Finite Group Databasebeta

← #37544765

Subgroups — #37544765

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#26041848C211yesmaximal, characteristic, complemented
2#26041666C211yesmaximal, characteristic, complemented
3#26041655C211yesmaximal, characteristic, complemented
4#26041645C211yesmaximal, characteristic, complemented
5#26041636C211yesmaximal, characteristic, complemented
6#26041494C212yesmaximal, complemented
7#26041474C212yesmaximal, complemented
8#26041422C212yesmaximal, complemented
9#26041412C212yesmaximal, complemented
10#26041340C211yesmaximal, characteristic, complemented
11#540×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×D724C2211yescharacteristic
14C2×#15668437C2212yescomplemented
15C2×#15668434C2212yes—
16C2×#15668434C2212yes—
17C2×#15668434C2212yescomplemented
18#19153813C2211yescharacteristic
19#19153813C2211yescharacteristic
20#19153804C2211yescharacteristic
21#19153804C2211yescharacteristic
22#19153791—22no—
23#19153786—22no—
24C4⋊#13089678C2212yescomplemented
25#19153739C2212yescomplemented
26#19153736C2212yes—
27#19153736C2212yes—
28#19153736C2212yescomplemented
29C22⋊#13089678C2212yes—
30#19153727C2211yescharacteristic, complemented
31C4×#13089678C2211yescharacteristic, complemented
32#127×C181C2211yescharacteristic, complemented
33#125×C181C2211yescharacteristic, complemented
34#123×C181—22no—
35#121×C181—22no—
36#121×C181C2212yes—
37C4×D4×C181C2212yescomplemented
38#117×C181C2211yescharacteristic
39C23×D181—21no—
40#15668449—42no—
41C2×Dic362C2311yescharacteristic
42C2×D724—21no—
43C2×D724—22no—
44C2×D724—22no—
45C2×D724—42no—
46C2×D724C2311yescharacteristic
47C2×D724C2311yescharacteristic
48C2×D724C2311yescharacteristic
49C2×C22⋊D181—21no—
50C2×C4×D181—21no—
51#15668437—42no—
52#15668434—42no—
53#15668434—42no—
54#15668434—42no—
55C2×#13089678C2312yes—
56C2×#13089678C2312yes—
57#15668426—22no—
58#15668426—22no—
59C2×D4×C181—22no—
60C2×D4×C181—22no—
61C2×D4×C181C2312yes—
62C22×C4×C181—22no—
63C22×C4×C181C2311yescharacteristic
64C4⋊C4×C181—22no—
65C4⋊C4×C181D412yes—
66C4⋊C4×C181C2312yes—
67C22⋊C4×C181—22no—
68C22⋊C4×C181—22no—
69C22⋊C4×C181D412yes—
70C42×C181C2311yescharacteristic
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181—21no—
74C22×D181—21no—
75C22×D181—42no—
76C22×D181—82no—
77Dic362—22no—
78D724—22no—
79D724—22no—
80D724—22no—
81D724—42no—
82D724—82no—
83D724—82no—
84C22⋊D181—42no—
85C2×Dic181—21no—
86C4×D181—42no—
87#13089678—22no—
88#13089678—22no—
89C23×C181—22no—
90C23×C181C2×D411yescharacteristic
91D4×C181—22no—
92D4×C181—22no—
93D4×C181—82no—
94D4×C181—82no—
95C2×C4×C181—22no—
96C2×C4×C181—22no—
97C2×C4×C181—82no—
98C2×C4×C181C2×D412yes—
99C2×C4×C181C2×D412yes—
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2411yescharacteristic
102C22×C181—21no—
103C22×C181—22no—
104C22×C181—42no—
105C22×C181—42no—
106C22×C181—42no—
107C22×C181C22×D411yescharacteristic
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—41no—
111C2×D181—42no—
112C2×D181—42no—
113C2×D181—42no—
114C2×D181—84no—
115C4×C181—21no—
116C4×C181—22no—
117C4×C181—22no—
118C4×C181—42no—
119C4×C181H(5,𝔽2)12yes—
120Dic181—41no—
121C2×C181—41no—
122C2×C181—82no—
123C2×C181#53511yescharacteristic
124C2×C181#57612yes—
125D181—81no—
126D181—81no—
127D181—81no—
128C181#322011yescharacteristic, complemented, Sylow
129#3220—1811nomaximal, Sylow
130#540D18111yescharacteristic, complemented
131#535—1811no—
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×#41—1811no—
142C22×D4—1811no—
143C2×SD16—1812no—
144C2×D8—1812no—
145C2×D8—1812no—
146C2×D8—1812no—
147#127—1811no—
148#127—1811no—
149#127C2×D18111yescharacteristic, complemented
150#125C2×D18111yescharacteristic, complemented
151#124—1811no—
152#124—1811no—
153#123—22no—
154#121—22no—
155#121—3622no—
156#121C2×D18112yes—
157#120—3622no—
158C4×D4C2×D18112yescomplemented
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—3621no—
169#41—7242no—
170C2×Q8—1811no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—3621no—
174C2×D4—3622no—
175C2×D4—3622no—
176C2×D4—3621no—
177C2×D4—1811no—
178C2×D4—1811no—
179C2×D4—1811no—
180C2×D4—7242no—
181C2×D4C22×D18112yes—
182C22×C4—22no—
183C22×C4—3621no—
184C22×C4C22×D18111yescharacteristic
185SD16—7242no—
186D8—7242no—
187D8—7242no—
188D8—7242no—
189C2×C8—1812no—
190C2×C8—1812no—
191C4⋊C4—22no—
192C4⋊C4C22⋊D18112yes—
193C4⋊C4C22×D18112yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—3622no—
197C22⋊C4—3622no—
198C22⋊C4C22⋊D18112yes—
199C42C22×D18111yescharacteristic
200C42C22×D18111yescharacteristic
201C23—22no—
202C23—3621no—
203C23—3621no—
204C23—3621no—
205C23—14482no—
206C23—7242no—
207C23C2×C22⋊D18111yescharacteristic
208Q8—3622no—
209D4—22no—
210D4—22no—
211D4—82no—
212D4—82no—
213D4—3622no—
214D4—3622no—
215D4—3622no—
216D4—14482no—
217D4—14482no—
218D4—7242no—
219D4—7242no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—82no—
223C2×C4—3621no—
224C2×C4—7242no—
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—14484no—
237C22—7242no—
238C22—7242no—
239C22—7242no—
240C22—7241no—
241C22—7241no—
242C22—7241no—
243C22C22×C22⋊D18111yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—42no—
248C4—7241no—
249C4#1915386212yes—
250C2—41no—
251C2—82no—
252C2—14481no—
253C2—14481no—
254C2—14481no—
255C2#2604184311yescharacteristic
256C2#2604208212yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#540×C181C2
socleC22×C181C22×D4
hypercenter#540D181
hypocenterC181#3220
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #26041848C2semidirect
2. #26041666C2semidirect
3. #26041655C2semidirect
4. #26041645C2semidirect
5. #26041636C2semidirect
6. #26041494C2semidirect
7. #26041474C2semidirect
8. #26041422C2semidirect
9. #26041412C2semidirect
10. #26041340C2semidirect
11. #540×C181C2semidirect
14. C2×#15668437C22semidirect
17. C2×#15668434C22semidirect
24. C4⋊#13089678C22semidirect
25. #19153739C22semidirect
28. #19153736C22semidirect
30. #19153727C22semidirect
31. C4×#13089678C22semidirect
32. #127×C181C22semidirect
33. #125×C181C22semidirect
37. C4×D4×C181C22semidirect
128. C181#3220semidirect
130. #540D181semidirect
149. #127C2×D181semidirect
150. #125C2×D181semidirect
158. C4×D4C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418482
#260416662
#260416552
#260416452
#260416362
#260414942
#260414742
#260414222
#260414122
#260413402
#540×C1812
#3220181