Finite Group Databasebeta

← #36362473

Subgroups — #36362473

Order 22144.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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#26012258C212yesmaximal, complemented
2#26011613C214yesmaximal, complemented
3#26011608C214yesmaximal, complemented
4#26011540C211yesmaximal, characteristic, complemented
5C2×#19149606C212yesmaximal, complemented
6M16×Dic173C211yesmaximal, characteristic, complemented
7#575×C173C211yesmaximal, characteristic
8#19149686—24no—
9#19149686C2218yescomplemented
10C2×#14583810—24no—
11C2×#14583810C2212yes—
12C2×#14583810C2211yescharacteristic
13C2×C4×Dic173C2211yescharacteristic
14#19149615C2214yescomplemented
15#19149606C2214yescomplemented
16#19149606C2214yescomplemented
17#19149589C2212yescomplemented
18C8×Dic173C2212yescomplemented
19C2×#41×C173C2212yes—
20#137×C173C414yescomplemented
21AGL(1,ℤ/8)×C173C414yescomplemented
22#135×C173C2214yes—
23C2×M16×C173C2211yescharacteristic
24C22×Dic173—21no—
25#14583810—28no—
26#14583810—22no—
27#14583810D414yes—
28#14583810C2314yes—
29C2×#13063436—28no—
30C2×#13063436—22no—
31C2×#13063436C2312yes—
32C4×Dic173—21no—
33C4×Dic173C2311yescharacteristic
34C4×Dic173C2311yescharacteristic
35#41×C173—22no—
36#41×C173—22no—
37#41×C173C2×C414yes—
38#41×C173C2314yes—
39C2×Q8×C173C2×C412yes—
40C2×D4×C173—22no—
41C2×D4×C173C2×C412yes—
42C22×C4×C173—22no—
43C22×C4×C173C2311yescharacteristic
44Q16×C173C2×C414yescomplemented
45SD16×C173C2×C418yescomplemented
46D8×C173C2×C414yescomplemented
47M16×C173C2×C414yes—
48C2×C8×C173C2312yes—
49C2×Dic173—21no—
50C2×Dic173—21no—
51C2×Dic173—42no—
52#13063436—24no—
53#13063436C2×D414yes—
54C23×C173—22no—
55C23×C173#4111yescharacteristic
56Q8×C173—22no—
57Q8×C173C22×C414yes—
58D4×C173—24no—
59D4×C173—24no—
60D4×C173—22no—
61D4×C173C22×C414yes—
62C2×C4×C173—22no—
63C2×C4×C173—24no—
64C2×C4×C173—24no—
65C2×C4×C173C22×C412yes—
66C2×C4×C173C2×D412yes—
67C2×C4×C173#4111yescharacteristic
68C2×C4×C173C2411yescharacteristic
69C8×C173C22×C414yes—
70C22×C173—24no—
71C22×C173—41no—
72C22×C173—82no—
73C22×C173C4×D412yes—
74C22×C173C2×#4111yescharacteristic
75C4×C173—24no—
76C4×C173C4×D412yes—
77C4×C173C23×C411yescharacteristic
78C4×C173C22×D411yescharacteristic
79Dic173—22no—
80Dic173—41no—
81C2×C173—21no—
82C2×C173—22no—
83C2×C173—44no—
84C2×C173C2×C4×D411yescharacteristic
85C173#295611yescharacteristic, complemented, Sylow
86#2956—1731nomaximal, Sylow
87#575D17311yescharacteristic
88#568—1732no—
89#444—1734no—
90#443—1734no—
91#430—1731no—
92C2×#104—1732no—
93C4×M16—1731no—
94C2×#41C2×D17312yes—
95#137Dic17314yescomplemented
96AGL(1,ℤ/8)Dic17314yescomplemented
97#135C2×D17314yes—
98#131—3464no—
99#131—1738no—
100C2×M16—3464no—
101C2×M16—1732no—
102C2×M16—1731no—
103C2×M16C2×D17311yescharacteristic
104C2×C42—1731no—
105#108—1734no—
106#104—1734no—
107#104—1734no—
108#97—1732no—
109C4×C8—1732no—
110#41—22no—
111#41—22no—
112#41C2×Dic17314yes—
113#41C22×D17314yes—
114C2×Q8C2×Dic17312yes—
115C2×D4—22no—
116C2×D4C2×Dic17312yes—
117C22×C4—22no—
118C22×C4—3461no—
119C22×C4C22×D17311yescharacteristic
120Q16C2×Dic17314yescomplemented
121SD16C2×Dic17318yescomplemented
122D8C2×Dic17314yescomplemented
123M16—3462no—
124M16—3468no—
125M16—1734no—
126M16—1734no—
127M16C2×Dic17314yes—
128C2×C8—3462no—
129C2×C8—3468no—
130C2×C8—1732no—
131C2×C8C22×D17312yes—
132C42—3461no—
133C42—1731no—
134C42—1731no—
135C23—22no—
136C23#1458382511yescharacteristic
137Q8—22no—
138Q8C22×Dic17314yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic17314yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—3461no—
147C2×C4—3461no—
148C2×C4—6922no—
149C2×C4C22×Dic17312yes—
150C2×C4D4×D17312yes—
151C2×C4#1458382511yescharacteristic
152C2×C4C23×D17311yescharacteristic
153C8—3464no—
154C8—1734no—
155C8C22×Dic17314yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic17312yes—
160C22C2×#1458382511yescharacteristic
161C4—24no—
162C4—3462no—
163C4—6921no—
164C4D4×Dic17312yes—
165C4C23×Dic17311yescharacteristic
166C4C2×D4×D17311yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic17311yescharacteristic
subgroupG/H
centerC4C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting, prime radical#575×C173C2
socleC2×C173C2×C4×D4
cosocleC2×C4×C173C24
hypercenter#575D173
hypocenterC173#2956
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #26012258C2semidirect
2. #26011613C2semidirect
3. #26011608C2semidirect
4. #26011540C2semidirect
5. C2×#19149606C2semidirect
6. M16×Dic173C2semidirect
9. #19149686C22semidirect
14. #19149615C22semidirect
15. #19149606C22semidirect
16. #19149606C22semidirect
17. #19149589C22semidirect
18. C8×Dic173C22semidirect
20. #137×C173C4semidirect
21. AGL(1,ℤ/8)×C173C4semidirect
44. Q16×C173C2×C4semidirect
45. SD16×C173C2×C4semidirect
46. D8×C173C2×C4semidirect
85. C173#2956semidirect
95. #137Dic173semidirect
96. AGL(1,ℤ/8)Dic173semidirect
120. Q16C2×Dic173semidirect
121. SD16C2×Dic173semidirect
122. D8C2×Dic173semidirect

Maximal subgroups

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

subgroupindex
#260122582
#260116132
#260116082
#260115402
C2×#191496062
M16×Dic1732
#575×C1732
#2956173