Finite Group Databasebeta

← #36362023

Subgroups — #36362023

Order 22144.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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×#19149724C211yesmaximal, characteristic
2#26011966C211yesmaximal, characteristic, complemented
3#26011464C214yesmaximal, complemented
4C2×#19149593C218yesmaximal, complemented
5#518×C173C211yesmaximal, characteristic, complemented
6#19149724—24no—
7#19149724—24no—
8C2×#14583826C2212yes—
9C2×#14583825C414yescomplemented
10C22×C22⋊D173C2214yes—
11C2×D4×D173C414yescomplemented
12D4×Dic173—24no—
13#19149638C2212yescomplemented
14C2×C22⋊Dic173C2214yescomplemented
15#19149593C22116yescomplemented
16C22×D4×C173C2211yescharacteristic
17C4×D4×C173—24no—
18#117×C173C2212yescomplemented
19C2×C22⋊C4×C173C2214yescomplemented
20C23×D173C2×C414yescomplemented
21#14583826—24no—
22#14583826—24no—
23#14583825—216no—
24C2×Dic346C2×C412yescomplemented
25C2×D692C2×C412yescomplemented
26C2×C22⋊D173—28no—
27C2×C22⋊D173—28no—
28C2×C22⋊D173C2×C418yescomplemented
29C2×C22⋊D173C2314yes—
30D4×D173—216no—
31C22×Dic173—24no—
32C22×Dic173C2×C414yescomplemented
33C2×C4×D173C2×C414yescomplemented
34C4⋊Dic173—22no—
35C22⋊Dic173—24no—
36C22⋊Dic173C2314yes—
37C4×Dic173—22no—
38C24×C173C2312yes—
39C2×D4×C173C2×C414yes—
40C2×D4×C173D414yes—
41C2×D4×C173D414yes—
42C22×C4×C173—24no—
43C22×C4×C173C2311yescharacteristic
44C4⋊C4×C173—22no—
45C22⋊C4×C173—24no—
46C22⋊C4×C173C2314yes—
47C42×C173—22no—
48C22×D173—28no—
49C22×D173—84no—
50C22×D173C22×C414yes—
51Dic346—24no—
52D692—24no—
53C22⋊D173—216no—
54C22⋊D173—24no—
55C22⋊D173—24no—
56C2×Dic173—28no—
57C2×Dic173—24no—
58C2×Dic173—84no—
59C2×Dic173C22×C414yes—
60C4×D173—28no—
61C23×C173—82no—
62C23×C173C22×C414yes—
63C23×C173C2×D414yes—
64C23×C173C2×D414yes—
65C23×C173C2411yescharacteristic
66D4×C173C22⋊C4116yes—
67C2×C4×C173—24no—
68C2×C4×C173—84no—
69C2×C4×C173C22×C412yes—
70C2×C4×C173C2×D412yes—
71C2×C4×C173C2×D412yes—
72C22×C173—44no—
73C22×C173—41no—
74C22×C173—44no—
75C22×C173—44no—
76C22×C173C2×C22⋊C418yes—
77C22×C173C23×C411yescharacteristic
78C22×C173C22×D411yescharacteristic
79C22×C173C22×D411yescharacteristic
80C2×D173—24no—
81C2×D173—44no—
82C2×D173—88no—
83C4×C173—44no—
84C4×C173C2×C22⋊C414yes—
85Dic173—24no—
86Dic173—44no—
87C2×C173—21no—
88C2×C173—21no—
89C2×C173—21no—
90C2×C173—28no—
91C2×C173C22×C22⋊C411yescharacteristic
92D173—44no—
93C173#288311yescharacteristic, complemented, Sylow
94#2883—1731nomaximal, Sylow
95C2×H(5,𝔽2)—1731no—
96#518—1731no—
97#518D17311yescharacteristic, complemented
98#410—1734no—
99C2×#99—1738no—
100H(5,𝔽2)—3464no—
101H(5,𝔽2)—3464no—
102C2×#41—1734no—
103C2×#41—1732no—
104C22×D4—1734no—
105C22×D4—1734no—
106C22×D4C2×D17311yescharacteristic
107C4×D4—24no—
108C4×D4—3464no—
109#117—1732no—
110#117C2×D17312yescomplemented
111C2×C22⋊C4—1734no—
112C2×C22⋊C4C2×D17314yescomplemented
113#99—17316no—
114C24—1734no—
115C24C22×D17312yes—
116#41—34616no—
117#41—3464no—
118#41—3464no—
119C2×Q8—1732no—
120C2×D4—3468no—
121C2×D4—3468no—
122C2×D4—34616no—
123C2×D4—1738no—
124C2×D4—1734no—
125C2×D4—1732no—
126C2×D4C4×D17314yes—
127C2×D4C22⋊D17314yes—
128C2×D4D69214yes—
129C22×C4—24no—
130C22×C4—3464no—
131C22×C4—1734no—
132C22×C4—1734no—
133C22×C4C22×D17311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—3462no—
136C22⋊C4—24no—
137C22⋊C4—3464no—
138C22⋊C4—1734no—
139C22⋊C4C22×D17314yes—
140C42—22no—
141C42—3462no—
142C23—82no—
143C23—3468no—
144C23—13844no—
145C23—1734no—
146C23C2×C4×D17314yes—
147C23C2×C22⋊D17314yes—
148C23C2×D69214yes—
149C23C23×D17311yescharacteristic
150Q8—3464no—
151D4—34616no—
152D4—3464no—
153D4—3464no—
154D4—3464no—
155D4#14583803116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—3468no—
159C2×C4—3468no—
160C2×C4—3464no—
161C2×C4—13844no—
162C2×C4—1734no—
163C2×C4C2×C4×D17312yes—
164C2×C4C2×C22⋊D17312yes—
165C2×C4C2×D69212yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—3464no—
171C22—13848no—
172C22—6924no—
173C22C2×#1458380318yes—
174C22C22×C4×D17311yescharacteristic
175C22C22×C22⋊D17311yescharacteristic
176C22C22×D69211yescharacteristic
177C4—44no—
178C4—3464no—
179C4—6924no—
180C4C2×#1458380314yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—6924no—
186C2C22×#1458380311yescharacteristic
subgroupG/H
centerC2C22×#14583803
commutatorC22×C173C23×C4
FrattiniC23C23×D173
Fitting#518×C173C2
socleC2×C173C22×C22⋊C4
prime radicalC2×#19149724C2
cosocleC23×C173C24
hypercenter#518D173
hypocenterC173#2883
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #26011966C2semidirect
3. #26011464C2semidirect
4. C2×#19149593C2semidirect
5. #518×C173C2semidirect
9. C2×#14583825C4semidirect
11. C2×D4×D173C4semidirect
13. #19149638C22semidirect
14. C2×C22⋊Dic173C22semidirect
15. #19149593C22semidirect
18. #117×C173C22semidirect
19. C2×C22⋊C4×C173C22semidirect
20. C23×D173C2×C4semidirect
24. C2×Dic346C2×C4semidirect
25. C2×D692C2×C4semidirect
28. C2×C22⋊D173C2×C4semidirect
32. C22×Dic173C2×C4semidirect
33. C2×C4×D173C2×C4semidirect
93. C173#2883semidirect
97. #518D173semidirect
110. #117C2×D173semidirect
112. C2×C22⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
#260119662
#260114642
C2×#191495932
#518×C1732
#2883173