Finite Group Databasebeta

← #22602118

Subgroups — #22602118

Order 9088.

1748 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 557 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
1C2×#14556174C212yesmaximal, complemented
2#17986671C214yesmaximal, complemented
3D4×D284C2112yesmaximal, complemented
4#17986568C216yesmaximal, complemented
5#17986549C213yesmaximal, complemented
6C2×#14556125C213yesmaximal, complemented
7C4×#41×C71C211yesmaximal, characteristic, complemented
8#14556174—216no—
9C2×#11765866C2214yescomplemented
10C2×#11765865C2216yescomplemented
11C22×D284C2216yescomplemented
12C22×D284C22112yescomplemented
13C2×D4×D71C22112yescomplemented
14#14556128C2211yescharacteristic, complemented
15#14556125C2216yescomplemented
16#14556125C22112yescomplemented
17#14556125C2213yescomplemented
18#14556116C2216yescomplemented
19#14556103C22124yescomplemented
20#14556102C22112yescomplemented
21#14556098C22124yescomplemented
22C4×D284C22112yescomplemented
23C2×#41×C71C2211yescharacteristic
24C4×Q8×C71C2212yescomplemented
25C4×D4×C71C2216yescomplemented
26#117×C71C2213yescomplemented
27C2×C42×C71C2213yescomplemented
28C23×D71C23112yes—
29#11765866—216no—
30#11765865—224no—
31C2×Dic142C2312yes—
32C2×D284—224no—
33C2×D284—424no—
34C2×D284C23112yes—
35C2×D284C2316yes—
36C2×D284C23112yes—
37C2×D284C23124yes—
38C2×C22⋊D71C23112yes—
39D4×D71—248no—
40C2×C4×D71C23112yes—
41C4⋊Dic71C2314yes—
42(C2×D71)⋊C4C23124yes—
43#41×C71D418yescomplemented
44C2×Q8×C71C2311yescharacteristic
45C2×D4×C71C2313yes—
46C22×C4×C71C2316yes—
47C22×C4×C71C2313yes—
48C4⋊C4×C71C2316yes—
49C22⋊C4×C71C2316yes—
50C42×C71C2311yescharacteristic
51C42×C71C2313yes—
52C42×C71C2316yes—
53C23×C71C2413yes—
54C22×D71—224no—
55C22×D71—424no—
56C22×D71C24112yes—
57Q8×C71C2×D414yescomplemented
58D4×C71C2×D4112yescomplemented
59C2×C4×C71—212no—
60C2×C4×C71C2×D4112yescomplemented
61C2×C4×C71C2416yes—
62C2×C4×C71C2411yescharacteristic
63C2×C4×C71C2413yes—
64C2×C4×C71C2412yes—
65C22⋊D71—224no—
66C2×Dic71C2414yes—
67D284—224no—
68D284—212no—
69D284—224no—
70D284—424no—
71C4×D71—224no—
72Dic142—24no—
73C22×C71—26no—
74C22×C71C22×D416yes—
75C22×C71C2511yescharacteristic
76C2×D71—212no—
77C2×D71—224no—
78C2×D71—824no—
79C4×C71—26no—
80C4×C71C22×D412yes—
81C4×C71C22×D416yes—
82C4×C71H(5,𝔽2)14yescomplemented
83Dic71—24no—
84C2×C71—26no—
85C2×C71C23×D411yescharacteristic
86C2×C71C2×H(5,𝔽2)12yes—
87D71—412no—
88#3498—711nomaximal, Sylow
89C71#349811yescharacteristic, complemented, Sylow
90C2×H(5,𝔽2)—712no—
91#550—714no—
92D42—7112no—
93#535—716no—
94#532—713no—
95C2×#127—713no—
96C4×#41D7111yescharacteristic, complemented
97H(5,𝔽2)—14216no—
98C2×#41—714no—
99C2×#41—716no—
100C2×#41C2×D7111yescharacteristic
101C22×D4—7112no—
102C22×D4—7112no—
103C22×D4—716no—
104C4⋊Q8—711no—
105#127—713no—
106#127—716no—
107#127—7112no—
108#124—716no—
109#121—7112no—
110#121—7124no—
111#120—7124no—
112C4×Q8C2×D7112yescomplemented
113C4×D4—7112no—
114C4×D4C2×D7116yescomplemented
115#117C2×D7113yescomplemented
116C2×C42C2×D7113yescomplemented
117C24—7112no—
118#41—14216no—
119#41—14224no—
120#41D28418yescomplemented
121C2×Q8—712no—
122C2×Q8C22×D7111yescharacteristic
123C2×D4—28424no—
124C2×D4—7112no—
125C2×D4—7124no—
126C2×D4—7112no—
127C2×D4—7112no—
128C2×D4—716no—
129C2×D4—14248no—
130C2×D4—14224no—
131C2×D4C22×D7113yes—
132C22×C4—7112no—
133C22×C4C22×D7116yes—
134C22×C4C22×D7113yes—
135C4⋊C4—714no—
136C4⋊C4C22×D7116yes—
137C22⋊C4—7124no—
138C22⋊C4C22×D7116yes—
139C42C22×D7111yescharacteristic
140C42C22×D7113yes—
141C42C22×D7116yes—
142C23—28424no—
143C23—7112no—
144C23—14224no—
145C23C23×D7113yes—
146Q8—1424no—
147Q8C2×D28414yescomplemented
148D4—28424no—
149D4—14224no—
150D4—14224no—
151D4—14212no—
152D4—14224no—
153D4C2×D284112yescomplemented
154C2×C4—212no—
155C2×C4—714no—
156C2×C4—14224no—
157C2×C4C2×D284112yescomplemented
158C2×C4C23×D7116yes—
159C2×C4C23×D7111yescharacteristic
160C2×C4C23×D7113yes—
161C2×C4C23×D7112yes—
162C22—26no—
163C22—56824no—
164C22—14212no—
165C22—14224no—
166C22C22×D28416yes—
167C22C24×D7111yescharacteristic
168C4—26no—
169C4—1424no—
170C4C22×D28416yes—
171C4C22×D28412yes—
172C4#1455617414yescomplemented
173C2—26no—
174C2—28412no—
175C2C23×D28411yescharacteristic
176C2C2×#1455617412yes—
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
FittingC4×#41×C71C2
hypercenterC4×#41D71
hypocenterC71#3498
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#14556174C2semidirect
2. #17986671C2semidirect
3. D4×D284C2semidirect
4. #17986568C2semidirect
5. #17986549C2semidirect
6. C2×#14556125C2semidirect
7. C4×#41×C71C2semidirect
9. C2×#11765866C22semidirect
10. C2×#11765865C22semidirect
11. C22×D284C22semidirect
12. C22×D284C22semidirect
13. C2×D4×D71C22semidirect
14. #14556128C22semidirect
15. #14556125C22semidirect
16. #14556125C22semidirect
17. #14556125C22semidirect
18. #14556116C22semidirect
19. #14556103C22semidirect
20. #14556102C22semidirect
21. #14556098C22semidirect
22. C4×D284C22semidirect
24. C4×Q8×C71C22semidirect
25. C4×D4×C71C22semidirect
26. #117×C71C22semidirect
27. C2×C42×C71C22semidirect
43. #41×C71D4semidirect
57. Q8×C71C2×D4semidirect
58. D4×C71C2×D4semidirect
60. C2×C4×C71C2×D4semidirect
82. C4×C71H(5,𝔽2)semidirect
89. C71#3498semidirect
96. C4×#41D71semidirect
112. C4×Q8C2×D71semidirect
114. C4×D4C2×D71semidirect
115. #117C2×D71semidirect
116. C2×C42C2×D71semidirect
120. #41D284semidirect
147. Q8C2×D284semidirect
153. D4C2×D284semidirect
157. C2×C4C2×D284semidirect
172. C4#14556174semidirect