Finite Group Databasebeta

← #36365363

Subgroups — #36365363

Order 22144.

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#26012296C212yesmaximal, complemented
2#26012293C212yesmaximal, complemented
3#26011868C211yesmaximal, characteristic, complemented
4#26011762C212yesmaximal, complemented
5#26011761C212yesmaximal, complemented
6#26011660C214yesmaximal, complemented
7#26011609C211yesmaximal, characteristic, complemented
8#443×C173C211yesmaximal, characteristic, complemented
9#19149725C2214yescomplemented
10#19149707—24no—
11#19149707C2214yescomplemented
12#19149706—24no—
13#19149704C2214yescomplemented
14#19149702—22no—
15#19149701—22no—
16#19149701C2211yescharacteristic
17C2×D1384—22no—
18C2×D1384C2212yescomplemented
19C2×D1384C2212yescomplemented
20D8×D173—22no—
21#19149685C2212yescomplemented
22#19149676C2212yescomplemented
23#19149615C2211yescharacteristic, complemented
24#19149607C2212yescomplemented
25#19149595C2214yescomplemented
26#135×C173C2211yescharacteristic
27#131×C173C2212yescomplemented
28#108×C173C2211yescharacteristic, complemented
29#104×C173C2212yescomplemented
30C4×C8×C173C2211yescharacteristic, complemented
31#14583827—24no—
32#14583826—24no—
33#14583826C2312yes—
34C2×D692—24no—
35C2×D692—42no—
36C2×D692C2314yes—
37D4×D173—24no—
38D4×D173—44no—
39Dic692D411yescharacteristic, complemented
40#14583815—24no—
41#14583815D412yescomplemented
42#14583813—24no—
43D1384—24no—
44D1384—22no—
45D1384—22no—
46D1384—22no—
47D1384—22no—
48D1384D411yescharacteristic, complemented
49#14583811—24no—
50#14583810C2312yes—
51#14583809—22no—
52C8×D173—22no—
53#41×C173C2312yes—
54Q16×C173D411yescharacteristic, complemented
55SD16×C173D412yescomplemented
56D8×C173D411yescharacteristic, complemented
57M16×C173—22no—
58M16×C173C2312yes—
59C2×C8×C173—22no—
60C2×C8×C173C2311yescharacteristic
61C2×C8×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63C22×D173—24no—
64C22×D173—44no—
65Dic346C2×D412yes—
66D692—24no—
67D692—24no—
68D692—44no—
69D692—84no—
70D692C2×D412yes—
71C22⋊D173—22no—
72C22⋊D173—44no—
73C4×D173—22no—
74C4×D173—44no—
75#13063436—22no—
76Q8×C173C2×D412yes—
77D4×C173—22no—
78D4×C173C2×D412yes—
79C2×C4×C173—21no—
80C2×C4×C173—22no—
81C2×C4×C173C2411yescharacteristic
82C8×C173—22no—
83C8×C173C2×D411yescharacteristic
84C8×C173C2×D411yescharacteristic
85C8×C173C2×D411yescharacteristic
86C8×C173C2×D411yescharacteristic
87C22×C173—22no—
88C22×C173H(5,𝔽2)11yescharacteristic
89C2×D173—22no—
90C2×D173—44no—
91C2×D173—84no—
92C2×D173—164no—
93C4×C173—22no—
94C4×C173—22no—
95C4×C173C22×D411yescharacteristic
96C4×C173C22×D411yescharacteristic
97Dic173—22no—
98C2×C173—21no—
99C2×C173—42no—
100C2×C173D4211yescharacteristic
101D173—42no—
102D173—84no—
103C173#329211yescharacteristic, complemented, Sylow
104#3292—1731nomaximal, Sylow
105#576—1732no—
106#576—1732no—
107#493—1731no—
108#472—1732no—
109#472—1732no—
110#453—1734no—
111#443—1731no—
112#443D17311yescharacteristic, complemented
113H(5,𝔽2)—1734no—
114AGL(1,ℤ/8)—3464no—
115AGL(1,ℤ/8)—3464no—
116AGL(1,ℤ/8)—1734no—
117AGL(1,ℤ/8)—1734no—
118#135—3462no—
119#135—3462no—
120#135—1731no—
121#135C2×D17311yescharacteristic
122C2×D8—3462no—
123C2×D8—3462no—
124C2×D8—1732no—
125C2×D8—1732no—
126#131—1732no—
127#131C2×D17312yescomplemented
128#127—1732no—
129#108—1731no—
130#108C2×D17311yescharacteristic, complemented
131#104—1732no—
132#104C2×D17312yescomplemented
133#100—1734no—
134C4×C8C2×D17311yescharacteristic, complemented
135#41—3464no—
136#41—3464no—
137#41—1732no—
138#41C22×D17312yes—
139C2×D4—3464no—
140C2×D4—3464no—
141C2×D4—1734no—
142C2×D4—6922no—
143C2×D4—6924no—
144Q16—1731no—
145Q16D69211yescharacteristic, complemented
146SD16—3464no—
147SD16—3464no—
148SD16—1732no—
149SD16D69212yescomplemented
150D8—3464no—
151D8—3464no—
152D8—3462no—
153D8—3462no—
154D8—3462no—
155D8—3462no—
156D8—1731no—
157D8D69211yescharacteristic, complemented
158M16—22no—
159M16—3462no—
160M16—1732no—
161M16C22×D17312yes—
162C2×C8—22no—
163C2×C8—3462no—
164C2×C8C22×D17311yescharacteristic
165C2×C8C22×D17311yescharacteristic
166C42C22×D17311yescharacteristic
167C23—3464no—
168C23—6924no—
169Q8—1732no—
170Q8C2×D69212yes—
171D4—22no—
172D4—3462no—
173D4—3464no—
174D4—3464no—
175D4—13844no—
176D4—1732no—
177D4—6924no—
178D4—6924no—
179D4C2×D69212yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—3462no—
183C2×C4—6924no—
184C2×C4C23×D17311yescharacteristic
185C8—22no—
186C8—3462no—
187C8D4×D17311yescharacteristic
188C8D4×D17311yescharacteristic
189C8C2×D69211yescharacteristic
190C8C2×D69211yescharacteristic
191C22—22no—
192C22—3462no—
193C22—13844no—
194C22—6924no—
195C22—27684no—
196C22#1914972511yescharacteristic
197C4—22no—
198C4—22no—
199C4—3462no—
200C4C2×D4×D17311yescharacteristic
201C4C22×D69211yescharacteristic
202C2—21no—
203C2—42no—
204C2—13844no—
205C2—6922no—
206C2D4×D69211yescharacteristic
subgroupG/H
centerC2D4×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#443×C173C2
socleC2×C173D42
hypercenter#443D173
hypocenterC173#3292
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #26012296C2semidirect
2. #26012293C2semidirect
3. #26011868C2semidirect
4. #26011762C2semidirect
5. #26011761C2semidirect
6. #26011660C2semidirect
7. #26011609C2semidirect
8. #443×C173C2semidirect
9. #19149725C22semidirect
11. #19149707C22semidirect
13. #19149704C22semidirect
18. C2×D1384C22semidirect
19. C2×D1384C22semidirect
21. #19149685C22semidirect
22. #19149676C22semidirect
23. #19149615C22semidirect
24. #19149607C22semidirect
25. #19149595C22semidirect
27. #131×C173C22semidirect
28. #108×C173C22semidirect
29. #104×C173C22semidirect
30. C4×C8×C173C22semidirect
39. Dic692D4semidirect
41. #14583815D4semidirect
48. D1384D4semidirect
54. Q16×C173D4semidirect
55. SD16×C173D4semidirect
56. D8×C173D4semidirect
103. C173#3292semidirect
112. #443D173semidirect
127. #131C2×D173semidirect
130. #108C2×D173semidirect
132. #104C2×D173semidirect
134. C4×C8C2×D173semidirect
145. Q16D692semidirect
149. SD16D692semidirect
157. D8D692semidirect

Maximal subgroups

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

subgroupindex
#260122962
#260122932
#260118682
#260117622
#260117612
#260116602
#260116092
#443×C1732
#3292173