Finite Group Databasebeta

← #36362790

Subgroups — #36362790

Order 22144.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#19149705C212yesmaximal, complemented
2#26012002C211yesmaximal, characteristic, complemented
3#26011642C214yesmaximal, complemented
4#26011625C214yesmaximal, complemented
5#26011496C212yesmaximal, complemented
6#26011455C211yesmaximal, characteristic, complemented
7C23×D4×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic, complemented
9#19149705—28no—
10C2×#14583814C2214yescomplemented
11C2×#14583811C2214yescomplemented
12C2×#14583810C2212yescomplemented
13#19149662—21no—
14#19149658C2212yescomplemented
15#19149653C2212yescomplemented
16#19149648—21no—
17C4×C22⋊D173—22no—
18#19149638C2211yescharacteristic, complemented
19#19149600C2218yescomplemented
20C22⋊#13063436C2214yescomplemented
21C25×C173—21no—
22C22×D4×C173—28no—
23C22×D4×C173—22no—
24C22×D4×C173—21no—
25C22×D4×C173C2214yes—
26C22×D4×C173C2212yes—
27C23×C4×C173C2211yescharacteristic
28#14583826—24no—
29C2×Dic346C2311yescharacteristic, complemented
30C2×D692C2311yescharacteristic, complemented
31C2×C22⋊D173—22no—
32C2×C22⋊D173—21no—
33C2×C4×D173—21no—
34#14583814—28no—
35#14583811—28no—
36#14583810—24no—
37C2×#13063436C2314yescomplemented
38C4⋊Dic173C2312yescomplemented
39#14583805—22no—
40C22⋊Dic173—21no—
41C22⋊Dic173—22no—
42#14583803—22no—
43C4×Dic173—21no—
44C24×C173—28no—
45C24×C173—24no—
46C24×C173—22no—
47C24×C173—48no—
48C24×C173D411yescharacteristic
49C2×D4×C173—28no—
50C2×D4×C173—216no—
51C2×D4×C173—216no—
52C2×D4×C173—24no—
53C2×D4×C173—22no—
54C2×D4×C173—24no—
55C2×D4×C173D418yescomplemented
56C2×D4×C173C2314yes—
57C22×C4×C173—24no—
58C22×C4×C173D412yes—
59C22×C4×C173D411yescharacteristic
60C22×C4×C173C2312yes—
61C22×C4×C173C2311yescharacteristic
62C22×D173—21no—
63Dic346—22no—
64D692—22no—
65C22⋊D173—42no—
66C22⋊D173—44no—
67C2×Dic173—21no—
68C2×Dic173—22no—
69C4×D173—42no—
70#13063436—24no—
71C23×C173—22no—
72C23×C173—24no—
73C23×C173—216no—
74C23×C173—28no—
75C23×C173—24no—
76C23×C173—432no—
77C23×C173—48no—
78C23×C173—48no—
79C23×C173—48no—
80C23×C173C2×D411yescharacteristic
81C23×C173C2×D412yes—
82D4×C173—216no—
83D4×C173—28no—
84D4×C173—24no—
85D4×C173C2×D418yes—
86C2×C4×C173—22no—
87C2×C4×C173—24no—
88C2×C4×C173—24no—
89C2×C4×C173C2×D412yes—
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2×D414yes—
92C2×C4×C173C2411yescharacteristic
93C22×C173—28no—
94C22×C173—22no—
95C22×C173—22no—
96C22×C173—22no—
97C22×C173—28no—
98C22×C173—416no—
99C22×C173—432no—
100C22×C173—48no—
101C22×C173#12012yes—
102C22×C173AGL(1,ℤ/8)14yescomplemented
103C22×C173C22×D411yescharacteristic
104C2×D173—41no—
105C2×D173—42no—
106C4×C173—22no—
107C4×C173#12012yes—
108C4×C173C22×D412yes—
109Dic173—41no—
110Dic173—42no—
111C2×C173—22no—
112C2×C173—24no—
113C2×C173—48no—
114C2×C173C2×#12011yescharacteristic
115C2×C173C2×AGL(1,ℤ/8)12yes—
116D173—81no—
117C173#300311yescharacteristic, complemented, Sylow
118#3003—1731nomaximal, Sylow
119C23×D4D17311yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1732no—
121#525—1731no—
122#450—1734no—
123#447—1734no—
124#418—1732no—
125#407—1731no—
126C25—21no—
127C2×#41—1731no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D17314yes—
132C22×D4C2×D17312yes—
133C23×C4C2×D17311yescharacteristic
134AGL(1,ℤ/8)—3468no—
135C2×SD16—1734no—
136C2×D8—1734no—
137C2×M16—1732no—
138#123—3461no—
139C22⋊Q8—1732no—
140#121—1732no—
141#120—3461no—
142C4×D4—3462no—
143#117—1731no—
144#102—1738no—
145C22⋊C8—1734no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D17311yescharacteristic
151#41—3464no—
152C2×Q8—1731no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—3462no—
160C2×D4—3461no—
161C2×D4—1731no—
162C2×D4C22⋊D17318yescomplemented
163C2×D4C22×D17314yes—
164C22×C4—24no—
165C22×C4—3461no—
166C22×C4C22⋊D17312yes—
167C22×C4C22⋊D17311yescharacteristic
168C22×C4C22×D17312yes—
169C22×C4C22×D17311yescharacteristic
170SD16—3468no—
171D8—3468no—
172M16—3464no—
173C2×C8—1734no—
174C4⋊C4—3462no—
175C4⋊C4—1732no—
176C22⋊C4—3461no—
177C22⋊C4—3462no—
178C22⋊C4—3462no—
179C42—3461no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—3461no—
190C23C2×C22⋊D17311yescharacteristic
191C23C2×C22⋊D17312yes—
192Q8—3462no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—3462no—
197D4—6922no—
198D4—6924no—
199D4C2×C22⋊D17318yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—3462no—
204C2×C4—3461no—
205C2×C4—6922no—
206C2×C4C2×C22⋊D17312yes—
207C2×C4C2×C22⋊D17311yescharacteristic
208C2×C4C2×C22⋊D17314yes—
209C2×C4C23×D17311yescharacteristic
210C8—3464no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—6921no—
220C22—6922no—
221C22#1914964812yes—
222C22#1914970514yescomplemented
223C22C22×C22⋊D17311yescharacteristic
224C4—22no—
225C4—6921no—
226C4—6922no—
227C4#1914964812yes—
228C4C22×C22⋊D17312yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—13841no—
233C2C2×#1914964811yescharacteristic
234C2C2×#1914970512yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC23×D4×C173C2
socleC22×C173C22×D4
hypercenterC23×D4D173
hypocenterC173#3003
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. #26012002C2semidirect
3. #26011642C2semidirect
4. #26011625C2semidirect
5. #26011496C2semidirect
6. #26011455C2semidirect
7. C23×D4×C173C2semidirect
8. C2×#14583826C22semidirect
10. C2×#14583814C22semidirect
11. C2×#14583811C22semidirect
12. C2×#14583810C22semidirect
14. #19149658C22semidirect
15. #19149653C22semidirect
18. #19149638C22semidirect
19. #19149600C22semidirect
20. C22⋊#13063436C22semidirect
29. C2×Dic346C23semidirect
30. C2×D692C23semidirect
37. C2×#13063436C23semidirect
38. C4⋊Dic173C23semidirect
55. C2×D4×C173D4semidirect
102. C22×C173AGL(1,ℤ/8)semidirect
117. C173#3003semidirect
119. C23×D4D173semidirect
162. C2×D4C22⋊D173semidirect
222. C22#19149705semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
#260120022
#260116422
#260116252
#260114962
#260114552
C23×D4×C1732
#3003173