Finite Group Databasebeta

← #37544766

Subgroups — #37544766

Order 23168.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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#26041883C211yesmaximal, characteristic, complemented
2#26041847C211yesmaximal, characteristic, complemented
3#26041655C211yesmaximal, characteristic, complemented
4#26041644C211yesmaximal, characteristic, complemented
5#26041633C211yesmaximal, characteristic, complemented
6#26041496C212yesmaximal, complemented
7#26041472C212yesmaximal, complemented
8#26041423C212yesmaximal, complemented
9#26041411C212yesmaximal, complemented
10#26041341C211yesmaximal, characteristic, complemented
11#496×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14C2×#15668434C2212yes—
15C2×#15668434C2212yescomplemented
16#19153813C2211yescharacteristic
17#19153812C2211yescharacteristic
18#19153808C2211yescharacteristic, complemented
19#19153805C2211yescharacteristic
20#19153797—22no—
21#19153789—22no—
22#19153788—22no—
23#19153787C2212yes—
24#19153786—22no—
25D4×Dic181C2212yescomplemented
26#19153773C2211yescharacteristic
27#19153745C2212yescomplemented
28#19153740C2212yescomplemented
29#19153738C2212yes—
30#19153738C2212yescomplemented
31#19153737C2212yes—
32#19153728C2212yes—
33C4×#13089678C2211yescharacteristic, complemented
34C2×SD16×C181C2212yescomplemented
35C2×D8×C181C2212yes—
36#127×C181C2211yescharacteristic, complemented
37#124×C181C2211yescharacteristic
38#97×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40#15668450—42no—
41C2×D724—21no—
42C2×D724—22no—
43C2×D724C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—22no—
46C2×C22⋊D181—21no—
47C2×C22⋊D181—22no—
48D4×D181—42no—
49C22×Dic181—22no—
50C2×C4×D181—21no—
51C2×C4×D181C2311yescharacteristic
52#15668434—42no—
53#15668434—42no—
54C2×#13089678C2312yes—
55C4⋊Dic181C2312yes—
56#15668428—22no—
57#15668428D412yes—
58C22⋊Dic181—22no—
59C22⋊Dic181—22no—
60#15668426—22no—
61#15668426D412yes—
62C4×Dic181C2311yescharacteristic
63C2×Q8×C181C2311yescharacteristic
64C2×D4×C181—22no—
65C2×D4×C181C2311yescharacteristic
66C2×D4×C181C2312yes—
67SD16×C181—42no—
68D8×C181—42no—
69C2×C8×C181C2312yes—
70C22⋊C4×C181—22no—
71C42×C181C2311yescharacteristic
72C22×D181—21no—
73C22×D181—21no—
74C22×D181—42no—
75C22×D181—82no—
76C22×D181C2×D411yescharacteristic
77D724—22no—
78D724—22no—
79D724—42no—
80D724—82no—
81C22⋊D181—42no—
82C22⋊D181—82no—
83C22⋊D181—82no—
84C2×Dic181—22no—
85C2×Dic181—82no—
86C2×Dic181C2×D412yes—
87C2×Dic181C2×D411yescharacteristic
88C4×D181—22no—
89C4×D181—42no—
90#13089678—22no—
91C23×C181—21no—
92C23×C181—22no—
93Q8×C181—22no—
94D4×C181—22no—
95D4×C181—22no—
96D4×C181—22no—
97D4×C181—82no—
98C2×C4×C181—21no—
99C2×C4×C181C2×D412yes—
100C2×C4×C181C2411yescharacteristic
101C8×C181—22no—
102C22×C181—42no—
103C22×C181—41no—
104C22×C181—42no—
105C22×C181—42no—
106C22×C181—42no—
107C22×C181C22×D411yescharacteristic
108C2×D181—21no—
109C2×D181—22no—
110C2×D181—41no—
111C2×D181—41no—
112C2×D181—42no—
113C2×D181—42no—
114C2×D181—84no—
115C4×C181—22no—
116C4×C181—41no—
117C4×C181H(5,𝔽2)12yes—
118Dic181—21no—
119Dic181—22no—
120Dic181—42no—
121C2×C181—81no—
122C2×C181—82no—
123C2×C181#53511yescharacteristic
124C2×C181#57612yes—
125D181—41no—
126D181—81no—
127D181—81no—
128C181#322011yescharacteristic, complemented, Sylow
129#3220—1811nomaximal, Sylow
130#540—1811no—
131#535—1811no—
132#496D18111yescharacteristic, complemented
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×SD16C2×D18112yescomplemented
144C2×D8—1812no—
145C2×D8—1812no—
146C2×D8C2×D18112yes—
147#127—1811no—
148#127—1811no—
149#127C2×D18111yescharacteristic, complemented
150#125—1811no—
151#124—1811no—
152#124C2×D18111yescharacteristic
153#123—3622no—
154#121—3622no—
155#121—3622no—
156#121—1812no—
157#120—3622no—
158C4×D4—1812no—
159#117—1811no—
160C4⋊C8—1812no—
161#103—1812no—
162#102—1812no—
163#102—1812no—
164#102—1812no—
165C22⋊C8—1812no—
166#97C2×D18111yescharacteristic, complemented
167C4×C8—1811no—
168C24—3621no—
169#41—7242no—
170C2×Q8C22×D18111yescharacteristic
171C2×D4—22no—
172C2×D4—3621no—
173C2×D4—3622no—
174C2×D4—3622no—
175C2×D4—3621no—
176C2×D4—3622no—
177C2×D4—1811no—
178C2×D4—1811no—
179C2×D4—7242no—
180C2×D4C22×D18111yescharacteristic
181C2×D4C22×D18112yes—
182C22×C4—3621no—
183C22×C4—3622no—
184C22×C4—1811no—
185SD16—42no—
186D8—42no—
187D8—7242no—
188D8—7242no—
189C2×C8—1812no—
190C2×C8C22×D18112yes—
191C4⋊C4—3622no—
192C4⋊C4—1812no—
193C4⋊C4—1812no—
194C22⋊C4—22no—
195C22⋊C4—3622no—
196C22⋊C4—3622no—
197C22⋊C4—3622no—
198C22⋊C4—1812no—
199C42—1811no—
200C42C22×D18111yescharacteristic
201C23—21no—
202C23—22no—
203C23—3621no—
204C23—3621no—
205C23—14482no—
206C23—1811no—
207C23—7242no—
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—82no—
213D4—3622no—
214D4—3622no—
215D4—14482no—
216D4—14482no—
217D4—14482no—
218D4—7242no—
219D4—7242no—
220C2×C4—21no—
221C2×C4—3622no—
222C2×C4—3622no—
223C2×C4—14482no—
224C2×C4—1812no—
225C2×C4—1811no—
226C2×C4—7242no—
227C2×C4D4×D18112yes—
228C2×C4C23×D18111yescharacteristic
229C8—22no—
230C8—3622no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22—3622no—
237C22—3621no—
238C22—14484no—
239C22—7242no—
240C22—7242no—
241C22—7241no—
242C22—7241no—
243C22C2×D4×D18111yescharacteristic
244C4—22no—
245C4—41no—
246C4—3622no—
247C4—3621no—
248C4—7242no—
249C4#1915386112yes—
250C2—81no—
251C2—82no—
252C2—14481no—
253C2—14481no—
254C2—7241no—
255C2#2604184211yescharacteristic
256C2#2604208312yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC22×C181C22×D4
hypercenter#496D181
hypocenterC181#3220
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26041883C2semidirect
2. #26041847C2semidirect
3. #26041655C2semidirect
4. #26041644C2semidirect
5. #26041633C2semidirect
6. #26041496C2semidirect
7. #26041472C2semidirect
8. #26041423C2semidirect
9. #26041411C2semidirect
10. #26041341C2semidirect
11. #496×C181C2semidirect
15. C2×#15668434C22semidirect
18. #19153808C22semidirect
25. D4×Dic181C22semidirect
27. #19153745C22semidirect
28. #19153740C22semidirect
30. #19153738C22semidirect
33. C4×#13089678C22semidirect
34. C2×SD16×C181C22semidirect
36. #127×C181C22semidirect
38. #97×C181C22semidirect
128. C181#3220semidirect
132. #496D181semidirect
143. C2×SD16C2×D181semidirect
149. #127C2×D181semidirect
166. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418832
#260418472
#260416552
#260416442
#260416332
#260414962
#260414722
#260414232
#260414112
#260413412
#496×C1812
#3220181