Finite Group Databasebeta

← #36363815

Subgroups — #36363815

Order 22144.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 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#26012103C211yesmaximal, characteristic, complemented
2#26011832C212yesmaximal, complemented
3#26011791C212yesmaximal, complemented
4#26011696C212yesmaximal, complemented
5#26011642C212yesmaximal
6#26011591C212yesmaximal, complemented
7C4×#14583814C212yesmaximal, complemented
8#26011552C211yesmaximal, characteristic, complemented
9#518×C173C211yesmaximal, characteristic
10C2×#14583814C2212yes—
11C2×#14583814C2212yes—
12#19149679C2211yescharacteristic, complemented
13#19149672—22no—
14#19149671C2211yescharacteristic, complemented
15#19149669—22no—
16#19149658C2212yes—
17C4×Dic346C2212yescomplemented
18#19149612C2212yescomplemented
19#19149610C2212yescomplemented
20C4⋊#13063436C2212yescomplemented
21#19149602C2212yescomplemented
22#19149602C2212yescomplemented
23#19149600C2212yes—
24#19149600C2212yes—
25C22⋊#13063436C2212yes—
26#19149590C2211yescharacteristic, complemented
27C4×#13063436C2211yescharacteristic, complemented
28C22×D4×C173C2211yescharacteristic
29C4×D4×C173—22no—
30C4×D4×C173C2212yes—
31C4×D4×C173C2212yes—
32#117×C173C2211yescharacteristic
33#117×C173C2211yescharacteristic
34C2×C22⋊C4×C173—22no—
35C2×Dic346C2312yes—
36#14583814—24no—
37#14583814—24no—
38C2×#13063436C2312yes—
39C2×#13063436C2312yes—
40C4⋊Dic173—22no—
41C4⋊Dic173C2312yes—
42#14583805—22no—
43#14583805—22no—
44#14583805—22no—
45C22⋊Dic173—22no—
46C4×Dic173—22no—
47C24×C173—21no—
48C2×D4×C173—24no—
49C2×D4×C173—21no—
50C2×D4×C173C2312yes—
51C22×C4×C173—22no—
52C22×C4×C173—22no—
53C22×C4×C173C2311yescharacteristic
54C4⋊C4×C173D412yes—
55C4⋊C4×C173C2312yes—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—22no—
58C22⋊C4×C173—22no—
59C22⋊C4×C173D412yes—
60C42×C173—21no—
61C42×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63Dic346—22no—
64Dic346—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67#13063436—22no—
68#13063436—22no—
69C23×C173—24no—
70C23×C173—22no—
71C23×C173—82no—
72C23×C173C2×D411yescharacteristic
73D4×C173—24no—
74D4×C173—22no—
75D4×C173#4114yes—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173—82no—
79C2×C4×C173—82no—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D412yes—
82C2×C4×C173C2×D412yes—
83C2×C4×C173C2411yescharacteristic
84C22×C173—21no—
85C22×C173—24no—
86C22×C173—22no—
87C22×C173—42no—
88C22×C173—42no—
89C22×C173—84no—
90C22×C173C22×D411yescharacteristic
91C4×C173—21no—
92C4×C173—22no—
93C4×C173—22no—
94C4×C173—42no—
95C4×C173C2×#4112yes—
96Dic173—42no—
97Dic173—42no—
98C2×C173—41no—
99C2×C173—44no—
100C2×C173#52511yescharacteristic
101C2×C173#57712yes—
102C173#312311yescharacteristic, complemented, Sylow
103#3123—1731nomaximal, Sylow
104#541—1731no—
105#518D17311yescharacteristic
106#485—1732no—
107#478—1732no—
108#461—1732no—
109#450—1732no—
110#440—1732no—
111C4×SD16—1732no—
112#433—1731no—
113C22×D4C2×D17311yescharacteristic
114C2×SD16—1732no—
115C2×SD16—1732no—
116C4⋊Q8—1731no—
117#126—3462no—
118#125—3462no—
119#125—1731no—
120C22⋊Q8—1732no—
121C4×Q8—1732no—
122C4×D4—22no—
123C4×D4C2×D17312yes—
124C4×D4C2×D17312yes—
125#117C2×D17311yescharacteristic
126#117C2×D17311yescharacteristic
127C2×C22⋊C4—22no—
128#107—1732no—
129#106—1732no—
130C4⋊C8—1732no—
131#103—1732no—
132#103—1732no—
133#102—1732no—
134#102—1732no—
135C22⋊C8—1732no—
136#97—1731no—
137C4×C8—1731no—
138C24—21no—
139C2×Q8—1732no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D17312yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D17311yescharacteristic
146SD16—3464no—
147SD16—3464no—
148C2×C8—1732no—
149C2×C8—1732no—
150C4⋊C4—3462no—
151C4⋊C4—3462no—
152C4⋊C4—3462no—
153C4⋊C4—3462no—
154C4⋊C4—1732no—
155C4⋊C4C22⋊D17312yes—
156C4⋊C4C22×D17312yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—3462no—
161C22⋊C4C22⋊D17312yes—
162C42—21no—
163C42—3462no—
164C42C22×D17311yescharacteristic
165C42C22×D17311yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D17311yescharacteristic
170Q8—3462no—
171Q8—3462no—
172D4—24no—
173D4—22no—
174D4#1458382614yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4C2×C22⋊D17311yescharacteristic
182C2×C4C2×C22⋊D17312yes—
183C2×C4C2×C22⋊D17312yes—
184C2×C4C23×D17311yescharacteristic
185C8—3462no—
186C8—3462no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D17311yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—6922no—
199C4—6922no—
200C4C2×#1458382612yes—
201C2—41no—
202C2—44no—
203C2#2601200211yescharacteristic
204C2#2601230212yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#518×C173C2
socleC22×C173C22×D4
prime radicalC22×D4×C173C22
hypercenter#518D173
hypocenterC173#3123
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #26012103C2semidirect
2. #26011832C2semidirect
3. #26011791C2semidirect
4. #26011696C2semidirect
6. #26011591C2semidirect
7. C4×#14583814C2semidirect
8. #26011552C2semidirect
12. #19149679C22semidirect
14. #19149671C22semidirect
17. C4×Dic346C22semidirect
18. #19149612C22semidirect
19. #19149610C22semidirect
20. C4⋊#13063436C22semidirect
21. #19149602C22semidirect
22. #19149602C22semidirect
26. #19149590C22semidirect
27. C4×#13063436C22semidirect
102. C173#3123semidirect

Maximal subgroups

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

subgroupindex
#260121032
#260118322
#260117912
#260116962
#260116422
#260115912
C4×#145838142
#260115522
#518×C1732
#3123173