Finite Group Databasebeta

← #22598713

Subgroups — #22598713

Order 9088.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986555C211yesmaximal, characteristic, complemented
2#127×D71C211yesmaximal, characteristic
3#17986365C211yesmaximal, characteristic, complemented
4#17986364C211yesmaximal, characteristic, complemented
5#17986363C211yesmaximal, characteristic, complemented
6#17986344C214yesmaximal, complemented
7#17986002C214yesmaximal, complemented
8#17985951C211yesmaximal, characteristic, complemented
9#490×C71C211yesmaximal, characteristic, complemented
10C2×D4×D71—22no—
11C2×D4×D71C2212yes—
12C2×#11765848C2212yes—
13#14556127C2212yescomplemented
14#14556125C2211yescharacteristic
15#14556124—22no—
16#14556124C2212yes—
17#14556120C2211yescharacteristic, complemented
18#14556106—22no—
19#14556093—22no—
20#14556086C2212yes—
21C42×D71C2211yescharacteristic
22#14556050C2214yescomplemented
23#14556049C2214yescomplemented
24#14556048C2214yescomplemented
25#14556039C2211yescharacteristic, complemented
26C8×Dic71C2212yescomplemented
27#127×C71C2211yescharacteristic
28#125×C71C2211yescharacteristic, complemented
29#102×C71C2214yescomplemented
30#97×C71C2211yescharacteristic, complemented
31C23×D71—22no—
32C2×Dic142—22no—
33C2×D284—22no—
34C2×D284C2312yes—
35C2×C22⋊D71—24no—
36C2×C22⋊D71—22no—
37D4×D71—28no—
38D4×D71—48no—
39C2×C4×D71D412yes—
40C2×C4×D71C2311yescharacteristic
41#11765848—24no—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71C2312yes—
44Dic71⋊C4—22no—
45Dic71⋊C4—22no—
46(C2×D71)⋊C4—22no—
47C4×Dic71—22no—
48C4×Dic71D412yes—
49C4×Dic71C2311yescharacteristic
50C2×D4×C71—22no—
51C2×D4×C71C2312yes—
52C2×C8×C71C2312yes—
53C4⋊C4×C71—22no—
54C4⋊C4×C71C2312yes—
55C42×C71C2311yescharacteristic
56C23×C71—22no—
57C22×D71—22no—
58C22×D71—48no—
59C22×D71—44no—
60C22×D71C2×D411yescharacteristic
61D4×C71—24no—
62D4×C71—44no—
63C2×C4×C71—22no—
64C2×C4×C71C2×D412yes—
65C2×C4×C71C2411yescharacteristic
66C22⋊D71—44no—
67C22⋊D71—48no—
68C2×Dic71—22no—
69C2×Dic71C2×D411yescharacteristic
70C2×Dic71C2×D412yes—
71D284—24no—
72D284—44no—
73C4×D71—24no—
74C4×D71#4114yes—
75Dic142—44no—
76C8×C71—22no—
77C71⋊C8—22no—
78C22×C71—42no—
79C22×C71—44no—
80C22×C71C22×D411yescharacteristic
81C2×D71—22no—
82C2×D71—42no—
83C2×D71—44no—
84C2×D71—88no—
85C2×D71#12412yes—
86C4×C71—22no—
87C4×C71—42no—
88C4×C71C2×#4112yes—
89Dic71—42no—
90Dic71#12412yes—
91Dic71AGL(1,ℤ/8)14yes—
92C2×C71—82no—
93C2×C71C2×#12411yescharacteristic
94C2×C71C2×AGL(1,ℤ/8)12yes—
95D71—22no—
96D71—82no—
97#3138—711nomaximal, Sylow
98C71#313811yescharacteristic, complemented, Sylow
99#533—711no—
100C2×#127—711no—
101#490—711no—
102#490—711no—
103#490—711no—
104#490D7111yescharacteristic, complemented
105#486—714no—
106#418—714no—
107C4×M16—711no—
108C22×D4—712no—
109C22×D4—1422no—
110C2×M16—712no—
111C4⋊Q8—712no—
112#127—712no—
113#127—711no—
114#127—1422no—
115#127C2×D7111yescharacteristic
116#125—711no—
117#125C2×D7111yescharacteristic, complemented
118C22⋊Q8—1422no—
119C4×Q8—1422no—
120#117—712no—
121C2×C42—711no—
122#102—714no—
123#102—714no—
124#102—714no—
125#102C2×D7114yescomplemented
126#97—711no—
127#97C2×D7111yescharacteristic, complemented
128C4×C8—712no—
129C24—1422no—
130C2×Q8—1422no—
131C2×D4—22no—
132C2×D4—2848no—
133C2×D4—712no—
134C2×D4—1428no—
135C2×D4—1424no—
136C2×D4—1422no—
137C2×D4—1422no—
138C2×D4C22×D7112yes—
139C22×C4—711no—
140C22×C4—712no—
141M16—1424no—
142C2×C8—712no—
143C2×C8C22×D7112yes—
144C4⋊C4—22no—
145C4⋊C4—712no—
146C4⋊C4—1422no—
147C4⋊C4—1422no—
148C4⋊C4C22×D7112yes—
149C22⋊C4—1422no—
150C42—712no—
151C42—711no—
152C42—1422no—
153C42C22×D7111yescharacteristic
154C23—22no—
155C23—2844no—
156C23—2848no—
157C23—711no—
158C23—1422no—
159Q8—2844no—
160D4—24no—
161D4—44no—
162D4—2844no—
163D4—2848no—
164D4—2844no—
165D4—1424no—
166C2×C4—22no—
167C2×C4—711no—
168C2×C4—712no—
169C2×C4—714no—
170C2×C4—1424no—
171C2×C4—1422no—
172C2×C4D4×D7112yes—
173C2×C4C23×D7111yescharacteristic
174C8—22no—
175C8—1422no—
176C22—42no—
177C22—44no—
178C22—2842no—
179C22—2844no—
180C22—5688no—
181C22—712no—
182C22—1422no—
183C22C2×D4×D7111yescharacteristic
184C4—22no—
185C4—42no—
186C4—2842no—
187C4—712no—
188C4—714no—
189C4#41×D7112yes—
190C2—82no—
191C2—5682no—
192C2—1422no—
193C2#124×D7111yescharacteristic
194C2AGL(1,ℤ/8)×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#490×C71C2
socleC22×C71C22×D4
hypercenter#490D71
hypocenterC71#3138
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #17986555C2semidirect
3. #17986365C2semidirect
4. #17986364C2semidirect
5. #17986363C2semidirect
6. #17986344C2semidirect
7. #17986002C2semidirect
8. #17985951C2semidirect
9. #490×C71C2semidirect
13. #14556127C22semidirect
17. #14556120C22semidirect
22. #14556050C22semidirect
23. #14556049C22semidirect
24. #14556048C22semidirect
25. #14556039C22semidirect
26. C8×Dic71C22semidirect
28. #125×C71C22semidirect
29. #102×C71C22semidirect
30. #97×C71C22semidirect
98. C71#3138semidirect
104. #490D71semidirect
117. #125C2×D71semidirect
125. #102C2×D71semidirect
127. #97C2×D71semidirect