—22no—
46D856—22no—
47D856—22no—
48D856D411yescharacteristic, complemented
49#13133803—24no—
50#13133802C2312yes—
51#13133801—22no—
52C8×D107—22no—
53#41×C107C2312yes—
54Q16×C107D411yescharacteristic, complemented
55SD16×C107D412yescomplemented
56D8×C107D411yescharacteristic, complemented
57M16×C107—22no—
58M16×C107C2312yes—
59C2×C8×C107—22no—
60C2×C8×C107C2311yescharacteristic
61C2×C8×C107C2311yescharacteristic
62C42×C107C2311yescharacteristic
63C22×D107—24no—
64C22×D107—44no—
65Q8×C107C2×D412yes—
66D4×C107—22no—
67D4×C107C2×D412yes—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107C2411yescharacteristic
71C22⋊D107—22no—
72C22⋊D107—44no—
73D428—24no—
74D428—24no—
75D428—44no—
76D428—84no—
77D428C2×D412yes—
78C4×D107—22no—
79C4×D107—44no—
80Dic214C2×D412yes—
81C8×C107—22no—
82C8×C107C2×D411yescharacteristic
83C8×C107C2×D411yescharacteristic
84C8×C107C2×D411yescharacteristic
85C8×C107C2×D411yescharacteristic
86C107⋊C8—22no—
87C22×C107—22no—
88C22×C107H(5,𝔽2)11yescharacteristic
89C2×D107—22no—
90C2×D107—44no—
91C2×D107—84no—
92C2×D107—164no—
93C4×C107—22no—
94C4×C107—22no—
95C4×C107C22×D411yescharacteristic
96C4×C107C22×D411yescharacteristic
97Dic107—22no—
98C2×C107—21no—
99C2×C107—42no—
100C2×C107D4211yescharacteristic
101D107—42no—
102D107—84no—
103#3292—1071nomaximal, Sylow
104C107#329211yescharacteristic, complemented, Sylow
105#576—1072no—
106#576—1072no—
107#493—1071no—
108#472—1072no—
109#472—1072no—
110#453—1074no—
111#443—1071no—
112#443D10711yescharacteristic, complemented
113H(5,𝔽2)—1074no—
114AGL(1,ℤ/8)—1074no—
115AGL(1,ℤ/8)—1074no—
116AGL(1,ℤ/8)—2144no—
117AGL(1,ℤ/8)—2144no—
118#135—1071no—
119#135—2142no—
120#135—2142no—
121#135C2×D10711yescharacteristic
122C2×D8—1072no—
123C2×D8—1072no—
124C2×D8—2142no—
125C2×D8—2142no—
126#131—1072no—
127#131C2×D10712yescomplemented
128#127—1072no—
129#108—1071no—
130#108C2×D10711yescharacteristic, complemented
131#104—1072no—
132#104C2×D10712yescomplemented
133#100—1074no—
134C4×C8C2×D10711yescharacteristic, complemented
135#41—1072no—
136#41—2144no—
137#41—2144no—
138#41C22×D10712yes—
139C2×D4—1074no—
140C2×D4—4282no—
141C2×D4—4284no—
142C2×D4—2144no—
143C2×D4—2144no—
144Q16—1071no—
145Q16D42811yescharacteristic, complemented
146SD16—1072no—
147SD16—2144no—
148SD16—2144no—
149SD16D42812yescomplemented
150D8—1071no—
151D8—2144no—
152D8—2144no—
153D8—2142no—
154D8—2142no—
155D8—2142no—
156D8—2142no—
157D8D42811yescharacteristic, complemented
158M16—22no—
159M16—1072no—
160M16—2142no—
161M16C22×D10712yes—
162C2×C8—22no—
163C2×C8—2142no—
164C2×C8C22×D10711yescharacteristic
165C2×C8C22×D10711yescharacteristic
166C42C22×D10711yescharacteristic
167C23—4284no—
168C23—2144no—
169Q8—1072no—
170Q8C2×D42812yes—
171D4—22no—
172D4—8564no—
173D4—1072no—
174D4—4284no—
175D4—4284no—
176D4—2142no—
177D4—2144no—
178D4—2144no—
179D4C2×D42812yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—4284no—
183C2×C4—2142no—
184C2×C4C23×D10711yescharacteristic
185C8—22no—
186C8—2142no—
187C8D4×D10711yescharacteristic
188C8D4×D10711yescharacteristic
189C8C2×D42811yescharacteristic
190C8C2×D42811yescharacteristic
191C22—22no—
192C22—8564no—
193C22—4284no—
194C22—17124no—
195C22—2142no—
196C22#1681215511yescharacteristic
197C4—22no—
198C4—22no—
199C4—2142no—
200C4C2×D4×D10711yescharacteristic
201C4C22×D42811yescharacteristic
202C2—21no—
203C2—42no—
204C2—8564no—
205C2—4282no—
206C2D4×D42811yescharacteristic
subgroupG/H
centerC2D4×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#443×C107C2
socleC2×C107D42
hypercenter#443D107
hypocenterC107#3292
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20293432C2semidirect
2. #20293429C2semidirect
3. #20293004C2semidirect
4. #20292898C2semidirect
5. #20292897C2semidirect
6. #20292796C2semidirect
7. #20292745C2semidirect
8. #443×C107C2semidirect
9. #16812155C22semidirect
11. #16812137C22semidirect
13. #16812134C22semidirect
18. C2×D856C22semidirect
19. C2×D856C22semidirect
21. #16812115C22semidirect
22. #16812106C22semidirect
23. #16812045C22semidirect
24. #16812037C22semidirect
25. #16812025C22semidirect
27. #131×C107C22semidirect
28. #108×C107C22semidirect
29. #104×C107C22semidirect
30. C4×C8×C107C22semidirect
39. Dic428D4semidirect
41. #13133807D4semidirect
48. D856D4semidirect
54. Q16×C107D4semidirect
55. SD16×C107D4semidirect
56. D8×C107D4semidirect
104. C107#3292semidirect
112. #443D107semidirect
127. #131C2×D107semidirect
130. #108C2×D107semidirect
132. #104C2×D107semidirect
134. C4×C8C2×D107semidirect
145. Q16D428semidirect
149. SD16D428semidirect
157. D8D428semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202934322
#202934292
#202930042
#202928982
#202928972
#202927962
#202927452
#443×C1072
#3292107