Finite Group Databasebeta

← #37545320

Subgroups — #37545320

Order 23168.

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

normal subgroups only

This group has 546 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#26041932C211yesmaximal, characteristic, complemented
3#26041914C211yesmaximal, characteristic, complemented
4#26041522C212yesmaximal, complemented
5#26041481C211yesmaximal, characteristic, complemented
6#26041471C211yesmaximal, characteristic, complemented
7#26041428C212yesmaximal, complemented
8#26041423C212yesmaximal, complemented
9C4×D1448C211yesmaximal, characteristic, complemented
10#26041358C211yesmaximal, characteristic, complemented
11#492×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2212yescomplemented
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15#19153843—28no—
16C2×#15668436C2212yes—
17C2×D1448C2211yescharacteristic
18C2×#15668434C2212yescomplemented
19C2×#15668432C2212yescomplemented
20#19153801—22no—
21#19153792C2212yescomplemented
22#19153791—21no—
23#19153787—21no—
24#19153787C2212yes—
25#19153784—22no—
26C4×D724C2211yescharacteristic
27C4×D724C2211yescharacteristic
28C4×D724C2211yescharacteristic, complemented
29C4⋊C4×D181—21no—
30#19153750C2211yescharacteristic, complemented
31C4⋊#13089678C2211yescharacteristic, complemented
32#19153740C2212yescomplemented
33#19153738C2211yescharacteristic
34#19153738C2211yescharacteristic, complemented
35#19153737C2212yescomplemented
36#19153728C2212yescomplemented
37C2×SD16×C181C2212yescomplemented
38C2×SD16×C181C2212yescomplemented
39C4⋊Q8×C181C2211yescharacteristic, complemented
40#127×C181C2211yescharacteristic
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43#15668450—24no—
44#15668450—42no—
45C2×D724—22no—
46C2×D724D411yescharacteristic
47C2×D724C2311yescharacteristic
48C2×D724C2311yescharacteristic
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51D4×D181—24no—
52D4×D181—42no—
53D4×D181—44no—
54C2×C4×D181—22no—
55C2×C4×D181—21no—
56C2×C4×D181—21no—
57C2×C4×D181C2312yes—
58#15668436—24no—
59D1448D414yescomplemented
60#15668434—24no—
61#15668432—24no—
62C2×#13089678C2312yes—
63C4⋊Dic181D411yescharacteristic
64C4⋊Dic181C2311yescharacteristic
65C4⋊Dic181C2311yescharacteristic
66#15668428—22no—
67C22⋊Dic181—22no—
68#15668426—22no—
69#15668426—21no—
70#15668426—21no—
71#15668426D412yes—
72C2×Q8×C181C2312yes—
73C2×D4×C181—21no—
74C2×D4×C181—21no—
75C2×D4×C181C2312yes—
76SD16×C181—24no—
77SD16×C181—42no—
78C2×C8×C181C2311yescharacteristic
79C2×C8×C181C2311yescharacteristic
80C4⋊C4×C181—21no—
81C4⋊C4×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—21no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88C22×D181C2×D412yes—
89D724—22no—
90D724—21no—
91D724—21no—
92D724—44no—
93D724C2×D412yes—
94D724C2×D412yescomplemented
95C22⋊D181—44no—
96C22⋊D181—44no—
97C2×Dic181—21no—
98C2×Dic181—21no—
99C2×Dic181C2×D412yes—
100C4×D181—22no—
101C4×D181—22no—
102C4×D181—44no—
103C4×D181—42no—
104C4×D181—42no—
105#13089678—22no—
106C23×C181—22no—
107Q8×C181—22no—
108Q8×C181—22no—
109D4×C181—22no—
110D4×C181—22no—
111D4×C181—42no—
112D4×C181—81no—
113C2×C4×C181—22no—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D411yescharacteristic
116C2×C4×C181C2411yescharacteristic
117C8×C181—21no—
118C8×C181C2×D412yescomplemented
119C22×C181—42no—
120C22×C181—42no—
121C22×C181—42no—
122C22×C181C22×D411yescharacteristic
123C2×D181—22no—
124C2×D181—22no—
125C2×D181—22no—
126C2×D181—22no—
127C2×D181—22no—
128C2×D181—41no—
129C2×D181—41no—
130C2×D181—41no—
131C2×D181—41no—
132C2×D181—88no—
133C4×C181—21no—
134C4×C181—42no—
135C4×C181AGL(1,ℤ/8)12yescomplemented
136C4×C181C22×D411yescharacteristic
137C4×C181H(5,𝔽2)11yescharacteristic
138Dic181—22no—
139Dic181—41no—
140Dic181—41no—
141C2×C181—82no—
142C2×C181D4211yescharacteristic
143C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C181#57711yescharacteristic
145D181—42no—
146D181—42no—
147D181—42no—
148C181#327211yescharacteristic, complemented, Sylow
149#3272—1811nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1812no—
151#547—1811no—
152D42—1811no—
153#492D18111yescharacteristic, complemented
154#468—1812no—
155#461—1811no—
156#459—1811no—
157#450—1812no—
158#449—1812no—
159C4×D8—1811no—
160#436—1811no—
161C2×#41—1812no—
162C22×D4—3621no—
163C22×D4—1812no—
164AGL(1,ℤ/8)—3628no—
165C2×SD16—1812no—
166C2×SD16C2×D18112yescomplemented
167C2×SD16C2×D18112yescomplemented
168C2×D8—1812no—
169C2×D8—1811no—
170C2×M16—1812no—
171C4⋊Q8C2×D18111yescharacteristic, complemented
172#127C2×D18111yescharacteristic
173#123—3622no—
174C22⋊Q8—1812no—
175#121—3621no—
176#121—3621no—
177#121—1812no—
178#120—3622no—
179C4×D4—1811no—
180C4×D4—1811no—
181C4×D4—1811no—
182C2×C4⋊C4—3621no—
183#107—1811no—
184C4⋊C8—1811no—
185#103—1812no—
186#102—1812no—
187#102—1811no—
188#102—1811no—
189C22⋊C8—1812no—
190C4×C8C2×D18111yescharacteristic, complemented
191C24—3622no—
192#41—3624no—
193#41—7242no—
194C2×Q8C22×D18112yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—3622no—
198C2×D4—3624no—
199C2×D4—3622no—
200C2×D4—3622no—
201C2×D4—1811no—
202C2×D4—1811no—
203C2×D4—1811no—
204C2×D4—7242no—
205C2×D4—7244no—
206C2×D4C22×D18112yes—
207C22×C4—3622no—
208C22×C4—3621no—
209C22×C4—3621no—
210C22×C4—1812no—
211SD16—24no—
212SD16—42no—
213SD16—3624no—
214D8—3624no—
215D8—1814no—
216M16—3624no—
217C2×C8—1812no—
218C2×C8C22×D18111yescharacteristic
219C2×C8C22×D18111yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3622no—
223C4⋊C4—1811no—
224C4⋊C4—1811no—
225C4⋊C4—1811no—
226C22⋊C4—3622no—
227C22⋊C4—3622no—
228C22⋊C4—3621no—
229C22⋊C4—3621no—
230C22⋊C4—1812no—
231C42C22×D18111yescharacteristic
232C23—22no—
233C23—3621no—
234C23—3621no—
235C23—14482no—
236C23—1812no—
237C23—7244no—
238C23—7244no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—3622no—
246D4—3621no—
247D4—3621no—
248D4—1812no—
249D4—1812no—
250D4—7244no—
251D4—7244no—
252D4—7244no—
253C2×C4—22no—
254C2×C4—3622no—
255C2×C4—3622no—
256C2×C4—3621no—
257C2×C4—3621no—
258C2×C4—1812no—
259C2×C4—7244no—
260C2×C4—7242no—
261C2×C4—7242no—
262C2×C4D4×D18111yescharacteristic
263C2×C4D4×D18111yescharacteristic
264C2×C4C23×D18111yescharacteristic
265C8—21no—
266C8—3622no—
267C8D4×D18112yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—3622no—
272C22—3622no—
273C22—3622no—
274C22—3622no—
275C22—3622no—
276C22—14488no—
277C22—7241no—
278C22—7241no—
279C22—7241no—
280C22—7241no—
281C22C2×D4×D18111yescharacteristic
282C4—21no—
283C4—42no—
284C4—3622no—
285C4—7241no—
286C4—7241no—
287C4#1915384312yescomplemented
288C4C2×D4×D18111yescharacteristic
289C4#1915386111yescharacteristic
290C2—82no—
291C2—7242no—
292C2—7242no—
293C2—7242no—
294C2#2604191411yescharacteristic
295C2C2×#1915384311yescharacteristic
296C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#492×C181C2
socleC22×C181C22×D4
hypercenter#492D181
hypocenterC181#3272
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#19153843C2semidirect
2. #26041932C2semidirect
3. #26041914C2semidirect
4. #26041522C2semidirect
5. #26041481C2semidirect
6. #26041471C2semidirect
7. #26041428C2semidirect
8. #26041423C2semidirect
9. C4×D1448C2semidirect
10. #26041358C2semidirect
11. #492×C181C2semidirect
12. C2×#15668450C22semidirect
18. C2×#15668434C22semidirect
19. C2×#15668432C22semidirect
21. #19153792C22semidirect
28. C4×D724C22semidirect
30. #19153750C22semidirect
31. C4⋊#13089678C22semidirect
32. #19153740C22semidirect
34. #19153738C22semidirect
35. #19153737C22semidirect
36. #19153728C22semidirect
37. C2×SD16×C181C22semidirect
38. C2×SD16×C181C22semidirect
39. C4⋊Q8×C181C22semidirect
41. C4×C8×C181C22semidirect
59. D1448D4semidirect
94. D724C2×D4semidirect
118. C8×C181C2×D4semidirect
135. C4×C181AGL(1,ℤ/8)semidirect
148. C181#3272semidirect
153. #492D181semidirect
166. C2×SD16C2×D181semidirect
167. C2×SD16C2×D181semidirect
171. C4⋊Q8C2×D181semidirect
190. C4×C8C2×D181semidirect
267. C8D4×D181semidirect
287. C4#19153843semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
#260419322
#260419142
#260415222
#260414812
#260414712
#260414282
#260414232
C4×D14482
#260413582
#492×C1812
#3272181