Finite Group Databasebeta

← #37545515

Subgroups — #37545515

Order 23168.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#26042091C212yesmaximal, complemented
2#26042086C212yesmaximal, complemented
3#26041652C211yesmaximal, characteristic, complemented
4#26041543C212yesmaximal, complemented
5#26041542C212yesmaximal, complemented
6#26041463C212yesmaximal, complemented
7#26041448C212yesmaximal, complemented
8#26041397C211yesmaximal, characteristic, complemented
9#443×C181C211yesmaximal, characteristic, complemented
10#19153863C2212yes—
11#19153862C2212yescomplemented
12#19153849—22no—
13#19153849C2212yescomplemented
14#19153848—22no—
15#19153846C2212yescomplemented
16#19153844—22no—
17#19153844C2212yes—
18#19153841C2212yes—
19#19153840—22no—
20#19153838—22no—
21#19153838C2211yescharacteristic
22#19153837—22no—
23C2×#15668438—22no—
24C2×#15668438C2212yescomplemented
25C2×#15668438C2212yescomplemented
26SD16×D181—22no—
27#19153822C2212yescomplemented
28#19153816C2211yescharacteristic, complemented
29#19153813C2211yescharacteristic
30#19153752C2211yescharacteristic, complemented
31#19153744C2212yescomplemented
32#19153734C2212yescomplemented
33#19153732C2212yescomplemented
34#135×C181C2211yescharacteristic
35#131×C181C2212yescomplemented
36#108×C181C2211yescharacteristic, complemented
37#104×C181C2212yescomplemented
38C4×C8×C181C2211yescharacteristic, complemented
39#15668450—22no—
40#15668449—22no—
41#15668449—22no—
42#15668449C2312yes—
43#15668448—22no—
44#15668448—42no—
45C2×Dic362—22no—
46C2×Dic362C2312yes—
47Q8×D181—22no—
48C2×D724—22no—
49C2×D724—41no—
50C2×D724C2312yes—
51D4×D181—22no—
52D4×D181—42no—
53Dic724—22no—
54Dic724D411yescharacteristic, complemented
55#15668439—22no—
56#15668438—22no—
57#15668438—22no—
58#15668438—22no—
59#15668438—22no—
60#15668438—22no—
61#15668438—22no—
62#15668438D412yescomplemented
63#15668437—22no—
64#15668436—22no—
65D1448—22no—
66D1448D411yescharacteristic, complemented
67#15668434—22no—
68#15668433C2312yes—
69#15668432—22no—
70C8×D181—22no—
71C4⋊Dic181—41no—
72#41×C181C2312yes—
73Q16×C181D411yescharacteristic, complemented
74SD16×C181D412yescomplemented
75D8×C181D411yescharacteristic, complemented
76M16×C181—22no—
77M16×C181C2312yes—
78C2×C8×C181—22no—
79C2×C8×C181C2311yescharacteristic
80C2×C8×C181C2311yescharacteristic
81C42×C181C2311yescharacteristic
82C22×D181—22no—
83C22×D181—42no—
84Dic362—22no—
85Dic362—22no—
86Dic362—42no—
87Dic362C2×D412yes—
88D724—22no—
89D724—22no—
90D724—42no—
91D724—82no—
92D724C2×D412yes—
93C22⋊D181—22no—
94C22⋊D181—42no—
95C22⋊D181—42no—
96C2×Dic181—22no—
97C2×Dic181—42no—
98C4×D181—22no—
99C4×D181—42no—
100C4×D181—42no—
101#13089678—22no—
102Q8×C181C2×D412yes—
103D4×C181—22no—
104D4×C181C2×D412yes—
105C2×C4×C181—21no—
106C2×C4×C181—22no—
107C2×C4×C181C2411yescharacteristic
108C8×C181—22no—
109C8×C181C2×D411yescharacteristic
110C8×C181C2×D411yescharacteristic
111C8×C181C2×D411yescharacteristic
112C8×C181C2×D411yescharacteristic
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×C181C22×D411yescharacteristic
122C4×C181C22×D411yescharacteristic
123Dic181—22no—
124Dic181—42no—
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—1812no—
138#453—1812no—
139#443—1811no—
140#443D18111yescharacteristic, complemented
1412−1+4—1812no—
142H(5,𝔽2)—1812no—
143#137—3622no—
144#137—3622no—
145#137—1812no—
146#137—1812no—
147AGL(1,ℤ/8)—3622no—
148AGL(1,ℤ/8)—3622no—
149AGL(1,ℤ/8)—1812no—
150AGL(1,ℤ/8)—1812no—
151#135—3622no—
152#135—3622no—
153#135—1811no—
154#135C2×D18111yescharacteristic
155C2×SD16—3622no—
156C2×SD16—3622no—
157C2×SD16—1812no—
158C2×SD16—1812no—
159#131—1812no—
160#131C2×D18112yescomplemented
161C4⋊Q8—1811no—
162#127—1811no—
163#108—1811no—
164#108C2×D18111yescharacteristic, complemented
165#104—1812no—
166#104C2×D18112yescomplemented
167#101—1812no—
168#100—1812no—
169C4×C8C2×D18111yescharacteristic, complemented
170#41—3622no—
171#41—3622no—
172#41—3622no—
173#41—3622no—
174#41—1812no—
175#41—7242no—
176#41C22×D18112yes—
177C2×Q8—3622no—
178C2×Q8—3622no—
179C2×Q8—1812no—
180C2×D4—3622no—
181C2×D4—3622no—
182C2×D4—1812no—
183C2×D4—7241no—
184C2×D4—7242no—
185Q16—3622no—
186Q16—3622no—
187Q16—1811no—
188Q16D72411yescharacteristic, complemented
189SD16—3622no—
190SD16—3622no—
191SD16—3622no—
192SD16—3622no—
193SD16—3622no—
194SD16—3622no—
195SD16—3622no—
196SD16—3622no—
197SD16—1812no—
198SD16D72412yescomplemented
199D8—3622no—
200D8—3622no—
201D8—1811no—
202D8D72411yescharacteristic, complemented
203M16—22no—
204M16—3622no—
205M16—1812no—
206M16C22×D18112yes—
207C2×C8—22no—
208C2×C8—3622no—
209C2×C8C22×D18111yescharacteristic
210C2×C8C22×D18111yescharacteristic
211C4⋊C4—7241no—
212C42C22×D18111yescharacteristic
213C23—3622no—
214C23—7242no—
215Q8—3622no—
216Q8—3622no—
217Q8—1812no—
218Q8—7242no—
219Q8C2×D72412yes—
220D4—22no—
221D4—3622no—
222D4—3622no—
223D4—3622no—
224D4—14482no—
225D4—1812no—
226D4—7242no—
227D4—7242no—
228D4—7242no—
229D4C2×D72412yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—3622no—
233C2×C4—3622no—
234C2×C4—7242no—
235C2×C4—7242no—
236C2×C4—7242no—
237C2×C4C23×D18111yescharacteristic
238C8—22no—
239C8—3622no—
240C8D4×D18111yescharacteristic
241C8D4×D18111yescharacteristic
242C8C2×D72411yescharacteristic
243C8C2×D72411yescharacteristic
244C22—22no—
245C22—28962no—
246C22—3622no—
247C22—14482no—
248C22—7242no—
249C22#1915386211yescharacteristic
250C4—22no—
251C4—22no—
252C4—3622no—
253C4—7242no—
254C4C2×D4×D18111yescharacteristic
255C4C22×D72411yescharacteristic
256C2—21no—
257C2—42no—
258C2—14482no—
259C2—7242no—
260C2D4×D72411yescharacteristic
subgroupG/H
centerC2D4×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#443×C181C2
socleC2×C181D42
hypercenter#443D181
hypocenterC181#3291
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #26042091C2semidirect
2. #26042086C2semidirect
3. #26041652C2semidirect
4. #26041543C2semidirect
5. #26041542C2semidirect
6. #26041463C2semidirect
7. #26041448C2semidirect
8. #26041397C2semidirect
9. #443×C181C2semidirect
11. #19153862C22semidirect
13. #19153849C22semidirect
15. #19153846C22semidirect
24. C2×#15668438C22semidirect
25. C2×#15668438C22semidirect
27. #19153822C22semidirect
28. #19153816C22semidirect
30. #19153752C22semidirect
31. #19153744C22semidirect
32. #19153734C22semidirect
33. #19153732C22semidirect
35. #131×C181C22semidirect
36. #108×C181C22semidirect
37. #104×C181C22semidirect
38. C4×C8×C181C22semidirect
54. Dic724D4semidirect
62. #15668438D4semidirect
66. D1448D4semidirect
73. Q16×C181D4semidirect
74. SD16×C181D4semidirect
75. D8×C181D4semidirect
130. C181#3291semidirect
140. #443D181semidirect
160. #131C2×D181semidirect
164. #108C2×D181semidirect
166. #104C2×D181semidirect
169. C4×C8C2×D181semidirect
188. Q16D724semidirect
198. SD16D724semidirect
202. D8D724semidirect

Maximal subgroups

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

subgroupindex
#260420912
#260420862
#260416522
#260415432
#260415422
#260414632
#260414482
#260413972
#443×C1812
#3291181