Finite Group Databasebeta

← #37545516

Subgroups — #37545516

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#26042090C212yesmaximal, complemented
2#26042089C212yesmaximal, complemented
3#26041651C211yesmaximal, characteristic, complemented
4#26041541C212yesmaximal, complemented
5#26041540C212yesmaximal, complemented
6#26041463C212yesmaximal, complemented
7#26041446C211yesmaximal, characteristic, complemented
8#26041393C212yesmaximal, complemented
9#453×C181C211yesmaximal, characteristic, complemented
10#19153863C2212yes—
11#19153861C2211yescharacteristic, complemented
12#19153849C2212yescomplemented
13#19153846—22no—
14#19153846C2212yescomplemented
15#19153845—22no—
16#19153842—22no—
17#19153842C2212yes—
18#19153840—22no—
19#19153836—22no—
20#19153836C2212yes—
21#19153835—22no—
22C2×#15668437—22no—
23C2×#15668437C2212yescomplemented
24C2×#15668437C2212yescomplemented
25SD16×D181—22no—
26#19153824C2212yescomplemented
27#19153823C2212yescomplemented
28#19153816C2211yescharacteristic, complemented
29#19153751C2212yescomplemented
30#19153744C2212yescomplemented
31#19153734C2212yescomplemented
32#19153733C2211yescharacteristic, complemented
33C4×#13089678C2211yescharacteristic, complemented
34H(5,𝔽2)×C181C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C181C2212yes—
36#127×C181C2211yescharacteristic
37#104×C181C2212yescomplemented
38#100×C181C2211yescharacteristic, complemented
39#15668449—22no—
40#15668449C2312yes—
41#15668448—22no—
42#15668448—22no—
43#15668448—42no—
44C2×Dic362—22no—
45C2×Dic362C2312yes—
46Q8×D181—22no—
47C2×C22⋊D181—41no—
48D4×D181—22no—
49Dic724—22no—
50#15668439—22no—
51#15668439D412yescomplemented
52#15668438—22no—
53#15668437—22no—
54#15668437—22no—
55#15668437—22no—
56#15668437—22no—
57#15668437—22no—
58#15668437—22no—
59#15668437D412yescomplemented
60#15668436D412yescomplemented
61#15668434—22no—
62#15668434D412yescomplemented
63#15668433—22no—
64#15668433C2312yes—
65#15668432—22no—
66C2×#13089678—22no—
67C2×#13089678C2312yes—
68C8×D181—22no—
69C4⋊Dic181—41no—
70#41×C181—22no—
71#41×C181C2312yes—
72C2×D4×C181—22no—
73C2×D4×C181—41no—
74C2×D4×C181—41no—
75C2×D4×C181C2311yescharacteristic
76C2×D4×C181C2311yescharacteristic
77SD16×C181—22no—
78D8×C181—22no—
79M16×C181C2312yes—
80C42×C181C2311yescharacteristic
81C22×D181—41no—
82Dic362—22no—
83Dic362—22no—
84Dic362—42no—
85Dic362C2×D412yes—
86D724C2×D412yes—
87C22⋊D181—22no—
88C22⋊D181—42no—
89C22⋊D181—42no—
90C2×Dic181—22no—
91C2×Dic181—41no—
92C2×Dic181—42no—
93C4×D181—22no—
94C4×D181—42no—
95#13089678—22no—
96#13089678C2×D412yes—
97#13089678C2×D412yes—
98C23×C181—21no—
99C23×C181—21no—
100C23×C181—41no—
101Q8×C181C2×D412yes—
102D4×C181—22no—
103D4×C181—22no—
104D4×C181—22no—
105D4×C181—42no—
106D4×C181—82no—
107D4×C181C2×D412yes—
108C2×C4×C181—21no—
109C2×C4×C181—22no—
110C2×C4×C181—41no—
111C2×C4×C181C2411yescharacteristic
112C8×C181—22no—
113C22×C181—22no—
114C22×C181—41no—
115C22×C181—41no—
116C22×C181—81no—
117C22×C181—81no—
118C22×C181—161no—
119C22×C181H(5,𝔽2)11yescharacteristic
120C2×D181—22no—
121C2×D181—161no—
122C4×C181—22no—
123C4×C181—22no—
124C4×C181C22×D412yes—
125Dic181—22no—
126Dic181—42no—
127C2×C181—21no—
128C2×C181—42no—
129C2×C181—81no—
130C2×C181—81no—
131C2×C181D4211yescharacteristic
132D181—42no—
133C181#329111yescharacteristic, complemented, Sylow
134#3291—1811nomaximal, Sylow
135#577—1812no—
136#577—1812no—
137#492—1811no—
138#471—1812no—
139#471—1812no—
140#456—1812no—
141#453—1811no—
142#453D18111yescharacteristic, complemented
143#443—1812no—
1442−1+4—1812no—
145H(5,𝔽2)—1811no—
146H(5,𝔽2)C2×D18111yescharacteristic, complemented
147#137—3622no—
148#137—3622no—
149#137—1812no—
150#137—1812no—
151AGL(1,ℤ/8)—3622no—
152AGL(1,ℤ/8)—3622no—
153AGL(1,ℤ/8)—1812no—
154AGL(1,ℤ/8)C2×D18112yes—
155#135—3622no—
156#135—3622no—
157#135—1812no—
158C2×SD16—3622no—
159C2×SD16—3622no—
160C2×SD16—1812no—
161C2×SD16—1812no—
162#131—1812no—
163#131—1812no—
164C4⋊Q8—1811no—
165#127C2×D18111yescharacteristic
166#108—1812no—
167#104—1812no—
168#104C2×D18112yescomplemented
169#101—1812no—
170#100—1811no—
171#100C2×D18111yescharacteristic, complemented
172C4×C8—1811no—
173#41—22no—
174#41—3622no—
175#41—3622no—
176#41—3622no—
177#41—1812no—
178#41—7242no—
179#41C22×D18112yes—
180C2×Q8—3622no—
181C2×Q8—3622no—
182C2×Q8—1812no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3622no—
187C2×D4—7241no—
188C2×D4C22×D18111yescharacteristic
189C2×D4C22×D18111yescharacteristic
190Q16—3622no—
191Q16—3622no—
192Q16—1812no—
193SD16—22no—
194SD16—3622no—
195SD16—3622no—
196SD16—3622no—
197SD16—3622no—
198SD16—3622no—
199SD16—3622no—
200SD16—3622no—
201SD16—1812no—
202SD16—1812no—
203D8—22no—
204D8—3622no—
205D8—1812no—
206M16—3622no—
207M16—3622no—
208M16—1812no—
209M16C22×D18112yes—
210C2×C8—3622no—
211C2×C8—3622no—
212C2×C8—1812no—
213C4⋊C4—7241no—
214C42C22×D18111yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—7241no—
219Q8—3622no—
220Q8—3622no—
221Q8—1812no—
222Q8—7242no—
223Q8D4×D18112yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—3622no—
230D4—1812no—
231D4—7242no—
232D4—7242no—
233D4D4×D18112yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—3622no—
238C2×C4—3622no—
239C2×C4—7242no—
240C2×C4—7241no—
241C2×C4—7242no—
242C2×C4C23×D18111yescharacteristic
243C8—22no—
244C8—3622no—
245C8—1812no—
246C8—1812no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—28961no—
254C22—3622no—
255C22#1915386111yescharacteristic
256C4—22no—
257C4—22no—
258C4—3622no—
259C4—7242no—
260C4C2×D4×D18112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—7242no—
266C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#453×C181C2
socleC2×C181D42
hypercenter#453D181
hypocenterC181#3291
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26042090C2semidirect
2. #26042089C2semidirect
3. #26041651C2semidirect
4. #26041541C2semidirect
5. #26041540C2semidirect
6. #26041463C2semidirect
7. #26041446C2semidirect
8. #26041393C2semidirect
9. #453×C181C2semidirect
11. #19153861C22semidirect
12. #19153849C22semidirect
14. #19153846C22semidirect
23. C2×#15668437C22semidirect
24. C2×#15668437C22semidirect
26. #19153824C22semidirect
27. #19153823C22semidirect
28. #19153816C22semidirect
29. #19153751C22semidirect
30. #19153744C22semidirect
31. #19153734C22semidirect
32. #19153733C22semidirect
33. C4×#13089678C22semidirect
34. H(5,𝔽2)×C181C22semidirect
37. #104×C181C22semidirect
38. #100×C181C22semidirect
51. #15668439D4semidirect
59. #15668437D4semidirect
60. #15668436D4semidirect
62. #15668434D4semidirect
133. C181#3291semidirect
142. #453D181semidirect
146. H(5,𝔽2)C2×D181semidirect
168. #104C2×D181semidirect
171. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420902
#260420892
#260416512
#260415412
#260415402
#260414632
#260414462
#260413932
#453×C1812
#3291181