Finite Group Databasebeta

← #37545490

Subgroups — #37545490

Order 23168.

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

normal subgroups only

This group has 464 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#26042088C212yesmaximal, complemented
2#26042082C212yesmaximal, complemented
3#26041669C211yesmaximal, characteristic, complemented
4#26041548C212yesmaximal, complemented
5#26041544C212yesmaximal, complemented
6#26041451C211yesmaximal, characteristic, complemented
7#26041448C212yesmaximal, complemented
8#26041398C212yesmaximal, complemented
9#455×C181C211yesmaximal, characteristic, complemented
10#19153864C2211yescharacteristic, complemented
11#19153862C2212yescomplemented
12#19153848—22no—
13#19153844C2212yescomplemented
14#19153843—22no—
15#19153842—22no—
16#19153842C2212yescomplemented
17#19153841—22no—
18#19153841C2212yes—
19#19153839—22no—
20#19153836—22no—
21#19153836C2212yes—
22C2×#15668436C2212yescomplemented
23SD16×D181—22no—
24C2×#15668434—22no—
25C2×#15668434C2212yescomplemented
26#19153824C2212yescomplemented
27#19153823C2212yescomplemented
28#19153813C2211yescharacteristic, complemented
29#19153751C2212yescomplemented
30#19153744C2212yescomplemented
31#19153735C2211yescharacteristic, complemented
32#19153732C2212yescomplemented
33#19153727C2211yescharacteristic, complemented
34H(5,𝔽2)×C181C2211yescharacteristic
35#137×C181C2212yescomplemented
36#124×C181C2211yescharacteristic, complemented
37#104×C181C2212yescomplemented
38#100×C181C2211yescharacteristic, complemented
39#15668450—22no—
40#15668450—22no—
41#15668449—22no—
42#15668449—41no—
43#15668449C2312yes—
44Q8×D181—22no—
45C2×D724—22no—
46C2×D724—41no—
47C2×D724C2312yes—
48D4×D181—22no—
49D4×D181—42no—
50#15668439D412yescomplemented
51#15668438—22no—
52#15668437—22no—
53#15668437D412yescomplemented
54#15668436—22no—
55#15668436—22no—
56#15668436—22no—
57#15668436D412yescomplemented
58D1448—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×Q8×C181C2311yescharacteristic
73C2×D4×C181—22no—
74C2×D4×C181—41no—
75C2×D4×C181C2311yescharacteristic
76Q16×C181—22no—
77SD16×C181—22no—
78M16×C181C2312yes—
79C22⋊C4×C181—41no—
80C42×C181C2311yescharacteristic
81C22×D181—22no—
82C22×D181—42no—
83Dic362—41no—
84Dic362C2×D412yes—
85D724—22no—
86D724—22no—
87D724—42no—
88D724—41no—
89D724—82no—
90D724C2×D412yes—
91C22⋊D181—22no—
92C22⋊D181—42no—
93C4×D181—22no—
94C4×D181—42no—
95C4×D181—42no—
96#13089678—22no—
97#13089678C2×D412yes—
98#13089678C2×D412yes—
99C23×C181—21no—
100C23×C181—41no—
101Q8×C181—22no—
102Q8×C181C2×D412yes—
103D4×C181—22no—
104D4×C181—22no—
105D4×C181—42no—
106D4×C181C2×D412yes—
107C2×C4×C181—22no—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181—41no—
111C2×C4×C181C2411yescharacteristic
112C8×C181—22no—
113C22×C181—22no—
114C22×C181—41no—
115C22×C181—81no—
116C22×C181—161no—
117C22×C181H(5,𝔽2)11yescharacteristic
118C2×D181—22no—
119C2×D181—42no—
120C2×D181—82no—
121C2×D181—162no—
122C4×C181—22no—
123C4×C181—41no—
124C4×C181—41no—
125C4×C181C22×D412yes—
126Dic181—22no—
127C2×C181—21no—
128C2×C181—42no—
129C2×C181—81no—
130C2×C181D4211yescharacteristic
131D181—42no—
132D181—82no—
133C181#328811yescharacteristic, complemented, Sylow
134#3288—1811nomaximal, Sylow
135#577—1812no—
136#576—1812no—
137#496—1811no—
138#472—1812no—
139#471—1812no—
140#455D18111yescharacteristic, complemented
141#454—1811no—
142#453—1812no—
143#444—1812no—
1442−1+4—1811no—
145H(5,𝔽2)—1812no—
146H(5,𝔽2)C2×D18111yescharacteristic
147#137—3622no—
148#137C2×D18112yescomplemented
149AGL(1,ℤ/8)—3622no—
150AGL(1,ℤ/8)—3622no—
151AGL(1,ℤ/8)—3622no—
152AGL(1,ℤ/8)—1812no—
153AGL(1,ℤ/8)—1812no—
154AGL(1,ℤ/8)—1812no—
155#135—3622no—
156#135—3622no—
157#135—1812no—
158C2×SD16—3622no—
159C2×SD16—1812no—
160C2×D8—3622no—
161C2×D8—1812no—
162#131—1812no—
163#131—1812no—
164#127—1811no—
165#124C2×D18111yescharacteristic, complemented
166#108—1812no—
167#104—1812no—
168#104C2×D18112yescomplemented
169#101—1811no—
170#100—1812no—
171#100C2×D18111yescharacteristic, complemented
172#97—1811no—
173#41—22no—
174#41—3622no—
175#41—3622no—
176#41—3622no—
177#41—1812no—
178#41—7241no—
179#41C22×D18112yes—
180C2×Q8—3622no—
181C2×Q8C22×D18111yescharacteristic
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—3622no—
185C2×D4—3622no—
186C2×D4—1812no—
187C2×D4—7241no—
188C2×D4—7242no—
189C2×D4C22×D18111yescharacteristic
190Q16—22no—
191Q16—1812no—
192SD16—22no—
193SD16—3622no—
194SD16—3622no—
195SD16—3622no—
196SD16—3622no—
197SD16—3622no—
198SD16—1812no—
199SD16—1812no—
200D8—3622no—
201D8—3622no—
202D8—3622no—
203D8—3622no—
204D8—3622no—
205D8—1812no—
206M16—3622no—
207M16—3622no—
208M16—1812no—
209M16C22×D18112yes—
210C2×C8—3622no—
211C2×C8—3622no—
212C2×C8—1812no—
213C22⋊C4—41no—
214C42C22×D18111yescharacteristic
215C23—21no—
216C23—41no—
217C23—3622no—
218C23—7242no—
219Q8—22no—
220Q8—1812no—
221Q8—7241no—
222Q8D4×D18112yes—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—3622no—
227D4—3622no—
228D4—3622no—
229D4—14482no—
230D4—1812no—
231D4—7242no—
232D4—7241no—
233D4—7242no—
234D4D4×D18112yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—3622no—
240C2×C4—7242no—
241C2×C4—7242no—
242C2×C4C23×D18111yescharacteristic
243C8—22no—
244C8—3622no—
245C8—1812no—
246C8—1812no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—161no—
251C22—28962no—
252C22—3622no—
253C22—14482no—
254C22—7242no—
255C22#1915386111yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—3622no—
260C4C2×D4×D18112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—14482no—
265C2—7242no—
266C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#455×C181C2
socleC2×C181D42
hypercenter#455D181
hypocenterC181#3288
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #26042088C2semidirect
2. #26042082C2semidirect
3. #26041669C2semidirect
4. #26041548C2semidirect
5. #26041544C2semidirect
6. #26041451C2semidirect
7. #26041448C2semidirect
8. #26041398C2semidirect
9. #455×C181C2semidirect
10. #19153864C22semidirect
11. #19153862C22semidirect
13. #19153844C22semidirect
16. #19153842C22semidirect
22. C2×#15668436C22semidirect
25. C2×#15668434C22semidirect
26. #19153824C22semidirect
27. #19153823C22semidirect
28. #19153813C22semidirect
29. #19153751C22semidirect
30. #19153744C22semidirect
31. #19153735C22semidirect
32. #19153732C22semidirect
33. #19153727C22semidirect
35. #137×C181C22semidirect
36. #124×C181C22semidirect
37. #104×C181C22semidirect
38. #100×C181C22semidirect
50. #15668439D4semidirect
53. #15668437D4semidirect
57. #15668436D4semidirect
63. #15668434D4semidirect
133. C181#3288semidirect
140. #455D181semidirect
148. #137C2×D181semidirect
165. #124C2×D181semidirect
168. #104C2×D181semidirect
171. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420882
#260420822
#260416692
#260415482
#260415442
#260414512
#260414482
#260413982
#455×C1812
#3288181