Finite Group Databasebeta

← #22600140

Subgroups — #22600140

Order 9088.

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#17986804C212yesmaximal, complemented
2#17986801C212yesmaximal, complemented
3#17986376C211yesmaximal, characteristic, complemented
4#17986270C212yesmaximal, complemented
5#17986269C212yesmaximal, complemented
6#17986168C214yesmaximal, complemented
7#17986117C211yesmaximal, characteristic, complemented
8#443×C71C211yesmaximal, characteristic, complemented
9#14556174C2214yescomplemented
10#14556156—24no—
11#14556156C2214yescomplemented
12#14556155—24no—
13#14556153C2214yescomplemented
14#14556151—22no—
15#14556150—22no—
16#14556150C2211yescharacteristic
17C2×D568—22no—
18C2×D568C2212yescomplemented
19C2×D568C2212yescomplemented
20D8×D71—22no—
21#14556134C2212yescomplemented
22#14556125C2212yescomplemented
23#14556064C2211yescharacteristic, complemented
24#14556056C2212yescomplemented
25#14556044C2214yescomplemented
26#135×C71C2211yescharacteristic
27#131×C71C2212yescomplemented
28#108×C71C2211yescharacteristic, complemented
29#104×C71C2212yescomplemented
30C4×C8×C71C2211yescharacteristic, complemented
31#11765866—24no—
32#11765865—24no—
33#11765865C2312yes—
34C2×D284—24no—
35C2×D284—42no—
36C2×D284C2314yes—
37D4×D71—24no—
38D4×D71—44no—
39Dic284D411yescharacteristic, complemented
40#11765854—24no—
41#11765854D412yescomplemented
42#11765852—24no—
43D568—24no—
44D568—22no—
45D568—22no—
46D568—22no—
47D568—22no—
48D568D411yescharacteristic, complemented
49#11765850—24no—
50#11765849C2312yes—
51#11765848—22no—
52C8×D71—22no—
53#41×C71C2312yes—
54Q16×C71D411yescharacteristic, complemented
55SD16×C71D412yescomplemented
56D8×C71D411yescharacteristic, complemented
57M16×C71—22no—
58M16×C71C2312yes—
59C2×C8×C71—22no—
60C2×C8×C71C2311yescharacteristic
61C2×C8×C71C2311yescharacteristic
62C42×C71C2311yescharacteristic
63C22×D71—24no—
64C22×D71—44no—
65Q8×C71C2×D412yes—
66D4×C71—22no—
67D4×C71C2×D412yes—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71C2411yescharacteristic
71C22⋊D71—22no—
72C22⋊D71—44no—
73D284—24no—
74D284—24no—
75D284—44no—
76D284—84no—
77D284C2×D412yes—
78C4×D71—22no—
79C4×D71—44no—
80Dic142C2×D412yes—
81C8×C71—22no—
82C8×C71C2×D411yescharacteristic
83C8×C71C2×D411yescharacteristic
84C8×C71C2×D411yescharacteristic
85C8×C71C2×D411yescharacteristic
86C71⋊C8—22no—
87C22×C71—22no—
88C22×C71H(5,𝔽2)11yescharacteristic
89C2×D71—22no—
90C2×D71—44no—
91C2×D71—84no—
92C2×D71—164no—
93C4×C71—22no—
94C4×C71—22no—
95C4×C71C22×D411yescharacteristic
96C4×C71C22×D411yescharacteristic
97Dic71—22no—
98C2×C71—21no—
99C2×C71—42no—
100C2×C71D4211yescharacteristic
101D71—42no—
102D71—84no—
103#3292—711nomaximal, Sylow
104C71#329211yescharacteristic, complemented, Sylow
105#576—712no—
106#576—712no—
107#493—711no—
108#472—712no—
109#472—712no—
110#453—714no—
111#443—711no—
112#443D7111yescharacteristic, complemented
113H(5,𝔽2)—714no—
114AGL(1,ℤ/8)—714no—
115AGL(1,ℤ/8)—714no—
116AGL(1,ℤ/8)—1424no—
117AGL(1,ℤ/8)—1424no—
118#135—711no—
119#135—1422no—
120#135—1422no—
121#135C2×D7111yescharacteristic
122C2×D8—712no—
123C2×D8—712no—
124C2×D8—1422no—
125C2×D8—1422no—
126#131—712no—
127#131C2×D7112yescomplemented
128#127—712no—
129#108—711no—
130#108C2×D7111yescharacteristic, complemented
131#104—712no—
132#104C2×D7112yescomplemented
133#100—714no—
134C4×C8C2×D7111yescharacteristic, complemented
135#41—712no—
136#41—1424no—
137#41—1424no—
138#41C22×D7112yes—
139C2×D4—2842no—
140C2×D4—2844no—
141C2×D4—714no—
142C2×D4—1424no—
143C2×D4—1424no—
144Q16—711no—
145Q16D28411yescharacteristic, complemented
146SD16—712no—
147SD16—1424no—
148SD16—1424no—
149SD16D28412yescomplemented
150D8—711no—
151D8—1424no—
152D8—1424no—
153D8—1422no—
154D8—1422no—
155D8—1422no—
156D8—1422no—
157D8D28411yescharacteristic, complemented
158M16—22no—
159M16—712no—
160M16—1422no—
161M16C22×D7112yes—
162C2×C8—22no—
163C2×C8—1422no—
164C2×C8C22×D7111yescharacteristic
165C2×C8C22×D7111yescharacteristic
166C42C22×D7111yescharacteristic
167C23—2844no—
168C23—1424no—
169Q8—712no—
170Q8C2×D28412yes—
171D4—22no—
172D4—2844no—
173D4—2844no—
174D4—5684no—
175D4—712no—
176D4—1422no—
177D4—1424no—
178D4—1424no—
179D4C2×D28412yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—2844no—
183C2×C4—1422no—
184C2×C4C23×D7111yescharacteristic
185C8—22no—
186C8—1422no—
187C8D4×D7111yescharacteristic
188C8D4×D7111yescharacteristic
189C8C2×D28411yescharacteristic
190C8C2×D28411yescharacteristic
191C22—22no—
192C22—2844no—
193C22—5684no—
194C22—11364no—
195C22—1422no—
196C22#1455617411yescharacteristic
197C4—22no—
198C4—22no—
199C4—1422no—
200C4C2×D4×D7111yescharacteristic
201C4C22×D28411yescharacteristic
202C2—21no—
203C2—42no—
204C2—2842no—
205C2—5684no—
206C2D4×D28411yescharacteristic
subgroupG/H
centerC2D4×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#443×C71C2
socleC2×C71D42
hypercenter#443D71
hypocenterC71#3292
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #17986804C2semidirect
2. #17986801C2semidirect
3. #17986376C2semidirect
4. #17986270C2semidirect
5. #17986269C2semidirect
6. #17986168C2semidirect
7. #17986117C2semidirect
8. #443×C71C2semidirect
9. #14556174C22semidirect
11. #14556156C22semidirect
13. #14556153C22semidirect
18. C2×D568C22semidirect
19. C2×D568C22semidirect
21. #14556134C22semidirect
22. #14556125C22semidirect
23. #14556064C22semidirect
24. #14556056C22semidirect
25. #14556044C22semidirect
27. #131×C71C22semidirect
28. #108×C71C22semidirect
29. #104×C71C22semidirect
30. C4×C8×C71C22semidirect
39. Dic284D4semidirect
41. #11765854D4semidirect
48. D568D4semidirect
54. Q16×C71D4semidirect
55. SD16×C71D4semidirect
56. D8×C71D4semidirect
104. C71#3292semidirect
112. #443D71semidirect
127. #131C2×D71semidirect
130. #108C2×D71semidirect
132. #104C2×D71semidirect
134. C4×C8C2×D71semidirect
145. Q16D284semidirect
149. SD16D284semidirect
157. D8D284semidirect