Finite Group Databasebeta

← #37545517

Subgroups — #37545517

Order 23168.

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

normal subgroups only

This group has 476 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#26042083C212yesmaximal, complemented
2#26042082C212yesmaximal, complemented
3#26041655C211yesmaximal, characteristic, complemented
4#26041548C212yesmaximal, complemented
5#26041547C212yesmaximal, complemented
6#26041448C212yesmaximal, complemented
7#26041446C211yesmaximal, characteristic, complemented
8#26041393C212yesmaximal, complemented
9#453×C181C211yesmaximal, characteristic, complemented
10#19153862C2212yescomplemented
11#19153861C2211yescharacteristic, complemented
12#19153844C2212yescomplemented
13#19153843—22no—
14#19153842—22no—
15#19153842C2212yescomplemented
16#19153841—22no—
17#19153841C2212yescomplemented
18#19153840—22no—
19#19153837—22no—
20#19153836—22no—
21#19153836C2212yes—
22C2×#15668434—22no—
23C2×#15668434C2212yescomplemented
24C2×#15668434C2212yescomplemented
25D8×D181—22no—
26#19153824C2212yescomplemented
27#19153823C2212yescomplemented
28#19153813C2211yescharacteristic, complemented
29#19153751C2212yescomplemented
30#19153744C2212yescomplemented
31#19153733C2211yescharacteristic, complemented
32#19153732C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34H(5,𝔽2)×C181C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C181C2212yescomplemented
36#127×C181C2211yescharacteristic, complemented
37#104×C181C2212yescomplemented
38#100×C181C2211yescharacteristic, complemented
39#15668450—22no—
40#15668449—22no—
41#15668449C2312yes—
42#15668448—22no—
43C2×D724—22no—
44C2×D724—41no—
45C2×D724C2312yes—
46C2×C22⋊D181—41no—
47D4×D181—22no—
48D4×D181—22no—
49D4×D181—42no—
50#15668439D412yescomplemented
51#15668438—22no—
52#15668437—22no—
53#15668437D412yescomplemented
54#15668436—22no—
55#15668436D412yescomplemented
56D1448—22no—
57#15668434—22no—
58#15668434—22no—
59#15668434—22no—
60#15668434—22no—
61#15668434—22no—
62#15668434—22no—
63#15668434D412yescomplemented
64#15668433—22no—
65#15668433C2312yes—
66#15668432—22no—
67C2×#13089678—22no—
68C2×#13089678C2312yes—
69C8×D181—22no—
70#41×C181—22no—
71#41×C181C2312yes—
72C2×D4×C181—22no—
73C2×D4×C181—41no—
74C2×D4×C181—41no—
75C2×D4×C181C2311yescharacteristic
76C2×D4×C181C2311yescharacteristic
77SD16×C181—22no—
78D8×C181—22no—
79M16×C181C2312yes—
80C42×C181C2311yescharacteristic
81C22×D181—22no—
82C22×D181—41no—
83C22×D181—42no—
84Dic362C2×D412yes—
85D724—22no—
86D724—22no—
87D724—42no—
88D724—82no—
89D724C2×D412yes—
90C22⋊D181—22no—
91C22⋊D181—42no—
92C22⋊D181—42no—
93C2×Dic181—41no—
94C4×D181—22no—
95C4×D181—42no—
96#13089678—22no—
97#13089678C2×D412yes—
98#13089678C2×D412yes—
99C23×C181—21no—
100C23×C181—21no—
101C23×C181—41no—
102Q8×C181C2×D412yes—
103D4×C181—22no—
104D4×C181—22no—
105D4×C181—22no—
106D4×C181—42no—
107D4×C181—82no—
108D4×C181C2×D412yes—
109C2×C4×C181—21no—
110C2×C4×C181—22no—
111C2×C4×C181—41no—
112C2×C4×C181C2411yescharacteristic
113C8×C181—22no—
114C22×C181—22no—
115C22×C181—41no—
116C22×C181—41no—
117C22×C181—81no—
118C22×C181—81no—
119C22×C181—161no—
120C22×C181H(5,𝔽2)11yescharacteristic
121C2×D181—22no—
122C2×D181—42no—
123C2×D181—82no—
124C2×D181—161no—
125C2×D181—162no—
126C4×C181—22no—
127C4×C181—22no—
128C4×C181C22×D412yes—
129Dic181—22no—
130C2×C181—21no—
131C2×C181—42no—
132C2×C181—81no—
133C2×C181—81no—
134C2×C181D4211yescharacteristic
135D181—42no—
136D181—82no—
137C181#329211yescharacteristic, complemented, Sylow
138#3292—1811nomaximal, Sylow
139#576—1812no—
140#576—1812no—
141#493—1811no—
142#472—1812no—
143#472—1812no—
144#453—1811no—
145#453—1812no—
146#453D18111yescharacteristic, complemented
147#443—1812no—
148H(5,𝔽2)—1812no—
149H(5,𝔽2)—1811no—
150H(5,𝔽2)C2×D18111yescharacteristic, complemented
151AGL(1,ℤ/8)—3622no—
152AGL(1,ℤ/8)—3622no—
153AGL(1,ℤ/8)—3622no—
154AGL(1,ℤ/8)—3622no—
155AGL(1,ℤ/8)—1812no—
156AGL(1,ℤ/8)—1812no—
157AGL(1,ℤ/8)—1812no—
158AGL(1,ℤ/8)C2×D18112yescomplemented
159#135—3622no—
160#135—3622no—
161#135—1812no—
162C2×D8—3622no—
163C2×D8—3622no—
164C2×D8—1812no—
165C2×D8—1812no—
166#131—1812no—
167#131—1812no—
168#127—1811no—
169#127C2×D18111yescharacteristic, complemented
170#108—1812no—
171#104—1812no—
172#104C2×D18112yescomplemented
173#100—1811no—
174#100—1812no—
175#100C2×D18111yescharacteristic, complemented
176C4×C8—1811no—
177#41—22no—
178#41—3622no—
179#41—3622no—
180#41—3622no—
181#41—1812no—
182#41C22×D18112yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3622no—
187C2×D4—3622no—
188C2×D4—3622no—
189C2×D4—1812no—
190C2×D4—7241no—
191C2×D4—7242no—
192C2×D4—7241no—
193C2×D4C22×D18111yescharacteristic
194C2×D4C22×D18111yescharacteristic
195Q16—1812no—
196SD16—22no—
197SD16—3622no—
198SD16—3622no—
199SD16—3622no—
200SD16—1812no—
201SD16—1812no—
202D8—22no—
203D8—3622no—
204D8—3622no—
205D8—3622no—
206D8—3622no—
207D8—3622no—
208D8—3622no—
209D8—3622no—
210D8—1812no—
211M16—3622no—
212M16—3622no—
213M16—1812no—
214M16C22×D18112yes—
215C2×C8—3622no—
216C2×C8—3622no—
217C2×C8—1812no—
218C42C22×D18111yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—3622no—
223C23—7242no—
224C23—7241no—
225Q8—1812no—
226Q8D4×D18112yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—3622no—
233D4—3622no—
234D4—3622no—
235D4—14482no—
236D4—1812no—
237D4—7242no—
238D4—7242no—
239D4—7242no—
240D4D4×D18112yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—3622no—
245C2×C4—7242no—
246C2×C4—7241no—
247C2×C4C23×D18111yescharacteristic
248C8—22no—
249C8—3622no—
250C8—1812no—
251C8—1812no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—28962no—
259C22—28961no—
260C22—3622no—
261C22—14482no—
262C22—7242no—
263C22#1915386111yescharacteristic
264C4—22no—
265C4—22no—
266C4—3622no—
267C4C2×D4×D18112yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—14482no—
273C2—7242no—
274C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#453×C181C2
socleC2×C181D42
hypercenter#453D181
hypocenterC181#3292
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26042083C2semidirect
2. #26042082C2semidirect
3. #26041655C2semidirect
4. #26041548C2semidirect
5. #26041547C2semidirect
6. #26041448C2semidirect
7. #26041446C2semidirect
8. #26041393C2semidirect
9. #453×C181C2semidirect
10. #19153862C22semidirect
11. #19153861C22semidirect
12. #19153844C22semidirect
15. #19153842C22semidirect
17. #19153841C22semidirect
23. C2×#15668434C22semidirect
24. C2×#15668434C22semidirect
26. #19153824C22semidirect
27. #19153823C22semidirect
28. #19153813C22semidirect
29. #19153751C22semidirect
30. #19153744C22semidirect
31. #19153733C22semidirect
32. #19153732C22semidirect
33. C4×#13089678C22semidirect
34. H(5,𝔽2)×C181C22semidirect
35. AGL(1,ℤ/8)×C181C22semidirect
36. #127×C181C22semidirect
37. #104×C181C22semidirect
38. #100×C181C22semidirect
50. #15668439D4semidirect
53. #15668437D4semidirect
55. #15668436D4semidirect
63. #15668434D4semidirect
137. C181#3292semidirect
146. #453D181semidirect
150. H(5,𝔽2)C2×D181semidirect
158. AGL(1,ℤ/8)C2×D181semidirect
169. #127C2×D181semidirect
172. #104C2×D181semidirect
175. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420832
#260420822
#260416552
#260415482
#260415472
#260414482
#260414462
#260413932
#453×C1812
#3292181