Finite Group Databasebeta

← #37544771

Subgroups — #37544771

Order 23168.

420 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 117 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#26041884C211yesmaximal, characteristic, complemented
2#26041669C211yesmaximal, characteristic, complemented
3#26041655C211yesmaximal, characteristic, complemented
4#26041643C211yesmaximal, characteristic, complemented
5#26041633C211yesmaximal, characteristic, complemented
6#26041491C212yesmaximal, complemented
7#26041471C212yesmaximal, complemented
8#26041424C212yesmaximal, complemented
9#26041413C212yesmaximal, complemented
10#26041340C211yesmaximal, characteristic, complemented
11#535×C181C211yesmaximal, characteristic, complemented
12C2×#15668436C2212yescomplemented
13C2×#15668434C2212yes—
14C2×#15668434C2212yes—
15C2×#15668434C2212yescomplemented
16#19153813C2211yescharacteristic, complemented
17#19153808C2211yescharacteristic, complemented
18#19153801—22no—
19#19153790C2212yes—
20#19153789—22no—
21C4×D724C2212yescomplemented
22C4⋊#13089678C2212yescomplemented
23#19153740C2212yescomplemented
24#19153737C2212yes—
25#19153737C2212yes—
26#19153737C2212yescomplemented
27C22⋊#13089678C2212yes—
28#19153727C2211yescharacteristic, complemented
29C4×#13089678C2211yescharacteristic, complemented
30C2×#41×C181C2211yescharacteristic
31C22×D4×C181C2211yescharacteristic
32#127×C181C2211yescharacteristic
33#127×C181C2211yescharacteristic
34#124×C181C2211yescharacteristic
35#124×C181C2211yescharacteristic
36#121×C181—22no—
37#120×C181—22no—
38#117×C181C2211yescharacteristic
39C2×D724—22no—
40C2×D724C2312yes—
41C2×C22⋊D181—22no—
42C2×C4×D181—22no—
43#15668436—42no—
44#15668434—42no—
45#15668434—42no—
46#15668434—42no—
47C2×#13089678C2312yes—
48C2×#13089678C2312yes—
49C4⋊Dic181C2312yes—
50#15668428—22no—
51#15668426—22no—
52#15668426—22no—
53C24×C181—21no—
54#41×C181—42no—
55C2×Q8×C181C2311yescharacteristic
56C2×D4×C181—21no—
57C2×D4×C181—21no—
58C2×D4×C181—22no—
59C2×D4×C181—22no—
60C2×D4×C181—42no—
61C2×D4×C181C2311yescharacteristic
62C2×D4×C181C2311yescharacteristic
63C2×D4×C181C2311yescharacteristic
64C22×C4×C181—21no—
65C22×C4×C181C2311yescharacteristic
66C4⋊C4×C181D412yes—
67C22⋊C4×C181—22no—
68C22⋊C4×C181—22no—
69C22⋊C4×C181D412yes—
70C42×C181C2311yescharacteristic
71C42×C181C2311yescharacteristic
72C22×D181—22no—
73D724—22no—
74D724—22no—
75D724—82no—
76C22⋊D181—82no—
77C2×Dic181—22no—
78C4×D181—82no—
79#13089678—22no—
80#13089678—22no—
81C23×C181—21no—
82C23×C181—21no—
83C23×C181—21no—
84C23×C181—42no—
85C23×C181—82no—
86C23×C181C2×D411yescharacteristic
87Q8×C181—22no—
88D4×C181—22no—
89D4×C181—22no—
90D4×C181—22no—
91D4×C181—42no—
92D4×C181—42no—
93D4×C181—82no—
94D4×C181—82no—
95C2×C4×C181—22no—
96C2×C4×C181—21no—
97C2×C4×C181—42no—
98C2×C4×C181C2×D412yes—
99C2×C4×C181C2×D412yes—
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2411yescharacteristic
102C22×C181—21no—
103C22×C181—22no—
104C22×C181—41no—
105C22×C181—41no—
106C22×C181—41no—
107C22×C181—42no—
108C22×C181—42no—
109C22×C181—42no—
110C22×C181—84no—
111C22×C181C22×D411yescharacteristic
112C2×D181—42no—
113C2×D181—42no—
114C2×D181—42no—
115C4×C181—21no—
116C4×C181—22no—
117C4×C181—22no—
118C4×C181—41no—
119C4×C181H(5,𝔽2)12yes—
120Dic181—42no—
121C2×C181—41no—
122C2×C181—81no—
123C2×C181—81no—
124C2×C181—81no—
125C2×C181#53511yescharacteristic
126C2×C181#57612yes—
127D181—82no—
128C181#322011yescharacteristic, complemented, Sylow
129#3220—1811nomaximal, Sylow
130#540—1811no—
131#535D18111yescharacteristic, complemented
132#496—1811no—
133#493—1811no—
134#490—1811no—
135#488—1811no—
136#463—1812no—
137#459—1812no—
138#449—1812no—
139#447—1812no—
140#433—1811no—
141C2×#41C2×D18111yescharacteristic
142C22×D4C2×D18111yescharacteristic
143C2×SD16—1812no—
144C2×D8—1812no—
145C2×D8—1812no—
146C2×D8—1812no—
147#127—1811no—
148#127C2×D18111yescharacteristic
149#127C2×D18111yescharacteristic
150#125—1811no—
151#124C2×D18111yescharacteristic
152#124C2×D18111yescharacteristic
153#123—3622no—
154#121—22no—
155#121—3622no—
156#121—1812no—
157#120—22no—
158C4×D4—1812no—
159#117C2×D18111yescharacteristic
160C4⋊C8—1812no—
161#103—1812no—
162#102—1812no—
163#102—1812no—
164#102—1812no—
165C22⋊C8—1812no—
166#97—1811no—
167C4×C8—1811no—
168C24—21no—
169#41—42no—
170C2×Q8C22×D18111yescharacteristic
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—42no—
176C2×D4—3622no—
177C2×D4—3622no—
178C2×D4—1812no—
179C2×D4C22×D18111yescharacteristic
180C2×D4C22×D18111yescharacteristic
181C2×D4C22×D18111yescharacteristic
182C22×C4—21no—
183C22×C4—3622no—
184C22×C4C22×D18111yescharacteristic
185SD16—7242no—
186D8—7242no—
187D8—7242no—
188D8—7242no—
189C2×C8—1812no—
190C2×C8—1812no—
191C4⋊C4—3622no—
192C4⋊C4—1812no—
193C4⋊C4C22⋊D18112yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—3622no—
197C22⋊C4—3622no—
198C22⋊C4C22⋊D18112yes—
199C42C22×D18111yescharacteristic
200C42C22×D18111yescharacteristic
201C23—21no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23—82no—
206C23—3622no—
207C23C2×C22⋊D18111yescharacteristic
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—42no—
213D4—42no—
214D4—82no—
215D4—82no—
216D4—3622no—
217D4—3622no—
218D4—14482no—
219D4—14482no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—3622no—
224C2×C4—14482no—
225C2×C4C2×C22⋊D18112yes—
226C2×C4C2×C22⋊D18112yes—
227C2×C4C2×C22⋊D18111yescharacteristic
228C2×C4C23×D18111yescharacteristic
229C8—3622no—
230C8—3622no—
231C22—22no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—84no—
240C22—7242no—
241C22—7242no—
242C22—7242no—
243C22C22×C22⋊D18111yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—7242no—
249C4#1915386112yes—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—81no—
254C2—14482no—
255C2#2604184611yescharacteristic
256C2#2604208212yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#535×C181C2
socleC22×C181C22×D4
hypercenter#535D181
hypocenterC181#3220
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26041884C2semidirect
2. #26041669C2semidirect
3. #26041655C2semidirect
4. #26041643C2semidirect
5. #26041633C2semidirect
6. #26041491C2semidirect
7. #26041471C2semidirect
8. #26041424C2semidirect
9. #26041413C2semidirect
10. #26041340C2semidirect
11. #535×C181C2semidirect
12. C2×#15668436C22semidirect
15. C2×#15668434C22semidirect
16. #19153813C22semidirect
17. #19153808C22semidirect
21. C4×D724C22semidirect
22. C4⋊#13089678C22semidirect
23. #19153740C22semidirect
26. #19153737C22semidirect
28. #19153727C22semidirect
29. C4×#13089678C22semidirect
128. C181#3220semidirect
131. #535D181semidirect

Maximal subgroups

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

subgroupindex
#260418842
#260416692
#260416552
#260416432
#260416332
#260414912
#260414712
#260414242
#260414132
#260413402
#535×C1812
#3220181