Finite Group Databasebeta

← #37543239

Subgroups — #37543239

Order 23168.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19153847C211yesmaximal, characteristic, complemented
2#26041846C211yesmaximal, characteristic
3#26041757C211yesmaximal, characteristic, complemented
4#26041499C214yesmaximal, complemented
5#26041477C214yesmaximal, complemented
6#26041305C211yesmaximal, characteristic, complemented
7#26041284C211yesmaximal, characteristic, complemented
8#26041276C211yesmaximal, characteristic, complemented
9C22×SD16×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2211yescharacteristic
11#19153847—24no—
12C2×#15668439C2212yescomplemented
13C2×#15668436C2212yescomplemented
14C2×#15668433C2211yescharacteristic
15#19153812C2212yes—
16#19153803C2212yes—
17#19153787—22no—
18#19153785—22no—
19Q8×Dic181—22no—
20Q8×Dic181C2214yescomplemented
21#19153775—22no—
22#19153775C2211yescharacteristic
23#19153775C2211yescharacteristic
24#19153745C2214yescomplemented
25#19153741C2212yescomplemented
26#19153738C2212yescomplemented
27#19153737C2214yes—
28C22×Q8×C181C2211yescharacteristic
29C22×D4×C181C2211yescharacteristic
30C2×SD16×C181—24no—
31C2×SD16×C181C2214yes—
32C22×C8×C181C2211yescharacteristic
33#15668449—22no—
34#15668449—22no—
35C2×Dic362C2311yescharacteristic
36C2×D724C2311yescharacteristic
37C2×C22⋊D181—24no—
38C2×C22⋊D181—21no—
39C2×C4×D181—21no—
40#15668439—24no—
41#15668436—24no—
42#15668433—22no—
43C2×#13089678C2312yes—
44C4⋊Dic181—24no—
45C4⋊Dic181D412yes—
46C4⋊Dic181C2312yes—
47C22⋊Dic181—24no—
48C22⋊Dic181—21no—
49C22⋊Dic181D412yes—
50C4×Dic181—24no—
51C4×Dic181—21no—
52C4×Dic181C2312yes—
53C24×C181—21no—
54C2×Q8×C181—21no—
55C2×Q8×C181—22no—
56C2×Q8×C181D414yescomplemented
57C2×Q8×C181C2312yes—
58C2×D4×C181—22no—
59C2×D4×C181—21no—
60C2×D4×C181—22no—
61C2×D4×C181C2312yes—
62C22×C4×C181—21no—
63C22×C4×C181C2311yescharacteristic
64SD16×C181—28no—
65C2×C8×C181—22no—
66C2×C8×C181C2312yes—
67C22×D181—21no—
68Dic362—21no—
69Dic362—21no—
70D724—21no—
71D724—21no—
72C22⋊D181—42no—
73C22⋊D181—84no—
74C2×Dic181—22no—
75C2×Dic181—21no—
76C2×Dic181C2×D414yes—
77C4×D181—42no—
78#13089678—22no—
79C23×C181—22no—
80C23×C181—42no—
81C23×C181—82no—
82C23×C181C2×D411yescharacteristic
83Q8×C181—24no—
84Q8×C181—22no—
85Q8×C181C2×D414yes—
86D4×C181—22no—
87D4×C181—22no—
88D4×C181—42no—
89C2×C4×C181—24no—
90C2×C4×C181—22no—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2×D412yes—
93C2×C4×C181#4112yes—
94C2×C4×C181C2411yescharacteristic
95C8×C181—22no—
96C22×C181—22no—
97C22×C181—42no—
98C22×C181—44no—
99C22×C181—84no—
100C22×C181#12112yes—
101C22×C181C22×D411yescharacteristic
102C2×D181—41no—
103C2×D181—42no—
104C4×C181—24no—
105C4×C181#12112yes—
106C4×C181C22×D411yescharacteristic
107C4×C181C2×#4111yescharacteristic
108Dic181—24no—
109Dic181—41no—
110Dic181—42no—
111C2×C181—22no—
112C2×C181—82no—
113C2×C181C2×#12111yescharacteristic
114C2×C181#57712yes—
115D181—81no—
116C181#304311yescharacteristic, complemented, Sylow
117#3043—1811nomaximal, Sylow
118C2×#137—1811no—
119C22×SD16D18111yescharacteristic, complemented
120#535—1811no—
121#519—1811no—
122#464—1814no—
123#460—1814no—
124#424—1811no—
125#418—1811no—
126#416—1811no—
127C2×#41—1811no—
128C22×Q8C2×D18111yescharacteristic
129C22×D4C2×D18111yescharacteristic
130#137—3624no—
131C2×Q16—1812no—
132C2×SD16—24no—
133C2×SD16—1812no—
134C2×SD16C2×D18114yes—
135C2×M16—1811no—
136C22×C8C2×D18111yescharacteristic
137#127—1812no—
138#124—1812no—
139#121—3622no—
140#120—3622no—
141C4×Q8—3622no—
142C4×Q8—1814no—
143#117—3622no—
144#117—1811no—
145#117—1811no—
146C4⋊C8—1814no—
147#103—1812no—
148#102—1814no—
149#102—1812no—
150C24—21no—
151#41—3622no—
152#41—3622no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—1811no—
156C2×Q8C22⋊D18114yescomplemented
157C2×Q8C22×D18112yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—3621no—
162C2×D4—3624no—
163C2×D4—1811no—
164C2×D4C22×D18112yes—
165C22×C4—21no—
166C22×C4—3621no—
167C22×C4C22×D18111yescharacteristic
168Q16—3624no—
169SD16—28no—
170SD16—3624no—
171M16—3622no—
172C2×C8—22no—
173C2×C8—1812no—
174C2×C8C22×D18112yes—
175C4⋊C4—3624no—
176C4⋊C4—1812no—
177C4⋊C4—1812no—
178C22⋊C4—3621no—
179C22⋊C4—3624no—
180C22⋊C4—1812no—
181C42—3624no—
182C42—3621no—
183C42—1812no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—3621no—
188C23D4×D18111yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—3621no—
192Q8—3621no—
193Q8C2×C22⋊D18114yes—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—3621no—
198D4—3621no—
199D4—14484no—
200D4—7242no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—3622no—
204C2×C4—3621no—
205C2×C4—1814no—
206C2×C4—7242no—
207C2×C4D4×D18111yescharacteristic
208C2×C4C2×C22⋊D18112yes—
209C2×C4#1566844812yes—
210C2×C4C23×D18111yescharacteristic
211C8—22no—
212C8—3622no—
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—7241no—
218C22—7242no—
219C22#1915378812yes—
220C22C2×D4×D18111yescharacteristic
221C4—24no—
222C4—3624no—
223C4—7241no—
224C4—7242no—
225C4#1915378812yes—
226C4C22×C22⋊D18111yescharacteristic
227C4C2×#1566844811yescharacteristic
228C2—22no—
229C2—82no—
230C2—14481no—
231C2C2×#1915378811yescharacteristic
232C2#2604208512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC22×SD16×C181C2
socleC22×C181C22×D4
prime radical#26041846C2
hypercenterC22×SD16D181
hypocenterC181#3043
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153847C2semidirect
3. #26041757C2semidirect
4. #26041499C2semidirect
5. #26041477C2semidirect
6. #26041305C2semidirect
7. #26041284C2semidirect
8. #26041276C2semidirect
9. C22×SD16×C181C2semidirect
12. C2×#15668439C22semidirect
13. C2×#15668436C22semidirect
20. Q8×Dic181C22semidirect
24. #19153745C22semidirect
25. #19153741C22semidirect
26. #19153738C22semidirect
56. C2×Q8×C181D4semidirect
116. C181#3043semidirect
119. C22×SD16D181semidirect
156. C2×Q8C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
#260418462
#260417572
#260414992
#260414772
#260413052
#260412842
#260412762
C22×SD16×C1812
#3043181