Finite Group Databasebeta

← #37546866

Subgroups — #37546866

Order 23168.

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

normal subgroups only

This group has 360 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#26042092C211yesmaximal, characteristic, complemented
2#26042081C211yesmaximal, characteristic, complemented
3#26041721C212yesmaximal, complemented
4#26041717C211yesmaximal, characteristic, complemented
5#26041716C212yesmaximal, complemented
6#26041712C212yesmaximal, complemented
7#26041709C211yesmaximal, characteristic, complemented
8C2×#19153759C211yesmaximal, characteristic, complemented
9SD32×D181C212yesmaximal, complemented
10#26041697C211yesmaximal, characteristic, complemented
11#510×C181C211yesmaximal, characteristic, complemented
12#19153864—21no—
13#19153862—21no—
14#19153848—22no—
15#19153847—21no—
16#19153843—22no—
17#19153841—21no—
18#19153839C2212yes—
19#19153839C2212yescomplemented
20#19153838C2211yescharacteristic, complemented
21Q16×D181C2212yescomplemented
22C2×D1448C2211yescharacteristic
23D8×D181C2212yes—
24#19153822C2211yescharacteristic
25#19153762C2211yescharacteristic, complemented
26#19153761C2212yescomplemented
27#19153760C2211yescharacteristic, complemented
28#19153759C2212yescomplemented
29#19153759C2212yescomplemented
30#19153759C2211yescharacteristic, complemented
31D2896C2212yescomplemented
32#19153757C2211yescharacteristic, complemented
33#19153755C2212yescomplemented
34C2×#15668424C2211yescharacteristic, complemented
35C16×D181C2212yescomplemented
36#135×C181C2211yescharacteristic
37C2×Q16×C181C2211yescharacteristic, complemented
38Q32×C181C2212yescomplemented
39SD32×C181C2212yescomplemented
40M32×C181C2211yescharacteristic, complemented
41#15668450—22no—
42#15668450—22no—
43#15668450—41no—
44#15668449—21no—
45#15668449—41no—
46#15668449—42no—
47#15668449D411yescharacteristic
48Q8×D181—22no—
49Q8×D181—41no—
50C2×D724—21no—
51C2×D724—41no—
52D4×D181—22no—
53D4×D181—42no—
54Dic724C2311yescharacteristic
55#15668439—22no—
56#15668438—21no—
57#15668436—22no—
58#15668436—22no—
59D1448—21no—
60D1448C2311yescharacteristic
61D1448C2312yes—
62#15668434—22no—
63#15668433D411yescharacteristic
64#15668432D412yes—
65C8×D181C2312yes—
66#15668424C2311yescharacteristic
67#15668424C2311yescharacteristic
68#41×C181—21no—
69C2×Q8×C181—21no—
70Q16×C181—21no—
71Q16×C181C2312yes—
72Q16×C181C2311yescharacteristic
73SD16×C181—21no—
74D8×C181C2311yescharacteristic
75C2×C8×C181C2311yescharacteristic
76C16×C181C2312yes—
77C22×D181—41no—
78C22×D181—42no—
79Dic362—21no—
80Dic362—42no—
81Dic362D811yescharacteristic
82D724—22no—
83D724—21no—
84D724—42no—
85D724—42no—
86D724—41no—
87D724D811yescharacteristic
88C22⋊D181—41no—
89C22⋊D181—42no—
90C22⋊D181D812yes—
91C4×D181—42no—
92C4×D181—41no—
93C4×D181—42no—
94C4×D181—42no—
95C4×D181C2×D412yes—
96#13089678C2×D412yes—
97Q8×C181—21no—
98Q8×C181—22no—
99Q8×C181—41no—
100D4×C181—21no—
101D4×C181—41no—
102C2×C4×C181—41no—
103C2×C4×C181—41no—
104C2×C4×C181C2×D411yescharacteristic
105C8×C181C2×D411yescharacteristic
106C8×C181C2411yescharacteristic
107C22×C181—41no—
108C22×C181C2×D811yescharacteristic
109C2×D181—41no—
110C2×D181—42no—
111C2×D181—162no—
112C2×D181—161no—
113C2×D181C2×D812yes—
114C4×C181—41no—
115C4×C181—42no—
116C4×C181C2×D811yescharacteristic
117C4×C181C22×D411yescharacteristic
118Dic181—41no—
119Dic181C2×D812yes—
120C2×C181—21no—
121C2×C181—81no—
122C2×C181C22×D811yescharacteristic
123D181—22no—
124D181—81no—
125D181—82no—
126C181#341611yescharacteristic, complemented, Sylow
127#3416—1811nomaximal, Sylow
128#578—1811no—
129#576—1811no—
130#510—1812no—
131#510D18111yescharacteristic, complemented
132#509—1812no—
133#509—1811no—
134#508—1811no—
135#508—1812no—
136C2×SD32—1811no—
137C2×SD32—1812no—
138#504—1811no—
1392−1+4—3621no—
140H(5,𝔽2)—3621no—
141#137—3622no—
142#137—3621no—
143AGL(1,ℤ/8)—3622no—
144AGL(1,ℤ/8)—3621no—
145#135—1812no—
146#135—1812no—
147#135—1811no—
148#135C2×D18111yescharacteristic
149C2×Q16—1812no—
150C2×Q16C2×D18111yescharacteristic, complemented
151C2×D8—1812no—
152C2×D8—1811no—
153#131—1811no—
154Q32—1811no—
155Q32C2×D18112yescomplemented
156SD32—1812no—
157SD32—1812no—
158SD32—1811no—
159SD32—1811no—
160SD32—1812no—
161SD32C2×D18112yescomplemented
162D16—1811no—
163D16—1812no—
164M32—1812no—
165M32C2×D18111yescharacteristic, complemented
166C2×C16—1811no—
167C2×C16—1812no—
168#41—21no—
169#41—3622no—
170#41—3622no—
171#41—3621no—
172#41—1811no—
173#41—7242no—
174#41—7241no—
175#41—7241no—
176C2×Q8—21no—
177C2×Q8—3622no—
178C2×Q8—7241no—
179C2×D4—3622no—
180C2×D4—3621no—
181C2×D4—7241no—
182C2×D4—7242no—
183Q16—21no—
184Q16—3622no—
185Q16—1811no—
186Q16C22×D18112yes—
187Q16C22×D18111yescharacteristic
188SD16—21no—
189SD16—3622no—
190SD16—3622no—
191SD16—3621no—
192D8—3622no—
193D8—3621no—
194D8—1811no—
195D8—1812no—
196D8C22×D18111yescharacteristic
197M16—1811no—
198M16—1812no—
199C2×C8—1812no—
200C2×C8C22×D18111yescharacteristic
201C16—1811no—
202C16—1811no—
203C16C22×D18112yes—
204C23—7242no—
205C23—7241no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—3621no—
210Q8—1811no—
211Q8—7242no—
212D4—21no—
213D4—41no—
214D4—3622no—
215D4—3621no—
216D4—1812no—
217D4—1811no—
218D4—7242no—
219D4—7242no—
220D4—7242no—
221D4—7241no—
222D4—7241no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1812no—
226C2×C4—7242no—
227C2×C4—7242no—
228C2×C4—7242no—
229C2×C4—7241no—
230C2×C4D4×D18111yescharacteristic
231C8—1812no—
232C8D4×D18111yescharacteristic
233C8C23×D18111yescharacteristic
234C22—41no—
235C22—28961no—
236C22—28962no—
237C22—1812no—
238C22—7241no—
239C22—7242no—
240C22D8×D18111yescharacteristic
241C4—41no—
242C4—42no—
243C4—1812no—
244C4—7241no—
245C4D8×D18111yescharacteristic
246C4C2×D4×D18111yescharacteristic
247C2—21no—
248C2—81no—
249C2—3622no—
250C2—14482no—
251C2—14481no—
252C2C2×D8×D18111yescharacteristic
subgroupG/H
centerC2C2×D8×D181
commutator, cosocleC8×C181C24
FrattiniC8C23×D181
Fitting#510×C181C2
socleC2×C181C22×D8
hypercenter#510D181
hypocenterC181#3416
LCSG ▷ C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #26042092C2semidirect
2. #26042081C2semidirect
3. #26041721C2semidirect
4. #26041717C2semidirect
5. #26041716C2semidirect
6. #26041712C2semidirect
7. #26041709C2semidirect
8. C2×#19153759C2semidirect
9. SD32×D181C2semidirect
10. #26041697C2semidirect
11. #510×C181C2semidirect
19. #19153839C22semidirect
20. #19153838C22semidirect
21. Q16×D181C22semidirect
25. #19153762C22semidirect
26. #19153761C22semidirect
27. #19153760C22semidirect
28. #19153759C22semidirect
29. #19153759C22semidirect
30. #19153759C22semidirect
31. D2896C22semidirect
32. #19153757C22semidirect
33. #19153755C22semidirect
34. C2×#15668424C22semidirect
35. C16×D181C22semidirect
37. C2×Q16×C181C22semidirect
38. Q32×C181C22semidirect
39. SD32×C181C22semidirect
40. M32×C181C22semidirect
126. C181#3416semidirect
131. #510D181semidirect
150. C2×Q16C2×D181semidirect
155. Q32C2×D181semidirect
161. SD32C2×D181semidirect
165. M32C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420922
#260420812
#260417212
#260417172
#260417162
#260417122
#260417092
C2×#191537592
SD32×D1812
#260416972
#510×C1812
#3416181