Finite Group Databasebeta

← #37545487

Subgroups — #37545487

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#26042085C212yesmaximal, complemented
2#26042080C212yesmaximal, complemented
3#26041546C212yesmaximal, complemented
4#26041539C212yesmaximal, complemented
5#26041456C211yesmaximal, characteristic, complemented
6#26041451C211yesmaximal, characteristic, complemented
7#26041446C212yesmaximal, complemented
8#26041402C212yesmaximal, complemented
9#496×C181C211yesmaximal, characteristic, complemented
10#19153864C2211yescharacteristic, complemented
11#19153861C2211yescharacteristic
12#19153861C2212yescomplemented
13#19153847C2212yescomplemented
14#19153845—22no—
15#19153843—22no—
16#19153842C2212yes—
17#19153842C2212yescomplemented
18#19153842C2212yescomplemented
19#19153840—22no—
20#19153840—22no—
21#19153838C2212yes—
22#19153837—22no—
23#19153835—22no—
24SD16×D181—22no—
25D8×D181—22no—
26#19153822C2212yescomplemented
27#19153822C2212yescomplemented
28#19153752C2212yescomplemented
29#19153744C2212yescomplemented
30#19153744C2212yescomplemented
31#19153735C2211yescharacteristic, complemented
32#19153733C2212yescomplemented
33#19153733C2211yescharacteristic, complemented
34C2×SD16×C181C2212yescomplemented
35C2×D8×C181C2212yescomplemented
36#127×C181C2211yescharacteristic, complemented
37#124×C181C2211yescharacteristic, complemented
38#97×C181C2211yescharacteristic, complemented
39#15668450—22no—
40#15668449—41no—
41#15668449C2312yes—
42#15668449C2312yes—
43#15668448—22no—
44#15668448—22no—
45#15668448—22no—
46Q8×D181—22no—
47C2×C22⋊D181—41no—
48C2×C22⋊D181—42no—
49D4×D181—22no—
50D4×D181—22no—
51D4×D181—22no—
52Dic724D412yescomplemented
53#15668439—22no—
54#15668438D412yescomplemented
55#15668438D412yescomplemented
56#15668437—22no—
57#15668437—22no—
58#15668437—22no—
59#15668436—22no—
60D1448D412yescomplemented
61#15668434—22no—
62#15668434—22no—
63#15668434—22no—
64#15668433C2312yes—
65#15668433C2312yes—
66#15668432—22no—
67#15668432—22no—
68C8×D181—22no—
69C8×D181—22no—
70C2×Q8×C181C2311yescharacteristic
71C2×D4×C181—41no—
72C2×D4×C181C2311yescharacteristic
73C2×D4×C181C2312yes—
74SD16×C181—22no—
75SD16×C181—22no—
76D8×C181—22no—
77D8×C181—22no—
78C2×C8×C181C2312yes—
79C22⋊C4×C181—41no—
80C42×C181C2311yescharacteristic
81C22×D181—42no—
82C22×D181—41no—
83Dic362—41no—
84Dic362C2×D412yes—
85Dic362C2×D412yes—
86D724—41no—
87D724C2×D412yes—
88D724C2×D412yes—
89C22⋊D181—22no—
90C22⋊D181—22no—
91C22⋊D181—42no—
92C22⋊D181—42no—
93C22⋊D181—42no—
94C2×Dic181—42no—
95C2×Dic181—41no—
96C4×D181—22no—
97C4×D181—22no—
98C4×D181—42no—
99#13089678—22no—
100#13089678—22no—
101C23×C181—21no—
102C23×C181—22no—
103Q8×C181—22no—
104D4×C181—22no—
105D4×C181—22no—
106D4×C181—22no—
107D4×C181—82no—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181C2411yescharacteristic
111C8×C181C2×D412yes—
112C8×C181C2×D412yes—
113C22×C181—41no—
114C22×C181—42no—
115C22×C181—82no—
116C22×C181—81no—
117C22×C181H(5,𝔽2)11yescharacteristic
118C2×D181—22no—
119C2×D181—22no—
120C2×D181—162no—
121C2×D181—161no—
122C4×C181—41no—
123C4×C181—41no—
124C4×C181C22×D412yes—
125Dic181—22no—
126Dic181—22no—
127C2×C181—21no—
128C2×C181—82no—
129C2×C181—81no—
130C2×C181D4211yescharacteristic
131D181—42no—
132D181—42no—
133C181#328811yescharacteristic, complemented, Sylow
134#3288—1811nomaximal, Sylow
135#577—1812no—
136#576—1812no—
137#496D18111yescharacteristic, complemented
138#472—1812no—
139#471—1812no—
140#455—1811no—
141#454—1811no—
142#453—1812no—
143#444—1812no—
1442−1+4—1811no—
145H(5,𝔽2)—1811no—
146H(5,𝔽2)—1812no—
147#137—3622no—
148#137—1812no—
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×SD16C2×D18112yescomplemented
160C2×D8—3622no—
161C2×D8C2×D18112yescomplemented
162#131—1812no—
163#131—1812no—
164#127C2×D18111yescharacteristic, complemented
165#124C2×D18111yescharacteristic, complemented
166#108—1812no—
167#104—1812no—
168#104—1812no—
169#101—1811no—
170#100—1812no—
171#100—1811no—
172#97C2×D18111yescharacteristic, complemented
173#41—3622no—
174#41—3622no—
175#41—3622no—
176#41—3622no—
177#41—1812no—
178#41—1812no—
179#41—7241no—
180C2×Q8—3622no—
181C2×Q8C22×D18111yescharacteristic
182C2×D4—41no—
183C2×D4—3622no—
184C2×D4—3622no—
185C2×D4—3622no—
186C2×D4—7241no—
187C2×D4—7242no—
188C2×D4C22×D18111yescharacteristic
189C2×D4C22×D18112yes—
190Q16—3622no—
191Q16—1812no—
192SD16—22no—
193SD16—22no—
194SD16—3622no—
195SD16—3622no—
196SD16—3622no—
197SD16—3622no—
198SD16—1812no—
199SD16—1812no—
200D8—22no—
201D8—22no—
202D8—3622no—
203D8—3622no—
204D8—3622no—
205D8—1812no—
206M16—3622no—
207M16—3622no—
208M16—1812no—
209M16—1812no—
210C2×C8—3622no—
211C2×C8—3622no—
212C2×C8C22×D18112yes—
213C22⋊C4—41no—
214C42C22×D18111yescharacteristic
215C23—21no—
216C23—22no—
217C23—7242no—
218C23—7241no—
219Q8—22no—
220Q8—1812no—
221Q8—1812no—
222Q8—7241no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—3622no—
228D4—3622no—
229D4—1812no—
230D4—1812no—
231D4—7241no—
232D4—7242no—
233D4—7242no—
234D4—7242no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—3622no—
238C2×C4—3622no—
239C2×C4—7242no—
240C2×C4—7241no—
241C2×C4—7242no—
242C2×C4C23×D18111yescharacteristic
243C8—3622no—
244C8—3622no—
245C8D4×D18112yes—
246C8D4×D18112yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—28961no—
252C22—28962no—
253C22—3622no—
254C22—3622no—
255C22#1915386111yescharacteristic
256C4—41no—
257C4—41no—
258C4—3622no—
259C4—3622no—
260C4C2×D4×D18112yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—7242no—
265C2—7242no—
266C2#2604191411yescharacteristic
subgroupG/H
centerC2#26041914
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC2×C181D42
hypercenter#496D181
hypocenterC181#3288
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26042085C2semidirect
2. #26042080C2semidirect
3. #26041546C2semidirect
4. #26041539C2semidirect
5. #26041456C2semidirect
6. #26041451C2semidirect
7. #26041446C2semidirect
8. #26041402C2semidirect
9. #496×C181C2semidirect
10. #19153864C22semidirect
12. #19153861C22semidirect
13. #19153847C22semidirect
17. #19153842C22semidirect
18. #19153842C22semidirect
26. #19153822C22semidirect
27. #19153822C22semidirect
28. #19153752C22semidirect
29. #19153744C22semidirect
30. #19153744C22semidirect
31. #19153735C22semidirect
32. #19153733C22semidirect
33. #19153733C22semidirect
34. C2×SD16×C181C22semidirect
35. C2×D8×C181C22semidirect
36. #127×C181C22semidirect
37. #124×C181C22semidirect
38. #97×C181C22semidirect
52. Dic724D4semidirect
54. #15668438D4semidirect
55. #15668438D4semidirect
60. D1448D4semidirect
133. C181#3288semidirect
137. #496D181semidirect
159. C2×SD16C2×D181semidirect
161. C2×D8C2×D181semidirect
164. #127C2×D181semidirect
165. #124C2×D181semidirect
172. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420852
#260420802
#260415462
#260415392
#260414562
#260414512
#260414462
#260414022
#496×C1812
#3288181