Finite Group Databasebeta

← #37545511

Subgroups — #37545511

Order 23168.

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

normal subgroups only

This group has 460 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#26042087C212yesmaximal, complemented
3#26041653C211yesmaximal, characteristic, complemented
4#26041545C212yesmaximal, complemented
5#26041544C212yesmaximal, complemented
6#26041461C211yesmaximal, characteristic, complemented
7#26041448C212yesmaximal, complemented
8#26041393C212yesmaximal, complemented
9#456×C181C211yesmaximal, characteristic, complemented
10#19153864C2211yescharacteristic
11#19153862C2212yescomplemented
12#19153848—22no—
13#19153847—22no—
14#19153847C2212yescomplemented
15#19153844C2212yes—
16#19153843—22no—
17#19153841—22no—
18#19153841C2212yes—
19#19153839—22no—
20#19153836—22no—
21#19153836C2212yes—
22C2×#15668436—22no—
23C2×#15668436C2212yescomplemented
24C2×#15668436C2212yescomplemented
25SD16×D181—22no—
26#19153824C2212yescomplemented
27#19153823C2212yescomplemented
28#19153813C2211yescharacteristic
29#19153751C2212yescomplemented
30#19153744C2212yescomplemented
31#19153735C2211yescharacteristic, complemented
32#19153732C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
342−1+4×C181C2211yescharacteristic
35#137×C181C2212yescomplemented
36C4⋊Q8×C181C2211yescharacteristic, complemented
37#104×C181C2212yescomplemented
38#101×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#15668439—22no—
51#15668439D412yescomplemented
52#15668438—22no—
53#15668437D412yescomplemented
54#15668436—22no—
55#15668436—22no—
56#15668436—22no—
57#15668436—22no—
58#15668436—22no—
59#15668436—22no—
60#15668436D412yescomplemented
61D1448—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×C181—41no—
72#41×C181C2312yes—
73C2×Q8×C181—22no—
74C2×Q8×C181C2311yescharacteristic
75C2×Q8×C181C2311yescharacteristic
76Q16×C181—22no—
77SD16×C181—22no—
78M16×C181C2312yes—
79C4⋊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—
99Q8×C181—22no—
100Q8×C181—22no—
101Q8×C181—41no—
102Q8×C181C2×D412yes—
103D4×C181—22no—
104D4×C181—41no—
105D4×C181C2×D412yes—
106C2×C4×C181—21no—
107C2×C4×C181—22no—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181—42no—
111C2×C4×C181C2411yescharacteristic
112C8×C181—22no—
113C22×C181—22no—
114C22×C181H(5,𝔽2)11yescharacteristic
115C2×D181—22no—
116C2×D181—42no—
117C2×D181—82no—
118C2×D181—162no—
119C4×C181—22no—
120C4×C181—22no—
121C4×C181—41no—
122C4×C181—41no—
123C4×C181C22×D412yes—
124Dic181—22no—
125C2×C181—21no—
126C2×C181—42no—
127C2×C181D4211yescharacteristic
128D181—42no—
129D181—82no—
130C181#329111yescharacteristic, complemented, Sylow
131#3291—1811nomaximal, Sylow
132#577—1812no—
133#577—1812no—
134#492—1811no—
135#471—1812no—
136#471—1812no—
137#456—1811no—
138#456D18111yescharacteristic, complemented
139#453—1812no—
140#443—1812no—
1412−1+4—1811no—
1422−1+4C2×D18111yescharacteristic
143H(5,𝔽2)—1812no—
144#137—3622no—
145#137—3622no—
146#137—1812no—
147#137C2×D18112yescomplemented
148AGL(1,ℤ/8)—3622no—
149AGL(1,ℤ/8)—3622no—
150AGL(1,ℤ/8)—1812no—
151AGL(1,ℤ/8)—1812no—
152#135—3622no—
153#135—3622no—
154#135—1812no—
155C2×SD16—3622no—
156C2×SD16—3622no—
157C2×SD16—1812no—
158C2×SD16—1812no—
159#131—1812no—
160#131—1812no—
161C4⋊Q8C2×D18111yescharacteristic, complemented
162#127—1811no—
163#108—1812no—
164#104—1812no—
165#104C2×D18112yescomplemented
166#101—1811no—
167#101C2×D18111yescharacteristic, complemented
168#100—1812no—
169C4×C8—1811no—
170#41—22no—
171#41—41no—
172#41—3622no—
173#41—3622no—
174#41—3622no—
175#41—1812no—
176#41—7241no—
177#41C22×D18112yes—
178C2×Q8—22no—
179C2×Q8—3622no—
180C2×Q8C22×D18111yescharacteristic
181C2×Q8C22×D18111yescharacteristic
182C2×D4—3622no—
183C2×D4—3622no—
184C2×D4—1812no—
185C2×D4—7241no—
186C2×D4—7242no—
187Q16—22no—
188Q16—3622no—
189Q16—1812no—
190SD16—22no—
191SD16—3622no—
192SD16—3622no—
193SD16—3622no—
194SD16—3622no—
195SD16—3622no—
196SD16—3622no—
197SD16—3622no—
198SD16—1812no—
199SD16—1812no—
200D8—3622no—
201D8—3622no—
202D8—1812no—
203M16—3622no—
204M16—3622no—
205M16—1812no—
206M16C22×D18112yes—
207C2×C8—3622no—
208C2×C8—3622no—
209C2×C8—1812no—
210C4⋊C4—41no—
211C42C22×D18111yescharacteristic
212C23—3622no—
213C23—7242no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—1812no—
218Q8—7241no—
219Q8D4×D18112yes—
220D4—22no—
221D4—41no—
222D4—3622no—
223D4—3622no—
224D4—3622no—
225D4—14482no—
226D4—1812no—
227D4—7242no—
228D4—7241no—
229D4—7242no—
230D4D4×D18112yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3622no—
237C2×C4—7242no—
238C2×C4—7242no—
239C2×C4C23×D18111yescharacteristic
240C8—22no—
241C8—3622no—
242C8—1812no—
243C8—1812no—
244C22—22no—
245C22—28962no—
246C22—3622no—
247C22—14482no—
248C22—7242no—
249C22#1915386111yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—3622no—
255C4C2×D4×D18112yes—
256C2—21no—
257C2—42no—
258C2—14482no—
259C2—7242no—
260C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#456×C181C2
socleC2×C181D42
hypercenter#456D181
hypocenterC181#3291
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26042088C2semidirect
2. #26042087C2semidirect
3. #26041653C2semidirect
4. #26041545C2semidirect
5. #26041544C2semidirect
6. #26041461C2semidirect
7. #26041448C2semidirect
8. #26041393C2semidirect
9. #456×C181C2semidirect
11. #19153862C22semidirect
14. #19153847C22semidirect
23. C2×#15668436C22semidirect
24. C2×#15668436C22semidirect
26. #19153824C22semidirect
27. #19153823C22semidirect
29. #19153751C22semidirect
30. #19153744C22semidirect
31. #19153735C22semidirect
32. #19153732C22semidirect
33. C4×#13089678C22semidirect
35. #137×C181C22semidirect
36. C4⋊Q8×C181C22semidirect
37. #104×C181C22semidirect
38. #101×C181C22semidirect
51. #15668439D4semidirect
53. #15668437D4semidirect
60. #15668436D4semidirect
63. #15668434D4semidirect
130. C181#3291semidirect
138. #456D181semidirect
147. #137C2×D181semidirect
161. C4⋊Q8C2×D181semidirect
165. #104C2×D181semidirect
167. #101C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420882
#260420872
#260416532
#260415452
#260415442
#260414612
#260414482
#260413932
#456×C1812
#3291181