Finite Group Databasebeta

← #37545407

Subgroups — #37545407

Order 23168.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#19153835C211yesmaximal, characteristic, complemented
2C2×D8×D181C211yesmaximal, characteristic, complemented
3#26041924C212yesmaximal, complemented
4#26041663C211yesmaximal, characteristic, complemented
5#26041521C211yesmaximal, characteristic, complemented
6#26041516C211yesmaximal, characteristic, complemented
7#26041494C212yesmaximal, complemented
8#26041431C212yesmaximal, complemented
9#26041425C212yesmaximal, complemented
10C8×D724C211yesmaximal, characteristic, complemented
11C4×D8×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2212yescomplemented
13C2×D4×D181C2212yes—
14#19153835—24no—
15C2×Dic724C2211yescharacteristic, complemented
16C2×#15668438C2212yescomplemented
17C2×#15668437C2212yescomplemented
18C2×D1448C2211yescharacteristic
19C2×#15668434C2212yes—
20D8×D181—24no—
21C2×C8×D181C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic, complemented
23#19153804C2212yescomplemented
24#19153800—22no—
25#19153796C2212yes—
26#19153791C2212yescomplemented
27#19153790—22no—
28#19153786—22no—
29C4×D724C2211yescharacteristic
30C2×#15668426—22no—
31C4⋊#13089678C2211yescharacteristic, complemented
32#19153739C2212yescomplemented
33#19153736C2212yescomplemented
34#19153728C2211yescharacteristic
35#19153728C2211yescharacteristic, complemented
36C2×D8×C181C2211yescharacteristic
37C4×D4×C181C2212yescomplemented
38#107×C181C2211yescharacteristic, complemented
39#102×C181C2212yescomplemented
40C4×C8×C181C2211yescharacteristic, complemented
41C23×D181—22no—
42#15668448—24no—
43#15668448—42no—
44C2×Dic362C2312yes—
45C2×D724—22no—
46C2×D724D411yescharacteristic
47C2×D724C2312yes—
48C2×C22⋊D181—22no—
49C2×C22⋊D181—22no—
50D4×D181—24no—
51D4×D181—42no—
52C22×Dic181—22no—
53C2×C4×D181C2311yescharacteristic
54C2×C4×D181C2311yescharacteristic
55Dic724—22no—
56#15668438—42no—
57#15668437—24no—
58D1448—22no—
59#15668434—24no—
60C2×#13089678C2311yescharacteristic
61C2×#13089678C2311yescharacteristic
62C8×D181—22no—
63C8×D181—22no—
64C4⋊Dic181—22no—
65C4⋊Dic181D411yescharacteristic
66#15668426—22no—
67#15668426—22no—
68#15668426—22no—
69#15668426—22no—
70#15668426D411yescharacteristic
71#15668426D411yescharacteristic
72C2×D4×C181C2312yes—
73C22×C4×C181—22no—
74D8×C181D414yescomplemented
75C2×C8×C181C2311yescharacteristic
76C2×C8×C181C2311yescharacteristic
77C4⋊C4×C181C2312yes—
78C22⋊C4×C181—22no—
79C42×C181C2311yescharacteristic
80C22×D181—22no—
81C22×D181—44no—
82C22×D181—44no—
83C22×D181—82no—
84C22×D181C2×D411yescharacteristic
85C22×D181C2×D411yescharacteristic
86Dic362—22no—
87Dic362—22no—
88D724—22no—
89D724—22no—
90D724—22no—
91D724—82no—
92C22⋊D181—44no—
93C22⋊D181—44no—
94C2×Dic181—22no—
95C2×Dic181—44no—
96C2×Dic181C2×D411yescharacteristic
97C2×Dic181C2×D411yescharacteristic
98C4×D181—21no—
99C4×D181—21no—
100C4×D181—21no—
101C4×D181—21no—
102#13089678—21no—
103#13089678—21no—
104C23×C181—22no—
105D4×C181—22no—
106D4×C181C2×D414yes—
107C2×C4×C181—22no—
108C2×C4×C181—82no—
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C8×C181—21no—
113C8×C181C2×D412yes—
114C22×C181—24no—
115C22×C181—42no—
116C22×C181—42no—
117C22×C181C22×D411yescharacteristic
118C2×D181—21no—
119C2×D181—21no—
120C2×D181—21no—
121C2×D181—21no—
122C2×D181—21no—
123C2×D181—42no—
124C2×D181—42no—
125C2×D181—42no—
126C2×D181—88no—
127C2×D181#13512yescomplemented
128C4×C181—21no—
129C4×C181—21no—
130C4×C181—42no—
131C4×C181C22×D411yescharacteristic
132C4×C181H(5,𝔽2)11yescharacteristic
133Dic181—21no—
134Dic181—21no—
135Dic181—42no—
136C2×C181—44no—
137C2×C181D4211yescharacteristic
138C2×C181C2×#13511yescharacteristic
139C2×C181#57611yescharacteristic
140D181—22no—
141D181—41no—
142D181—82no—
143C181#328011yescharacteristic, complemented, Sylow
144#3280—1811nomaximal, Sylow
145C2×#135—1811no—
146C22×D8—1811no—
147#546—1812no—
148#495—1811no—
149#467—1811no—
150#466—1811no—
151#463—1812no—
152#450—1812no—
153#449—1812no—
154C4×D8D18111yescharacteristic, complemented
155C8×D4—1811no—
156C2×#41—1812no—
157C22×D4—1812no—
158#135—3624no—
159C2×Q16—1811no—
160C2×SD16—1812no—
161C2×SD16—1812no—
162C2×D8—3624no—
163C2×D8—1812no—
164C2×D8—1811no—
165C2×D8C2×D18111yescharacteristic
166C22×C8—1811no—
167C22×C8—1811no—
168#124—1812no—
169#123—3622no—
170C22⋊Q8—1812no—
171#121—3622no—
172#121—1812no—
173#120—3622no—
174C4×D4—1811no—
175C4×D4C2×D18112yescomplemented
176C2×C22⋊C4—3622no—
177#107C2×D18111yescharacteristic, complemented
178C4⋊C8—1811no—
179#103—1812no—
180#102—1812no—
181#102C2×D18112yescomplemented
182C22⋊C8—1811no—
183C22⋊C8—1811no—
184C4×C8C2×D18111yescharacteristic, complemented
185C24—3622no—
186#41—3624no—
187#41—7242no—
188C2×Q8—1812no—
189C2×D4—3622no—
190C2×D4—3624no—
191C2×D4—3622no—
192C2×D4—3622no—
193C2×D4—1811no—
194C2×D4—1812no—
195C2×D4—7242no—
196C2×D4C22×D18112yes—
197C22×C4—22no—
198C22×C4—3622no—
199C22×C4—1811no—
200C22×C4—1811no—
201Q16—3622no—
202SD16—3624no—
203SD16—7242no—
204D8—3624no—
205D8—3622no—
206D8D72414yescomplemented
207C2×C8—3622no—
208C2×C8—3622no—
209C2×C8—1811no—
210C2×C8—1811no—
211C2×C8C22×D18111yescharacteristic
212C2×C8C22×D18111yescharacteristic
213C4⋊C4—3622no—
214C4⋊C4—1811no—
215C4⋊C4C22×D18112yes—
216C22⋊C4—22no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4—3622no—
220C22⋊C4—3622no—
221C22⋊C4—1811no—
222C22⋊C4—1811no—
223C42C22×D18111yescharacteristic
224C23—22no—
225C23—3622no—
226C23—14482no—
227C23—1811no—
228C23—1811no—
229C23—7244no—
230C23—7244no—
231Q8—3622no—
232Q8—3622no—
233D4—22no—
234D4—3622no—
235D4—3622no—
236D4—3622no—
237D4—14482no—
238D4—7244no—
239D4—7244no—
240D4C2×D72414yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—3621no—
244C2×C4—3621no—
245C2×C4—3621no—
246C2×C4—3621no—
247C2×C4—3622no—
248C2×C4—1811no—
249C2×C4—1811no—
250C2×C4—7244no—
251C2×C4D4×D18111yescharacteristic
252C2×C4D4×D18111yescharacteristic
253C2×C4C23×D18111yescharacteristic
254C8—21no—
255C8—3621no—
256C8—3621no—
257C8C2×D72412yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—3621no—
262C22—3621no—
263C22—3621no—
264C22—3621no—
265C22—3621no—
266C22—14488no—
267C22—1812no—
268C22—7242no—
269C22—7242no—
270C22—7242no—
271C22C2×D4×D18111yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—3621no—
276C4—3621no—
277C4—7242no—
278C4C22×D72411yescharacteristic
279C4#1915386211yescharacteristic
280C2—44no—
281C2—3622no—
282C2—14482no—
283C2—7241no—
284C2D4×D72411yescharacteristic
285C2#135×D18111yescharacteristic
286C2#2604208011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×D8×C181C2
socleC22×C181C22×D4
hypercenterC4×D8D181
hypocenterC181#3280
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19153835C2semidirect
2. C2×D8×D181C2semidirect
3. #26041924C2semidirect
4. #26041663C2semidirect
5. #26041521C2semidirect
6. #26041516C2semidirect
7. #26041494C2semidirect
8. #26041431C2semidirect
9. #26041425C2semidirect
10. C8×D724C2semidirect
11. C4×D8×C181C2semidirect
12. C2×#15668448C22semidirect
15. C2×Dic724C22semidirect
16. C2×#15668438C22semidirect
17. C2×#15668437C22semidirect
22. C2×C8×D181C22semidirect
23. #19153804C22semidirect
26. #19153791C22semidirect
31. C4⋊#13089678C22semidirect
32. #19153739C22semidirect
33. #19153736C22semidirect
35. #19153728C22semidirect
37. C4×D4×C181C22semidirect
38. #107×C181C22semidirect
39. #102×C181C22semidirect
40. C4×C8×C181C22semidirect
74. D8×C181D4semidirect
127. C2×D181#135semidirect
143. C181#3280semidirect
154. C4×D8D181semidirect
175. C4×D4C2×D181semidirect
177. #107C2×D181semidirect
181. #102C2×D181semidirect
184. C4×C8C2×D181semidirect
206. D8D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538352
C2×D8×D1812
#260419242
#260416632
#260415212
#260415162
#260414942
#260414312
#260414252
C8×D7242
C4×D8×C1812
#3280181