Finite Group Databasebeta

← #37544170

Subgroups — #37544170

Order 23168.

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

normal subgroups only

This group has 656 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×#19153843C212yesmaximal, complemented
2C2×#19153840C212yesmaximal, complemented
3#26041828C211yesmaximal, characteristic, complemented
4#127×D181C211yesmaximal, characteristic, complemented
5#26041669C211yesmaximal, characteristic, complemented
6#26041668C211yesmaximal, characteristic, complemented
7#26041666C211yesmaximal, characteristic, complemented
8#26041654C212yesmaximal, complemented
9#26041650C212yesmaximal, complemented
10#26041231C211yesmaximal, characteristic, complemented
11#496×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×#15668448C2211yescharacteristic
14C2×D4×D181—22no—
15C2×D4×D181C2212yes—
16#19153843—28no—
17#19153840—28no—
18C2×#15668438C2212yescomplemented
19C2×#15668437C2212yescomplemented
20C2×#15668436C2212yescomplemented
21C2×D1448C2212yescomplemented
22C2×#15668434C2212yescomplemented
23C2×#15668434C2212yescomplemented
24C2×#15668432C2212yes—
25#19153814C2211yescharacteristic, complemented
26#19153813C2211yescharacteristic
27#19153812—22no—
28#19153812C2212yes—
29#19153812C2211yescharacteristic, complemented
30#19153804C2211yescharacteristic, complemented
31#19153789—22no—
32#19153776—22no—
33C42×D181C2211yescharacteristic
34#19153727C2211yescharacteristic, complemented
35C8×Dic181C2212yescomplemented
36C2×SD16×C181C2212yescomplemented
37C2×D8×C181C2212yescomplemented
38#127×C181C2211yescharacteristic
39#124×C181C2211yescharacteristic, complemented
40#97×C181C2211yescharacteristic, complemented
41C23×D181—22no—
42#15668450—24no—
43#15668448—24no—
44C2×Dic362C2311yescharacteristic
45C2×D724—21no—
46C2×D724—22no—
47C2×D724C2311yescharacteristic
48C2×D724C2312yes—
49C2×C22⋊D181—21no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52C2×C22⋊D181—24no—
53D4×D181—24no—
54D4×D181—24no—
55D4×D181—48no—
56C22×Dic181—21no—
57C2×C4×D181—21no—
58C2×C4×D181D412yes—
59C2×C4×D181C2311yescharacteristic
60#15668438—24no—
61#15668437—24no—
62#15668436—24no—
63D1448—24no—
64#15668434—24no—
65#15668434—24no—
66#15668432D418yescomplemented
67C2×#13089678C2312yes—
68#15668428—22no—
69#15668426—22no—
70C4×Dic181D412yes—
71C4×Dic181C2311yescharacteristic
72C2×Q8×C181C2311yescharacteristic
73C2×D4×C181—22no—
74C2×D4×C181C2311yescharacteristic
75C2×D4×C181C2312yes—
76SD16×C181—24no—
77D8×C181—24no—
78C2×C8×C181C2312yes—
79C22⋊C4×C181—22no—
80C42×C181C2311yescharacteristic
81C22×D181—21no—
82C22×D181—22no—
83C22×D181—48no—
84C22×D181—44no—
85C22×D181C2×D411yescharacteristic
86Dic362—22no—
87D724—22no—
88D724—22no—
89D724—22no—
90D724—44no—
91D724—42no—
92C22⋊D181—42no—
93C22⋊D181—44no—
94C22⋊D181—48no—
95C22⋊D181—44no—
96C2×Dic181—21no—
97C2×Dic181—42no—
98C2×Dic181C2×D411yescharacteristic
99C2×Dic181C2×D412yes—
100C4×D181—24no—
101C4×D181—42no—
102C4×D181C2×D414yes—
103#13089678C2×D414yescomplemented
104C23×C181—21no—
105C23×C181—22no—
106Q8×C181—22no—
107D4×C181—22no—
108D4×C181—22no—
109D4×C181—22no—
110D4×C181—44no—
111C2×C4×C181—21no—
112C2×C4×C181C2×D412yes—
113C2×C4×C181C2411yescharacteristic
114C8×C181C2×D414yescomplemented
115C22×C181—41no—
116C22×C181—42no—
117C22×C181—44no—
118C22×C181—42no—
119C22×C181C22×D411yescharacteristic
120C2×D181—22no—
121C2×D181—41no—
122C2×D181—42no—
123C2×D181—44no—
124C2×D181—42no—
125C2×D181—88no—
126C2×D181#12712yes—
127C4×C181—22no—
128C4×C181—41no—
129C4×C181C22×D412yes—
130Dic181—41no—
131Dic181#12712yes—
132Dic181AGL(1,ℤ/8)14yescomplemented
133C2×C181—82no—
134C2×C181—81no—
135C2×C181C2×#12711yescharacteristic
136C2×C181C2×AGL(1,ℤ/8)12yes—
137D181—22no—
138D181—81no—
139D181—82no—
140C181#315111yescharacteristic, complemented, Sylow
141#3151—1811nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1812no—
143C2×AGL(1,ℤ/8)—1812no—
144#532—1811no—
145C2×#127—1811no—
146#496—1811no—
147#496—1811no—
148#496—1811no—
149#496D18111yescharacteristic, complemented
150#493—1812no—
151#492—1812no—
152C4×M16—1811no—
153C2×#41—1811no—
154C2×#41—1811no—
155C22×D4—3622no—
156C22×D4—1812no—
157AGL(1,ℤ/8)—3628no—
158AGL(1,ℤ/8)—3628no—
159C2×SD16—1812no—
160C2×SD16—1812no—
161C2×SD16—1812no—
162C2×SD16C2×D18112yescomplemented
163C2×D8—1812no—
164C2×D8—1812no—
165C2×D8—1812no—
166C2×D8C2×D18112yescomplemented
167C2×M16—1812no—
168C4⋊Q8—1811no—
169#127—3622no—
170#127—1812no—
171#127—1811no—
172#127—1811no—
173#127C2×D18111yescharacteristic
174#124—1811no—
175#124C2×D18111yescharacteristic, complemented
176#121—3622no—
177C4×D4—3622no—
178C2×C42—1811no—
179#97—1811no—
180#97C2×D18111yescharacteristic, complemented
181C4×C8—1812no—
182C24—3622no—
183#41—3624no—
184#41—3624no—
185C2×Q8—1811no—
186C2×Q8C22×D18111yescharacteristic
187C2×D4—22no—
188C2×D4—3621no—
189C2×D4—3621no—
190C2×D4—3624no—
191C2×D4—3624no—
192C2×D4—3622no—
193C2×D4—3622no—
194C2×D4—3624no—
195C2×D4—3622no—
196C2×D4—1811no—
197C2×D4—1812no—
198C2×D4—7248no—
199C2×D4C22×D18111yescharacteristic
200C2×D4C22×D18112yes—
201C22×C4—3621no—
202C22×C4—3621no—
203C22×C4—1812no—
204C22×C4—1811no—
205SD16—24no—
206SD16—3624no—
207SD16—3624no—
208SD16—3624no—
209D8—24no—
210D8—3624no—
211D8—3624no—
212D8—3624no—
213M16—1818no—
214C2×C8—1812no—
215C2×C8C22×D18112yes—
216C4⋊C4—3622no—
217C22⋊C4—22no—
218C22⋊C4—3622no—
219C42—1812no—
220C42—1811no—
221C42C22×D18111yescharacteristic
222C23—21no—
223C23—22no—
224C23—3621no—
225C23—3622no—
226C23—1811no—
227C23—7248no—
228C23—7244no—
229Q8—22no—
230Q8—3622no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3622no—
236D4—3622no—
237D4—3622no—
238D4—7244no—
239D4—7242no—
240D4—7244no—
241D4—7248no—
242D4—7244no—
243D4—7242no—
244C2×C4—21no—
245C2×C4—3624no—
246C2×C4—3621no—
247C2×C4—1812no—
248C2×C4—1811no—
249C2×C4—1814no—
250C2×C4—7242no—
251C2×C4—7242no—
252C2×C4D4×D18112yes—
253C2×C4C23×D18111yescharacteristic
254C8—1814no—
255C8D4×D18114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—3622no—
261C22—14488no—
262C22—1812no—
263C22—7244no—
264C22—7242no—
265C22—7241no—
266C22—7242no—
267C22C2×D4×D18111yescharacteristic
268C4—22no—
269C4—41no—
270C4—1814no—
271C4—1812no—
272C4—7241no—
273C4C2×D4×D18112yes—
274C2—82no—
275C2—81no—
276C2—3622no—
277C2—14482no—
278C2—14481no—
279C2#127×D18111yescharacteristic
280C2AGL(1,ℤ/8)×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC22×C181C22×D4
hypercenter#496D181
hypocenterC181#3151
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. C2×#19153843C2semidirect
2. C2×#19153840C2semidirect
3. #26041828C2semidirect
4. #127×D181C2semidirect
5. #26041669C2semidirect
6. #26041668C2semidirect
7. #26041666C2semidirect
8. #26041654C2semidirect
9. #26041650C2semidirect
10. #26041231C2semidirect
11. #496×C181C2semidirect
18. C2×#15668438C22semidirect
19. C2×#15668437C22semidirect
20. C2×#15668436C22semidirect
21. C2×D1448C22semidirect
22. C2×#15668434C22semidirect
23. C2×#15668434C22semidirect
25. #19153814C22semidirect
29. #19153812C22semidirect
30. #19153804C22semidirect
34. #19153727C22semidirect
35. C8×Dic181C22semidirect
36. C2×SD16×C181C22semidirect
37. C2×D8×C181C22semidirect
39. #124×C181C22semidirect
40. #97×C181C22semidirect
66. #15668432D4semidirect
103. #13089678C2×D4semidirect
114. C8×C181C2×D4semidirect
132. Dic181AGL(1,ℤ/8)semidirect
140. C181#3151semidirect
149. #496D181semidirect
162. C2×SD16C2×D181semidirect
166. C2×D8C2×D181semidirect
175. #124C2×D181semidirect
180. #97C2×D181semidirect
255. C8D4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
C2×#191538402
#260418282
#127×D1812
#260416692
#260416682
#260416662
#260416542
#260416502
#260412312
#496×C1812
#3151181