Finite Group Databasebeta

← #22599205

Subgroups — #22599205

Order 9088.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986563C212yesmaximal, complemented
2#17986324C214yesmaximal, complemented
3#17986252C212yesmaximal, complemented
4#17986235C212yesmaximal, complemented
5#17986135C214yesmaximal, complemented
6#408×C71C211yesmaximal, characteristic, complemented
7C22×D284C2212yescomplemented
8C2×D568C2212yes—
9C2×D568C2212yes—
10#14556124—22no—
11#14556117—22no—
12#14556102C2214yes—
13#14556102C2214yescomplemented
14#14556096—22no—
15#14556096—22no—
16#14556087C2212yescomplemented
17#14556062C2212yescomplemented
18#14556060C2212yescomplemented
19#14556050C2214yescomplemented
20#14556050C2214yescomplemented
21C2×#41×C71C2211yescharacteristic
22C2×M16×C71C2211yescharacteristic, complemented
23C22×C8×C71C2211yescharacteristic, complemented
24C22⋊C8×C71C2214yescomplemented
25C23×D71—22no—
26C2×D284—22no—
27C2×D284—42no—
28C2×D284—42no—
29C2×D284C2314yes—
30C2×C22⋊D71—24no—
31C2×C22⋊D71—24no—
32D568—42no—
33D568—42no—
34C4⋊Dic71C2314yes—
35C22⋊Dic71—22no—
36(C2×D71)⋊C4—24no—
37(C2×D71)⋊C4—24no—
38C4×Dic71—22no—
39#41×C71—41no—
40#41×C71—41no—
41C2×Q8×C71D411yescharacteristic
42C2×D4×C71D411yescharacteristic
43C2×D4×C71D412yes—
44C22×C4×C71C2312yes—
45C22×C4×C71C2311yescharacteristic
46M16×C71—41no—
47C2×C8×C71—41no—
48C2×C8×C71C2312yes—
49C2×C8×C71C2312yes—
50C23×C71C2×D412yes—
51C23×C71C2×D411yescharacteristic
52C22×D71—24no—
53C22×D71—44no—
54C22×D71—84no—
55Q8×C71—41no—
56D4×C71—42no—
57D4×C71—41no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—22no—
61C2×C4×C71—22no—
62C2×C4×C71C2×D412yes—
63C2×C4×C71C2×D411yescharacteristic
64C2×C4×C71C2411yescharacteristic
65C22⋊D71—84no—
66C22⋊D71—84no—
67C2×Dic71—24no—
68D284—24no—
69D284—24no—
70D284—44no—
71C8×C71—22no—
72C8×C71—22no—
73C22×C71—22no—
74C22×C71—21no—
75C22×C71—22no—
76C22×C71—24no—
77C22×C71C22×D411yescharacteristic
78C2×D71—44no—
79C2×D71—44no—
80C2×D71—44no—
81C2×D71—88no—
82C4×C71—22no—
83C4×C71—21no—
84C4×C71H(5,𝔽2)11yescharacteristic
85C4×C71H(5,𝔽2)11yescharacteristic
86Dic71—44no—
87C2×C71—42no—
88C2×C71—41no—
89C2×C71#53411yescharacteristic
90C2×C71#57612yes—
91D71—84no—
92#3199—711nomaximal, Sylow
93C71#319911yescharacteristic, complemented, Sylow
94#535—712no—
95#482—714no—
96#469—712no—
97#466—712no—
98#447—714no—
99#408D7111yescharacteristic, complemented
100C2×#41C2×D7111yescharacteristic
101C22×D4—712no—
102C2×D8—712no—
103C2×D8—712no—
104C2×M16C2×D7111yescharacteristic, complemented
105C22×C8C2×D7111yescharacteristic, complemented
106#127—1422no—
107#124—1422no—
108#121—714no—
109#121—714no—
110#120—1422no—
111#120—1422no—
112#117—712no—
113#107—712no—
114#106—712no—
115#102—714no—
116#102—714no—
117C22⋊C8C2×D7114yescomplemented
118C24—1422no—
119#41—41no—
120#41—41no—
121C2×Q8D28411yescharacteristic
122C2×D4—2842no—
123C2×D4—2842no—
124C2×D4—714no—
125C2×D4—1424no—
126C2×D4—1424no—
127C2×D4—1422no—
128C2×D4D28411yescharacteristic
129C2×D4D28412yes—
130C22×C4C22×D7112yes—
131C22×C4C22×D7111yescharacteristic
132D8—2842no—
133D8—2842no—
134M16—41no—
135C2×C8—41no—
136C2×C8C22×D7112yes—
137C2×C8C22×D7112yes—
138C4⋊C4—714no—
139C22⋊C4—1422no—
140C22⋊C4—1424no—
141C22⋊C4—1424no—
142C42—1422no—
143C23—2844no—
144C23—5684no—
145C23—1424no—
146C23C2×D28412yes—
147C23C2×D28411yescharacteristic
148Q8—41no—
149D4—42no—
150D4—41no—
151D4—2844no—
152D4—5684no—
153D4—5684no—
154D4—1424no—
155D4—1424no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—1424no—
161C2×C4C2×D28412yes—
162C2×C4C2×D28411yescharacteristic
163C2×C4C23×D7111yescharacteristic
164C8—22no—
165C8—22no—
166C22—22no—
167C22—21no—
168C22—22no—
169C22—24no—
170C22—2844no—
171C22—2844no—
172C22—2844no—
173C22—5688no—
174C22C22×D28411yescharacteristic
175C4—22no—
176C4—21no—
177C4—2844no—
178C4#1455617311yescharacteristic
179C4#1455617311yescharacteristic
180C2—42no—
181C2—41no—
182C2—5684no—
183C2#1798656111yescharacteristic
184C2#1798680412yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#408×C71C2
socleC22×C71C22×D4
hypercenter#408D71
hypocenterC71#3199
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #17986563C2semidirect
2. #17986324C2semidirect
3. #17986252C2semidirect
4. #17986235C2semidirect
5. #17986135C2semidirect
6. #408×C71C2semidirect
7. C22×D284C22semidirect
13. #14556102C22semidirect
16. #14556087C22semidirect
17. #14556062C22semidirect
18. #14556060C22semidirect
19. #14556050C22semidirect
20. #14556050C22semidirect
22. C2×M16×C71C22semidirect
23. C22×C8×C71C22semidirect
24. C22⋊C8×C71C22semidirect
93. C71#3199semidirect
99. #408D71semidirect
104. C2×M16C2×D71semidirect
105. C22×C8C2×D71semidirect
117. C22⋊C8C2×D71semidirect