Finite Group Databasebeta

← #37543860

Subgroups — #37543860

Order 23168.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#26041815C211yesmaximal, characteristic, complemented
2C2×D4×Dic181C211yesmaximal, characteristic, complemented
3#26041607C211yesmaximal, characteristic, complemented
4#26041593C211yesmaximal, characteristic, complemented
5#26041577C212yesmaximal, complemented
6#26041496C212yesmaximal, complemented
7#26041413C211yesmaximal, characteristic
8#26041388C212yesmaximal, complemented
9#26041380C212yesmaximal, complemented
10#26041333C211yesmaximal, characteristic, complemented
11#450×C181C211yesmaximal, characteristic, complemented
12C23×Dic181—21no—
13C2×#15668434C2212yes—
14#19153811—21no—
15#19153807C2211yescharacteristic, complemented
16#19153805C2211yescharacteristic, complemented
17#19153800—21no—
18#19153790C2211yescharacteristic
19Q8×Dic181C2211yescharacteristic, complemented
20D4×Dic181—21no—
21D4×Dic181—21no—
22D4×Dic181C2212yes—
23D4×Dic181C2212yes—
24#19153778C2211yescharacteristic, complemented
25#19153776—21no—
26#19153774—21no—
27C2×C4⋊Dic181C2211yescharacteristic
28C2×C22⋊Dic181—21no—
29C2×C4×Dic181C2211yescharacteristic
30#19153750C2212yescomplemented
31#19153747C2212yescomplemented
32#19153745C2212yescomplemented
33#19153740C2212yescomplemented
34#19153738C2212yescomplemented
35#19153737C2212yes—
36C22⋊#13089678C2211yescharacteristic
37#19153726C2212yescomplemented
38C22×D4×C181C2211yescharacteristic
39C2×SD16×C181C2212yes—
40C22⋊Q8×C181C2211yescharacteristic
41#102×C181C2212yes—
42C22⋊C8×C181C2211yescharacteristic
43C2×D724C2311yescharacteristic
44C2×C22⋊D181—21no—
45C22×Dic181—22no—
46C22×Dic181—22no—
47C22×Dic181—21no—
48C22×Dic181—44no—
49C22×Dic181D411yescharacteristic
50C22×Dic181D411yescharacteristic
51C2×C4×D181—21no—
52#15668434—24no—
53C2×#13089678C2312yes—
54C4⋊Dic181—21no—
55C4⋊Dic181—21no—
56C4⋊Dic181—21no—
57C4⋊Dic181C2311yescharacteristic
58C4⋊Dic181C2312yes—
59#15668428—21no—
60C22⋊Dic181—22no—
61C22⋊Dic181—22no—
62#15668426—21no—
63#15668426—21no—
64#15668426—21no—
65#15668426—21no—
66C4×Dic181—21no—
67C4×Dic181D411yescharacteristic
68C4×Dic181D411yescharacteristic
69C4×Dic181C2311yescharacteristic
70C4×Dic181C2311yescharacteristic
71C24×C181—21no—
72C2×Q8×C181C2311yescharacteristic
73C2×D4×C181—24no—
74C2×D4×C181—21no—
75C2×D4×C181C2312yes—
76C22×C4×C181C2311yescharacteristic
77SD16×C181—24no—
78C2×C8×C181C2312yes—
79C4⋊C4×C181—21no—
80C4⋊C4×C181C2311yescharacteristic
81C22⋊C4×C181—21no—
82C22×D181—21no—
83D724—22no—
84C22⋊D181—42no—
85C2×Dic181—22no—
86C2×Dic181—21no—
87C2×Dic181—22no—
88C2×Dic181—41no—
89C2×Dic181—44no—
90C2×Dic181—44no—
91C2×Dic181—42no—
92C2×Dic181C2×D411yescharacteristic
93C2×Dic181C2×D411yescharacteristic
94C2×Dic181C2×D411yescharacteristic
95C2×Dic181C2×D411yescharacteristic
96C4×D181—42no—
97#13089678—22no—
98C23×C181—24no—
99C23×C181—22no—
100C23×C181—42no—
101C23×C181—42no—
102C23×C181C2×D411yescharacteristic
103Q8×C181—22no—
104D4×C181—24no—
105D4×C181—22no—
106D4×C181#4114yes—
107C2×C4×C181—22no—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C8×C181—22no—
113C22×C181—21no—
114C22×C181—21no—
115C22×C181—24no—
116C22×C181—48no—
117C22×C181—42no—
118C22×C181—42no—
119C22×C181AGL(1,ℤ/8)12yes—
120C22×C181C22×D411yescharacteristic
121C2×D181—41no—
122C2×D181—41no—
123C2×D181—41no—
124C4×C181—21no—
125C4×C181—41no—
126C4×C181—41no—
127C4×C181C2×#4112yes—
128Dic181—21no—
129Dic181—21no—
130Dic181—21no—
131Dic181—21no—
132Dic181—42no—
133Dic181—41no—
134C2×C181—22no—
135C2×C181—44no—
136C2×C181#52511yescharacteristic
137C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C181#57511yescharacteristic
139D181—81no—
140C181#311411yescharacteristic, complemented, Sylow
141#3114—1811nomaximal, Sylow
142#529—1811no—
143C2×C4×D4—1811no—
144#483—1811no—
145#480—1811no—
146#478—1812no—
147#463—1812no—
148#450D18111yescharacteristic, complemented
149#447—1811no—
150#442—1812no—
151#440—1812no—
152#432—1811no—
153C22×D4C2×D18111yescharacteristic
154C23×C4—3621no—
155C2×SD16C2×D18112yes—
156C2×D8—1812no—
157#126—3621no—
158#125—1811no—
159#124—1811no—
160#123—3621no—
161C22⋊Q8C2×D18111yescharacteristic
162#121—1811no—
163C4×Q8—1811no—
164C4×D4—3621no—
165C4×D4—3621no—
166C4×D4—3621no—
167C4×D4—1812no—
168C4×D4—1812no—
169C4×D4—1811no—
170#117—3621no—
171C2×C4⋊C4—1811no—
172C2×C22⋊C4—3621no—
173C2×C42—1811no—
174#107—1812no—
175#106—1812no—
176C4⋊C8—1812no—
177#103—1812no—
178#102—1812no—
179#102—1812no—
180#102C2×D18112yes—
181C22⋊C8—1811no—
182C22⋊C8C2×D18111yescharacteristic
183#97—1812no—
184C24—21no—
185C2×Q8C22×D18111yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3621no—
189C2×D4—1811no—
190C2×D4C22×D18112yes—
191C22×C4—3621no—
192C22×C4—3621no—
193C22×C4—3622no—
194C22×C4—3622no—
195C22×C4—1811no—
196C22×C4—1811no—
197C22×C4—7244no—
198C22×C4C22×D18111yescharacteristic
199SD16—24no—
200D8—3624no—
201C2×C8—1812no—
202C2×C8C22×D18112yes—
203C4⋊C4—21no—
204C4⋊C4—3621no—
205C4⋊C4—3621no—
206C4⋊C4—3621no—
207C4⋊C4—3621no—
208C4⋊C4—1811no—
209C4⋊C4—1812no—
210C4⋊C4C22×D18111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—3621no—
213C22⋊C4—3621no—
214C22⋊C4—3621no—
215C22⋊C4—3622no—
216C22⋊C4—3622no—
217C22⋊C4—3621no—
218C42—3621no—
219C42—1811no—
220C42—1811no—
221C42—1811no—
222C42—1811no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3621no—
228C23D4×D18111yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—3622no—
233D4—7242no—
234D4#1566844814yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—3622no—
239C2×C4—3621no—
240C2×C4—3622no—
241C2×C4—1811no—
242C2×C4—1811no—
243C2×C4—1811no—
244C2×C4—1811no—
245C2×C4—7242no—
246C2×C4—7241no—
247C2×C4—7244no—
248C2×C4—7244no—
249C2×C4—7242no—
250C2×C4D4×D18111yescharacteristic
251C2×C4C23×D18111yescharacteristic
252C8—22no—
253C8—3622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—7241no—
261C22—7241no—
262C22—7241no—
263C22#1915384312yes—
264C22C2×D4×D18111yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—3621no—
269C4—3621no—
270C4—3621no—
271C4—3621no—
272C4—7242no—
273C4—7241no—
274C4C2×#1566844812yes—
275C2—22no—
276C2—44no—
277C2—14481no—
278C2#2604178511yescharacteristic
279C2C2×#1915384311yescharacteristic
280C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
prime radical#26041413C2
hypercenter#450D181
hypocenterC181#3114
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26041815C2semidirect
2. C2×D4×Dic181C2semidirect
3. #26041607C2semidirect
4. #26041593C2semidirect
5. #26041577C2semidirect
6. #26041496C2semidirect
8. #26041388C2semidirect
9. #26041380C2semidirect
10. #26041333C2semidirect
11. #450×C181C2semidirect
15. #19153807C22semidirect
16. #19153805C22semidirect
19. Q8×Dic181C22semidirect
24. #19153778C22semidirect
30. #19153750C22semidirect
31. #19153747C22semidirect
32. #19153745C22semidirect
33. #19153740C22semidirect
34. #19153738C22semidirect
37. #19153726C22semidirect
140. C181#3114semidirect
148. #450D181semidirect

Maximal subgroups

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

subgroupindex
#260418152
C2×D4×Dic1812
#260416072
#260415932
#260415772
#260414962
#260414132
#260413882
#260413802
#260413332
#450×C1812
#3114181