Finite Group Databasebeta

← #37545336

Subgroups — #37545336

Order 23168.

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

normal subgroups only

This group has 556 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
1C2×#19153843C212yesmaximal, complemented
2#26041951C211yesmaximal, characteristic, complemented
3#26041914C211yesmaximal, characteristic, complemented
4#26041655C211yesmaximal, characteristic, complemented
5#26041531C212yesmaximal, complemented
6#26041475C211yesmaximal, characteristic, complemented
7#26041423C212yesmaximal, complemented
8#26041414C212yesmaximal, complemented
9C4×#15668436C211yesmaximal, characteristic, complemented
10#26041356C211yesmaximal, characteristic, complemented
11#460×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2212yescomplemented
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15#19153843—28no—
16C2×#15668436C2211yescharacteristic
17C2×D1448C2212yes—
18C2×#15668434C2212yescomplemented
19C2×#15668434C2212yescomplemented
20C2×#15668432C2212yescomplemented
21#19153813—21no—
22#19153813C2211yescharacteristic, complemented
23#19153791—22no—
24#19153791C2212yescomplemented
25#19153787—21no—
26#19153787C2212yes—
27#19153784—22no—
28C4×D724—21no—
29C4×D724C2211yescharacteristic
30C4×D724C2211yescharacteristic
31#19153747C2211yescharacteristic, complemented
32#19153740C2211yescharacteristic, complemented
33#19153738C2212yescomplemented
34#19153736C2211yescharacteristic
35#19153728C2212yescomplemented
36C4×#13089678C2211yescharacteristic, complemented
37C2×SD16×C181C2212yescomplemented
38#127×C181C2211yescharacteristic
39C4×Q8×C181C2211yescharacteristic, complemented
40C4⋊C8×C181C2211yescharacteristic, complemented
41#102×C181C2212yescomplemented
42C23×D181—22no—
43#15668450—24no—
44#15668450—42no—
45C2×D724—22no—
46C2×D724—22no—
47C2×D724—21no—
48C2×D724—21no—
49C2×D724D411yescharacteristic
50C2×D724C2312yes—
51C2×D724C2311yescharacteristic
52C2×C22⋊D181—22no—
53C2×C22⋊D181—22no—
54D4×D181—24no—
55D4×D181—42no—
56D4×D181—44no—
57C2×C4×D181—22no—
58C2×C4×D181—21no—
59C2×C4×D181C2312yes—
60#15668436D414yescomplemented
61D1448—24no—
62#15668434—24no—
63#15668434—42no—
64#15668432—24no—
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C4⋊Dic181D411yescharacteristic
68C4⋊Dic181C2311yescharacteristic
69C22⋊Dic181—22no—
70#15668426—22no—
71#15668426—21no—
72#15668426D412yes—
73C2×Q8×C181C2311yescharacteristic
74C2×D4×C181—21no—
75C2×D4×C181—21no—
76C2×D4×C181C2312yes—
77SD16×C181—24no—
78C2×C8×C181C2312yes—
79C4⋊C4×C181—21no—
80C4⋊C4×C181C2311yescharacteristic
81C42×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88C22×D181C2×D412yes—
89D724—22no—
90D724—22no—
91D724—21no—
92D724—22no—
93D724—44no—
94D724—42no—
95D724—81no—
96D724—82no—
97D724C2×D412yes—
98C22⋊D181—44no—
99C22⋊D181—44no—
100C2×Dic181—21no—
101C2×Dic181C2×D412yes—
102C4×D181—22no—
103C4×D181—22no—
104C4×D181—44no—
105C4×D181—42no—
106#13089678—21no—
107#13089678C2×D412yescomplemented
108C23×C181—22no—
109Q8×C181—21no—
110Q8×C181C2×D412yescomplemented
111D4×C181—22no—
112D4×C181—22no—
113D4×C181—42no—
114D4×C181—81no—
115C2×C4×C181—21no—
116C2×C4×C181—21no—
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2×D411yescharacteristic
119C2×C4×C181C2411yescharacteristic
120C8×C181—22no—
121C22×C181—42no—
122C22×C181—42no—
123C22×C181—42no—
124C22×C181C22×D411yescharacteristic
125C2×D181—22no—
126C2×D181—22no—
127C2×D181—22no—
128C2×D181—22no—
129C2×D181—41no—
130C2×D181—41no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—88no—
135C4×C181—22no—
136C4×C181—21no—
137C4×C181—41no—
138C4×C181AGL(1,ℤ/8)12yescomplemented
139C4×C181C22×D411yescharacteristic
140C4×C181H(5,𝔽2)11yescharacteristic
141Dic181—22no—
142Dic181—41no—
143C2×C181—82no—
144C2×C181D4211yescharacteristic
145C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C181#57611yescharacteristic
147D181—42no—
148D181—42no—
149D181—82no—
150C181#327411yescharacteristic, complemented, Sylow
151#3274—1811nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1812no—
153#550—1811no—
154D42—1811no—
155#493—1811no—
156#469—1812no—
157#460D18111yescharacteristic, complemented
158#459—1811no—
159#449—1812no—
160#447—1812no—
161C4×SD16—1811no—
162#436—1811no—
163C2×#41—1812no—
164C22×D4—3621no—
165C22×D4—1812no—
166AGL(1,ℤ/8)—3628no—
167C2×SD16—1811no—
168C2×SD16C2×D18112yescomplemented
169C2×D8—1812no—
170C2×D8—1812no—
171C2×D8—1812no—
172C2×M16—1812no—
173#127—3621no—
174#127—1811no—
175#127C2×D18111yescharacteristic
176#121—3621no—
177#121—3622no—
178#121—1812no—
179#121—1812no—
180#120—3622no—
181C4×Q8C2×D18111yescharacteristic, complemented
182C4×D4—3621no—
183C4×D4—1811no—
184C4×D4—1811no—
185#106—1811no—
186C4⋊C8C2×D18111yescharacteristic, complemented
187#103—1811no—
188#102—1812no—
189#102—1811no—
190#102C2×D18112yescomplemented
191C22⋊C8—1812no—
192C4×C8—1811no—
193C24—3622no—
194#41—3624no—
195#41—7242no—
196C2×Q8C22×D18111yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—3622no—
200C2×D4—3622no—
201C2×D4—3624no—
202C2×D4—3622no—
203C2×D4—3622no—
204C2×D4—3621no—
205C2×D4—3621no—
206C2×D4—1811no—
207C2×D4—1812no—
208C2×D4—1811no—
209C2×D4—7242no—
210C2×D4—7244no—
211C2×D4C22×D18112yes—
212C22×C4—3622no—
213C22×C4—3621no—
214C22×C4—1812no—
215SD16—24no—
216SD16—1814no—
217D8—3624no—
218D8—3624no—
219D8—7242no—
220M16—3624no—
221C2×C8—1811no—
222C2×C8—1811no—
223C2×C8C22×D18112yes—
224C4⋊C4—21no—
225C4⋊C4—1811no—
226C4⋊C4—1811no—
227C4⋊C4C22×D18111yescharacteristic
228C22⋊C4—3622no—
229C22⋊C4—3622no—
230C22⋊C4—3621no—
231C22⋊C4—1812no—
232C42—21no—
233C42C22×D18111yescharacteristic
234C23—22no—
235C23—3621no—
236C23—3622no—
237C23—14482no—
238C23—1812no—
239C23—7244no—
240C23—7244no—
241Q8—21no—
242Q8D4×D18112yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—3622no—
248D4—3622no—
249D4—3621no—
250D4—3622no—
251D4—14481no—
252D4—14482no—
253D4—1812no—
254D4—7244no—
255D4—7244no—
256D4—7244no—
257D4—7242no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—3622no—
261C2×C4—3622no—
262C2×C4—3621no—
263C2×C4—1812no—
264C2×C4—7244no—
265C2×C4—7242no—
266C2×C4D4×D18111yescharacteristic
267C2×C4D4×D18111yescharacteristic
268C2×C4C23×D18111yescharacteristic
269C8—22no—
270C8—3621no—
271C8—1812no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3622no—
276C22—3622no—
277C22—3622no—
278C22—3622no—
279C22—14488no—
280C22—7242no—
281C22—7242no—
282C22—7242no—
283C22—7241no—
284C22—7241no—
285C22C2×D4×D18111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—3622no—
290C4—7241no—
291C4#1915384312yescomplemented
292C4C2×D4×D18111yescharacteristic
293C4#1915386211yescharacteristic
294C2—82no—
295C2—14482no—
296C2—7242no—
297C2—7242no—
298C2#2604191311yescharacteristic
299C2C2×#1915384311yescharacteristic
300C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#460×C181C2
socleC22×C181C22×D4
hypercenter#460D181
hypocenterC181#3274
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19153843C2semidirect
2. #26041951C2semidirect
3. #26041914C2semidirect
4. #26041655C2semidirect
5. #26041531C2semidirect
6. #26041475C2semidirect
7. #26041423C2semidirect
8. #26041414C2semidirect
9. C4×#15668436C2semidirect
10. #26041356C2semidirect
11. #460×C181C2semidirect
12. C2×#15668450C22semidirect
18. C2×#15668434C22semidirect
19. C2×#15668434C22semidirect
20. C2×#15668432C22semidirect
22. #19153813C22semidirect
24. #19153791C22semidirect
31. #19153747C22semidirect
32. #19153740C22semidirect
33. #19153738C22semidirect
35. #19153728C22semidirect
36. C4×#13089678C22semidirect
37. C2×SD16×C181C22semidirect
39. C4×Q8×C181C22semidirect
40. C4⋊C8×C181C22semidirect
41. #102×C181C22semidirect
60. #15668436D4semidirect
107. #13089678C2×D4semidirect
110. Q8×C181C2×D4semidirect
138. C4×C181AGL(1,ℤ/8)semidirect
150. C181#3274semidirect
157. #460D181semidirect
168. C2×SD16C2×D181semidirect
181. C4×Q8C2×D181semidirect
186. C4⋊C8C2×D181semidirect
190. #102C2×D181semidirect
242. Q8D4×D181semidirect
291. C4#19153843semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
#260419512
#260419142
#260416552
#260415312
#260414752
#260414232
#260414142
C4×#156684362
#260413562
#460×C1812
#3274181