Finite Group Databasebeta

← #37546867

Subgroups — #37546867

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#26042093C211yesmaximal, characteristic, complemented
2#26042080C211yesmaximal, characteristic, complemented
3#26041722C211yesmaximal, characteristic, complemented
4#26041718C212yesmaximal, complemented
5#26041713C212yesmaximal, complemented
6#26041709C211yesmaximal, characteristic, complemented
7#26041708C212yesmaximal, complemented
8C2×#19153760C211yesmaximal, characteristic, complemented
9SD32×D181C212yesmaximal, complemented
10#26041697C211yesmaximal, characteristic, complemented
11#509×C181C211yesmaximal, characteristic, complemented
12#19153863—21no—
13#19153861—21no—
14#19153846—21no—
15#19153845—22no—
16#19153842—21no—
17#19153840—22no—
18#19153838C2211yescharacteristic
19#19153835C2212yes—
20#19153835C2212yescomplemented
21C2×Dic724C2211yescharacteristic, complemented
22Q16×D181C2212yescomplemented
23D8×D181C2212yes—
24#19153822C2211yescharacteristic
25Dic1448C2212yescomplemented
26#19153762C2211yescharacteristic, complemented
27#19153761C2212yescomplemented
28#19153760C2212yescomplemented
29#19153760C2212yescomplemented
30#19153760C2211yescharacteristic, complemented
31#19153759C2211yescharacteristic, complemented
32#19153757C2211yescharacteristic, complemented
33#19153755C2212yescomplemented
34C2×#15668424C2211yescharacteristic, complemented
35C16×D181C2212yescomplemented
36#135×C181C2211yescharacteristic, complemented
37C2×D8×C181C2211yescharacteristic
38SD32×C181C2212yescomplemented
39D16×C181C2212yescomplemented
40M32×C181C2211yescharacteristic, complemented
41#15668449—21no—
42#15668449—41no—
43#15668449D411yescharacteristic
44#15668448—22no—
45#15668448—22no—
46#15668448—41no—
47#15668448—42no—
48C2×Dic362—21no—
49C2×Dic362—41no—
50Q8×D181—22no—
51C2×C22⋊D181—42no—
52D4×D181—22no—
53D4×D181—41no—
54Dic724—21no—
55Dic724C2312yes—
56Dic724C2311yescharacteristic
57#15668439—22no—
58#15668438—21no—
59#15668437—22no—
60#15668437—22no—
61D1448C2311yescharacteristic
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—42no—
78Dic362—21no—
79Dic362—22no—
80Dic362—41no—
81Dic362—42no—
82Dic362D811yescharacteristic
83D724—21no—
84D724D811yescharacteristic
85C22⋊D181—41no—
86C22⋊D181—42no—
87C22⋊D181—42no—
88C22⋊D181—42no—
89C22⋊D181D812yes—
90C2×Dic181—41no—
91C2×Dic181—42no—
92C2×Dic181—42no—
93C4×D181—42no—
94C4×D181—41no—
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—162no—
113C2×D181C2×D812yes—
114C4×C181—41no—
115C4×C181C2×D811yescharacteristic
116C4×C181C22×D411yescharacteristic
117Dic181—41no—
118Dic181—42no—
119Dic181C2×D812yes—
120C2×C181—21no—
121C2×C181—82no—
122C2×C181—81no—
123C2×C181C22×D811yescharacteristic
124D181—22no—
125D181—81no—
126C181#341611yescharacteristic, complemented, Sylow
127#3416—1811nomaximal, Sylow
128#578—1811no—
129#576—1811no—
130#510—1812no—
131#510—1811no—
132#509—1812no—
133#509D18111yescharacteristic, complemented
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, complemented
149C2×Q16—1812no—
150C2×Q16—1811no—
151C2×D8—1812no—
152C2×D8C2×D18111yescharacteristic
153#131—1811no—
154Q32—1811no—
155Q32—1812no—
156SD32—1811no—
157SD32—1812no—
158SD32—1812no—
159SD32—1811no—
160SD32—1812no—
161SD32C2×D18112yescomplemented
162D16—1811no—
163D16C2×D18112yescomplemented
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—7241no—
174#41—7242no—
175#41—7241no—
176C2×Q8—3622no—
177C2×Q8—3621no—
178C2×Q8—7241no—
179C2×D4—21no—
180C2×D4—3622no—
181C2×D4—7241no—
182C2×D4—7242no—
183Q16—3622no—
184Q16—3621no—
185Q16—1812no—
186Q16—1811no—
187Q16C22×D18111yescharacteristic
188SD16—21no—
189SD16—3622no—
190SD16—3622no—
191SD16—3621no—
192D8—21no—
193D8—3622no—
194D8—1811no—
195D8C22×D18111yescharacteristic
196D8C22×D18112yes—
197M16—1811no—
198M16—1812no—
199C2×C8—1812no—
200C2×C8C22×D18111yescharacteristic
201C16—1811no—
202C16—1811no—
203C16C22×D18112yes—
204C23—41no—
205C23—7242no—
206Q8—21no—
207Q8—3621no—
208Q8—3622no—
209Q8—1811no—
210Q8—7242no—
211Q8—7241no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3621no—
217D4—1812no—
218D4—1811no—
219D4—7242no—
220D4—7242no—
221D4—7242no—
222D4—7241no—
223C2×C4—41no—
224C2×C4—1812no—
225C2×C4—7242no—
226C2×C4—7242no—
227C2×C4—7242no—
228C2×C4—7241no—
229C2×C4—7241no—
230C2×C4D4×D18111yescharacteristic
231C8—1812no—
232C8D4×D18111yescharacteristic
233C8C23×D18111yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—28962no—
238C22—1812no—
239C22—7241no—
240C22D8×D18111yescharacteristic
241C4—41no—
242C4—1812no—
243C4—7241no—
244C4—7242no—
245C4D8×D18111yescharacteristic
246C4C2×D4×D18111yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—3622no—
251C2—14481no—
252C2C2×D8×D18111yescharacteristic
subgroupG/H
centerC2C2×D8×D181
commutator, cosocleC8×C181C24
FrattiniC8C23×D181
Fitting#509×C181C2
socleC2×C181C22×D8
hypercenter#509D181
hypocenterC181#3416
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. #26042093C2semidirect
2. #26042080C2semidirect
3. #26041722C2semidirect
4. #26041718C2semidirect
5. #26041713C2semidirect
6. #26041709C2semidirect
7. #26041708C2semidirect
8. C2×#19153760C2semidirect
9. SD32×D181C2semidirect
10. #26041697C2semidirect
11. #509×C181C2semidirect
20. #19153835C22semidirect
21. C2×Dic724C22semidirect
22. Q16×D181C22semidirect
25. Dic1448C22semidirect
26. #19153762C22semidirect
27. #19153761C22semidirect
28. #19153760C22semidirect
29. #19153760C22semidirect
30. #19153760C22semidirect
31. #19153759C22semidirect
32. #19153757C22semidirect
33. #19153755C22semidirect
34. C2×#15668424C22semidirect
35. C16×D181C22semidirect
36. #135×C181C22semidirect
38. SD32×C181C22semidirect
39. D16×C181C22semidirect
40. M32×C181C22semidirect
126. C181#3416semidirect
133. #509D181semidirect
148. #135C2×D181semidirect
161. SD32C2×D181semidirect
163. D16C2×D181semidirect
165. M32C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420932
#260420802
#260417222
#260417182
#260417132
#260417092
#260417082
C2×#191537602
SD32×D1812
#260416972
#509×C1812
#3416181