Finite Group Databasebeta

← #37546861

Subgroups — #37546861

Order 23168.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#26042081C211yesmaximal, characteristic, complemented
2#26042080C211yesmaximal, characteristic, complemented
3#26041717C211yesmaximal, characteristic, complemented
4#26041716C212yesmaximal, complemented
5#26041713C212yesmaximal, complemented
6#26041710C212yesmaximal, complemented
7#26041709C211yesmaximal, characteristic, complemented
8C2×#19153757C211yesmaximal, characteristic, complemented
9D16×D181C212yesmaximal, complemented
10#26041697C211yesmaximal, characteristic, complemented
11#509×C181C211yesmaximal, characteristic, complemented
12#19153862—21no—
13#19153861—21no—
14#19153843—22no—
15#19153842—21no—
16#19153841—21no—
17#19153840—22no—
18#19153839C2212yescomplemented
19#19153838C2211yescharacteristic, complemented
20#19153835C2212yescomplemented
21C2×D1448C2211yescharacteristic, complemented
22D8×D181C2212yescomplemented
23D8×D181C2212yescomplemented
24#19153822C2211yescharacteristic
25#19153762C2211yescharacteristic, complemented
26#19153761C2212yescomplemented
27#19153760C2211yescharacteristic, complemented
28#19153759C2211yescharacteristic, complemented
29D2896C2212yescomplemented
30#19153757C2211yescharacteristic, complemented
31#19153757C2212yescomplemented
32#19153757C2212yescomplemented
33#19153755C2212yescomplemented
34C2×#15668424C2211yescharacteristic, complemented
35C16×D181C2212yescomplemented
36#135×C181C2211yescharacteristic, complemented
37C2×D8×C181C2211yescharacteristic, complemented
38SD32×C181C2212yescomplemented
39D16×C181C2212yescomplemented
40M32×C181C2211yescharacteristic, complemented
41#15668450—22no—
42#15668449—21no—
43#15668449—41no—
44#15668449D411yescharacteristic
45#15668448—22no—
46#15668448—41no—
47C2×D724—21no—
48C2×D724—41no—
49C2×C22⋊D181—42no—
50D4×D181—22no—
51D4×D181—22no—
52D4×D181—41no—
53D4×D181—42no—
54Dic724C2311yescharacteristic
55#15668438—21no—
56#15668437—22no—
57#15668436—22no—
58D1448—21no—
59D1448C2312yes—
60D1448C2311yescharacteristic
61#15668434—22no—
62#15668434—22no—
63#15668433D411yescharacteristic
64#15668432D412yes—
65C8×D181C2312yes—
66#15668424C2311yescharacteristic
67#15668424C2311yescharacteristic
68#41×C181—21no—
69C2×D4×C181—21no—
70Q16×C181C2311yescharacteristic
71SD16×C181—21no—
72D8×C181—21no—
73D8×C181C2311yescharacteristic
74D8×C181C2312yes—
75C2×C8×C181C2311yescharacteristic
76C16×C181C2312yes—
77C22×D181—41no—
78C22×D181—42no—
79C22×D181—42no—
80Dic362—21no—
81Dic362D811yescharacteristic
82D724—21no—
83D724—22no—
84D724—42no—
85D724—41no—
86D724D811yescharacteristic
87C22⋊D181—41no—
88C22⋊D181—42no—
89C22⋊D181—42no—
90C22⋊D181—42no—
91C22⋊D181D812yes—
92C2×Dic181—42no—
93C4×D181—41no—
94C4×D181—42no—
95C4×D181C2×D412yes—
96#13089678C2×D412yes—
97C23×C181—41no—
98Q8×C181—21no—
99D4×C181—22no—
100D4×C181—21no—
101D4×C181—41no—
102D4×C181—41no—
103C2×C4×C181—41no—
104C2×C4×C181C2×D411yescharacteristic
105C8×C181C2×D411yescharacteristic
106C8×C181C2411yescharacteristic
107C22×C181—41no—
108C22×C181—42no—
109C22×C181—161no—
110C22×C181C2×D811yescharacteristic
111C2×D181—41no—
112C2×D181—42no—
113C2×D181—162no—
114C2×D181—162no—
115C2×D181—161no—
116C2×D181C2×D812yes—
117C4×C181—41no—
118C4×C181C2×D811yescharacteristic
119C4×C181C22×D411yescharacteristic
120Dic181—41no—
121Dic181C2×D812yes—
122C2×C181—21no—
123C2×C181—81no—
124C2×C181—82no—
125C2×C181C22×D811yescharacteristic
126D181—22no—
127D181—81no—
128D181—82no—
129C181#341511yescharacteristic, complemented, Sylow
130#3415—1811nomaximal, Sylow
131#576—1811no—
132#576—1811no—
133#509—1812no—
134#509—1812no—
135#509—1811no—
136#509D18111yescharacteristic, complemented
137#508—1811no—
138#508—1812no—
139C2×D16—1811no—
140C2×D16—1812no—
141#504—1811no—
142H(5,𝔽2)—3621no—
143H(5,𝔽2)—3621no—
144AGL(1,ℤ/8)—3622no—
145AGL(1,ℤ/8)—3622no—
146AGL(1,ℤ/8)—3621no—
147AGL(1,ℤ/8)—3621no—
148#135—1812no—
149#135—1812no—
150#135—1811no—
151#135C2×D18111yescharacteristic, complemented
152C2×D8—1812no—
153C2×D8—1812no—
154C2×D8—1811no—
155C2×D8C2×D18111yescharacteristic, complemented
156#131—1811no—
157Q32—1811no—
158SD32—1811no—
159SD32—1811no—
160SD32—1812no—
161SD32C2×D18112yescomplemented
162D16—1812no—
163D16—1811no—
164D16—1812no—
165D16—1812no—
166D16C2×D18112yescomplemented
167M32—1812no—
168M32C2×D18111yescharacteristic, complemented
169C2×C16—1811no—
170C2×C16—1812no—
171#41—21no—
172#41—3622no—
173#41—3622no—
174#41—3621no—
175#41—1811no—
176#41—7241no—
177#41—7241no—
178C2×D4—21no—
179C2×D4—3622no—
180C2×D4—3622no—
181C2×D4—3621no—
182C2×D4—7241no—
183C2×D4—7242no—
184C2×D4—7241no—
185C2×D4—7242no—
186Q16—1811no—
187Q16C22×D18111yescharacteristic
188SD16—21no—
189SD16—3622no—
190SD16—3622no—
191SD16—3621no—
192D8—21no—
193D8—3622no—
194D8—3622no—
195D8—3621no—
196D8—1812no—
197D8—1811no—
198D8C22×D18112yes—
199D8C22×D18111yescharacteristic
200M16—1811no—
201M16—1812no—
202C2×C8—1812no—
203C2×C8C22×D18111yescharacteristic
204C16—1811no—
205C16—1811no—
206C16C22×D18112yes—
207C23—41no—
208C23—7242no—
209C23—7242no—
210C23—7241no—
211Q8—21no—
212Q8—3621no—
213Q8—1811no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—3621no—
219D4—3622no—
220D4—1812no—
221D4—1811no—
222D4—7242no—
223D4—7242no—
224D4—7242no—
225D4—7242no—
226D4—7241no—
227D4—7241no—
228C2×C4—41no—
229C2×C4—1812no—
230C2×C4—7242no—
231C2×C4—7242no—
232C2×C4—7241no—
233C2×C4D4×D18111yescharacteristic
234C8—1812no—
235C8D4×D18111yescharacteristic
236C8C23×D18111yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—28961no—
241C22—28962no—
242C22—28962no—
243C22—1812no—
244C22—7242no—
245C22—7241no—
246C22D8×D18111yescharacteristic
247C4—41no—
248C4—1812no—
249C4—7241no—
250C4D8×D18111yescharacteristic
251C4C2×D4×D18111yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—3622no—
256C2—14482no—
257C2—14481no—
258C2C2×D8×D18111yescharacteristic
subgroupG/H
centerC2C2×D8×D181
commutator, cosocleC8×C181C24
FrattiniC8C23×D181
Fitting#509×C181C2
socleC2×C181C22×D8
hypercenter#509D181
hypocenterC181#3415
LCSG ▷ C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #26042081C2semidirect
2. #26042080C2semidirect
3. #26041717C2semidirect
4. #26041716C2semidirect
5. #26041713C2semidirect
6. #26041710C2semidirect
7. #26041709C2semidirect
8. C2×#19153757C2semidirect
9. D16×D181C2semidirect
10. #26041697C2semidirect
11. #509×C181C2semidirect
18. #19153839C22semidirect
19. #19153838C22semidirect
20. #19153835C22semidirect
21. C2×D1448C22semidirect
22. D8×D181C22semidirect
23. D8×D181C22semidirect
25. #19153762C22semidirect
26. #19153761C22semidirect
27. #19153760C22semidirect
28. #19153759C22semidirect
29. D2896C22semidirect
30. #19153757C22semidirect
31. #19153757C22semidirect
32. #19153757C22semidirect
33. #19153755C22semidirect
34. C2×#15668424C22semidirect
35. C16×D181C22semidirect
36. #135×C181C22semidirect
37. C2×D8×C181C22semidirect
38. SD32×C181C22semidirect
39. D16×C181C22semidirect
40. M32×C181C22semidirect
129. C181#3415semidirect
136. #509D181semidirect
151. #135C2×D181semidirect
155. C2×D8C2×D181semidirect
161. SD32C2×D181semidirect
166. D16C2×D181semidirect
168. M32C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420812
#260420802
#260417172
#260417162
#260417132
#260417102
#260417092
C2×#191537572
D16×D1812
#260416972
#509×C1812
#3415181