Finite Group Databasebeta

← #37542948

Subgroups — #37542948

Order 23168.

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×#19153842C212yesmaximal, complemented
2#26041790C211yesmaximal, characteristic, complemented
3#26041430C214yesmaximal, complemented
4#26041413C214yesmaximal, complemented
5#26041284C212yesmaximal, complemented
6#26041243C211yesmaximal, characteristic, complemented
7C23×D4×C181C211yesmaximal, characteristic, complemented
8C2×#15668449C2211yescharacteristic, complemented
9#19153842—28no—
10C2×#15668437C2214yescomplemented
11C2×#15668434C2214yescomplemented
12C2×#15668433C2212yescomplemented
13#19153799—21no—
14#19153795C2212yescomplemented
15#19153790C2212yescomplemented
16#19153785—21no—
17C4×C22⋊D181—22no—
18#19153775C2211yescharacteristic, complemented
19#19153737C2218yescomplemented
20C22⋊#13089678C2214yescomplemented
21C25×C181—21no—
22C22×D4×C181—28no—
23C22×D4×C181—22no—
24C22×D4×C181—21no—
25C22×D4×C181C2214yes—
26C22×D4×C181C2212yes—
27C23×C4×C181C2211yescharacteristic
28#15668449—24no—
29C2×Dic362C2311yescharacteristic, complemented
30C2×D724C2311yescharacteristic, complemented
31C2×C22⋊D181—22no—
32C2×C22⋊D181—21no—
33C2×C4×D181—21no—
34#15668437—28no—
35#15668434—28no—
36#15668433—24no—
37C2×#13089678C2314yescomplemented
38C4⋊Dic181C2312yescomplemented
39#15668428—22no—
40C22⋊Dic181—21no—
41C22⋊Dic181—22no—
42#15668426—22no—
43C4×Dic181—21no—
44C24×C181—28no—
45C24×C181—24no—
46C24×C181—22no—
47C24×C181—48no—
48C24×C181D411yescharacteristic
49C2×D4×C181—28no—
50C2×D4×C181—216no—
51C2×D4×C181—216no—
52C2×D4×C181—24no—
53C2×D4×C181—22no—
54C2×D4×C181—24no—
55C2×D4×C181D418yescomplemented
56C2×D4×C181C2314yes—
57C22×C4×C181—24no—
58C22×C4×C181D412yes—
59C22×C4×C181D411yescharacteristic
60C22×C4×C181C2312yes—
61C22×C4×C181C2311yescharacteristic
62C22×D181—21no—
63Dic362—22no—
64D724—22no—
65C22⋊D181—42no—
66C22⋊D181—44no—
67C2×Dic181—22no—
68C2×Dic181—21no—
69C4×D181—42no—
70#13089678—24no—
71C23×C181—22no—
72C23×C181—24no—
73C23×C181—216no—
74C23×C181—28no—
75C23×C181—24no—
76C23×C181—432no—
77C23×C181—48no—
78C23×C181—48no—
79C23×C181—48no—
80C23×C181C2×D411yescharacteristic
81C23×C181C2×D412yes—
82D4×C181—216no—
83D4×C181—28no—
84D4×C181—24no—
85D4×C181C2×D418yes—
86C2×C4×C181—22no—
87C2×C4×C181—24no—
88C2×C4×C181—24no—
89C2×C4×C181C2×D412yes—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D414yes—
92C2×C4×C181C2411yescharacteristic
93C22×C181—28no—
94C22×C181—22no—
95C22×C181—22no—
96C22×C181—22no—
97C22×C181—28no—
98C22×C181—416no—
99C22×C181—432no—
100C22×C181—48no—
101C22×C181#12012yes—
102C22×C181AGL(1,ℤ/8)14yescomplemented
103C22×C181C22×D411yescharacteristic
104C2×D181—41no—
105C2×D181—42no—
106C4×C181—22no—
107C4×C181#12012yes—
108C4×C181C22×D412yes—
109Dic181—41no—
110Dic181—42no—
111C2×C181—22no—
112C2×C181—24no—
113C2×C181—48no—
114C2×C181C2×#12011yescharacteristic
115C2×C181C2×AGL(1,ℤ/8)12yes—
116D181—81no—
117C181#300311yescharacteristic, complemented, Sylow
118#3003—1811nomaximal, Sylow
119C23×D4D18111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1812no—
121#525—1811no—
122#450—1814no—
123#447—1814no—
124#418—1812no—
125#407—1811no—
126C25—21no—
127C2×#41—1811no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D18114yes—
132C22×D4C2×D18112yes—
133C23×C4C2×D18111yescharacteristic
134AGL(1,ℤ/8)—3628no—
135C2×SD16—1814no—
136C2×D8—1814no—
137C2×M16—1812no—
138#123—3621no—
139C22⋊Q8—1812no—
140#121—1812no—
141#120—3621no—
142C4×D4—3622no—
143#117—1811no—
144#102—1818no—
145C22⋊C8—1814no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D18111yescharacteristic
151#41—3624no—
152C2×Q8—1811no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—3622no—
160C2×D4—3621no—
161C2×D4—1811no—
162C2×D4C22⋊D18118yescomplemented
163C2×D4C22×D18114yes—
164C22×C4—24no—
165C22×C4—3621no—
166C22×C4C22⋊D18112yes—
167C22×C4C22⋊D18111yescharacteristic
168C22×C4C22×D18112yes—
169C22×C4C22×D18111yescharacteristic
170SD16—3628no—
171D8—3628no—
172M16—3624no—
173C2×C8—1814no—
174C4⋊C4—3622no—
175C4⋊C4—1812no—
176C22⋊C4—3621no—
177C22⋊C4—3622no—
178C22⋊C4—3622no—
179C42—3621no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—3621no—
190C23C2×C22⋊D18111yescharacteristic
191C23C2×C22⋊D18112yes—
192Q8—3622no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—3622no—
197D4—7242no—
198D4—7244no—
199D4C2×C22⋊D18118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—3622no—
204C2×C4—3621no—
205C2×C4—7242no—
206C2×C4C2×C22⋊D18112yes—
207C2×C4C2×C22⋊D18111yescharacteristic
208C2×C4C2×C22⋊D18114yes—
209C2×C4C23×D18111yescharacteristic
210C8—3624no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—7241no—
220C22—7242no—
221C22#1915378512yes—
222C22#1915384214yescomplemented
223C22C22×C22⋊D18111yescharacteristic
224C4—22no—
225C4—7241no—
226C4—7242no—
227C4#1915378512yes—
228C4C22×C22⋊D18112yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—14481no—
233C2C2×#1915378511yescharacteristic
234C2C2×#1915384212yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC23×D4×C181C2
socleC22×C181C22×D4
hypercenterC23×D4D181
hypocenterC181#3003
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19153842C2semidirect
2. #26041790C2semidirect
3. #26041430C2semidirect
4. #26041413C2semidirect
5. #26041284C2semidirect
6. #26041243C2semidirect
7. C23×D4×C181C2semidirect
8. C2×#15668449C22semidirect
10. C2×#15668437C22semidirect
11. C2×#15668434C22semidirect
12. C2×#15668433C22semidirect
14. #19153795C22semidirect
15. #19153790C22semidirect
18. #19153775C22semidirect
19. #19153737C22semidirect
20. C22⋊#13089678C22semidirect
29. C2×Dic362C23semidirect
30. C2×D724C23semidirect
37. C2×#13089678C23semidirect
38. C4⋊Dic181C23semidirect
55. C2×D4×C181D4semidirect
102. C22×C181AGL(1,ℤ/8)semidirect
117. C181#3003semidirect
119. C23×D4D181semidirect
162. C2×D4C22⋊D181semidirect
222. C22#19153842semidirect

Maximal subgroups

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

subgroupindex
C2×#191538422
#260417902
#260414302
#260414132
#260412842
#260412432
C23×D4×C1812
#3003181