Finite Group Databasebeta

← #37545317

Subgroups — #37545317

Order 23168.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19153842C212yesmaximal, complemented
2#26041934C211yesmaximal, characteristic, complemented
3#26041651C211yesmaximal, characteristic, complemented
4#26041523C212yesmaximal, complemented
5#26041484C211yesmaximal, characteristic, complemented
6#26041471C211yesmaximal, characteristic, complemented
7#26041430C212yesmaximal, complemented
8#26041422C212yesmaximal, complemented
9C4×#15668434C211yesmaximal, characteristic, complemented
10#26041357C211yesmaximal, characteristic, complemented
11D42×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2212yescomplemented
13#19153842—28no—
14C2×#15668437C2212yes—
15C2×#15668437C2212yescomplemented
16C2×#15668437C2212yescomplemented
17C2×#15668434C2211yescharacteristic
18C2×#15668434C2212yescomplemented
19C2×#15668433C2212yescomplemented
20#19153816C2211yescharacteristic, complemented
21#19153798—22no—
22#19153795C2212yescomplemented
23#19153787—21no—
24C4×D724C2211yescharacteristic, complemented
25C4⋊C4×D181—21no—
26#19153749C2211yescharacteristic, complemented
27C4⋊#13089678C2211yescharacteristic, complemented
28#19153739C2212yescomplemented
29#19153737C2211yescharacteristic
30#19153737C2212yescomplemented
31#19153736C2211yescharacteristic, complemented
32C22⋊#13089678C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34C22×D4×C181—21no—
35C22×D4×C181C2212yes—
36#127×C181C2211yescharacteristic
37#121×C181—21no—
38#121×C181C2212yes—
39#120×C181—22no—
40C4×D4×C181C2211yescharacteristic
41C4×D4×C181C2211yescharacteristic
42#15668449—24no—
43#15668449—42no—
44C2×Dic362C2312yes—
45C2×D724C2311yescharacteristic
46C2×C22⋊D181—22no—
47C2×C4×D181—21no—
48C2×C4×D181—22no—
49#15668437—24no—
50#15668437—24no—
51#15668437—42no—
52#15668434—24no—
53#15668434D414yescomplemented
54#15668433—24no—
55C2×#13089678C2311yescharacteristic
56C2×#13089678C2312yes—
57C2×#13089678C2311yescharacteristic
58C4⋊Dic181—21no—
59C4⋊Dic181—21no—
60C4⋊Dic181C2311yescharacteristic
61#15668428—22no—
62C22⋊Dic181—22no—
63#15668426—21no—
64C24×C181—22no—
65C2×D4×C181—24no—
66C2×D4×C181—22no—
67C2×D4×C181—22no—
68C2×D4×C181—21no—
69C2×D4×C181—21no—
70C2×D4×C181—42no—
71C2×D4×C181—44no—
72C2×D4×C181D411yescharacteristic
73C2×D4×C181C2312yes—
74C2×D4×C181C2311yescharacteristic
75C22×C4×C181—21no—
76C22×C4×C181C2312yes—
77C4⋊C4×C181D411yescharacteristic
78C4⋊C4×C181C2311yescharacteristic
79C22⋊C4×C181—21no—
80C22⋊C4×C181—22no—
81C22⋊C4×C181D412yes—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84Dic362—22no—
85Dic362—22no—
86D724—21no—
87D724C2×D412yescomplemented
88C22⋊D181—44no—
89C2×Dic181—21no—
90C2×Dic181—22no—
91C4×D181—44no—
92C4×D181—42no—
93#13089678—22no—
94#13089678—21no—
95#13089678C2×D412yescomplemented
96C23×C181—21no—
97C23×C181—22no—
98C23×C181—44no—
99C23×C181—44no—
100C23×C181—82no—
101C23×C181C2×D412yes—
102D4×C181—22no—
103D4×C181—22no—
104D4×C181—21no—
105D4×C181—22no—
106D4×C181—44no—
107D4×C181—44no—
108D4×C181—42no—
109D4×C181—81no—
110D4×C181C2×D412yes—
111C2×C4×C181—22no—
112C2×C4×C181—22no—
113C2×C4×C181—21no—
114C2×C4×C181—42no—
115C2×C4×C181C2×D411yescharacteristic
116C2×C4×C181C2×D412yes—
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2411yescharacteristic
119C22×C181—22no—
120C22×C181—22no—
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—41no—
124C22×C181—41no—
125C22×C181—42no—
126C22×C181—42no—
127C22×C181—42no—
128C22×C181—88no—
129C22×C181C22×D411yescharacteristic
130C2×D181—22no—
131C2×D181—41no—
132C2×D181—41no—
133C4×C181—22no—
134C4×C181—21no—
135C4×C181—41no—
136C4×C181AGL(1,ℤ/8)12yescomplemented
137C4×C181C22×D411yescharacteristic
138C4×C181H(5,𝔽2)11yescharacteristic
139Dic181—42no—
140Dic181—41no—
141C2×C181—42no—
142C2×C181—42no—
143C2×C181—82no—
144C2×C181D4211yescharacteristic
145C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C181#57711yescharacteristic
147D181—42no—
148C181#327211yescharacteristic, complemented, Sylow
149#3272—1811nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1812no—
151#547—1811no—
152D42D18111yescharacteristic, complemented
153#492—1811no—
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—21no—
163C22×D4C2×D18112yes—
164AGL(1,ℤ/8)—3628no—
165C2×SD16—1812no—
166C2×SD16—1812no—
167C2×SD16—1812no—
168C2×D8—1812no—
169C2×D8—1811no—
170C2×M16—1812no—
171C4⋊Q8—1811no—
172#127C2×D18111yescharacteristic
173#123—3622no—
174C22⋊Q8—1812no—
175#121—21no—
176#121—3621no—
177#121C2×D18112yes—
178#120—22no—
179C4×D4—1811no—
180C4×D4C2×D18111yescharacteristic
181C4×D4C2×D18111yescharacteristic
182C2×C4⋊C4—3621no—
183#107—1811no—
184C4⋊C8—1811no—
185#103—1812no—
186#102—1811no—
187#102—1812no—
188#102—1811no—
189C22⋊C8—1812no—
190C4×C8—1811no—
191C24—22no—
192#41—3624no—
193#41—7242no—
194C2×Q8—1812no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—3622no—
203C2×D4—1811no—
204C2×D4C22⋊D18111yescharacteristic
205C2×D4C22×D18112yes—
206C2×D4C22×D18111yescharacteristic
207C22×C4—21no—
208C22×C4—3621no—
209C22×C4—3622no—
210C22×C4C22×D18112yes—
211SD16—3624no—
212SD16—3624no—
213SD16—7242no—
214D8—3624no—
215D8—1814no—
216M16—3624no—
217C2×C8—1811no—
218C2×C8—1812no—
219C2×C8—1811no—
220C4⋊C4—3621no—
221C4⋊C4—3622no—
222C4⋊C4—3621no—
223C4⋊C4—1811no—
224C4⋊C4C22⋊D18111yescharacteristic
225C4⋊C4C22×D18111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—3622no—
229C22⋊C4—3621no—
230C22⋊C4C22⋊D18112yes—
231C42C22×D18111yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—3621no—
238C23C2×C22⋊D18112yes—
239Q8—3622no—
240Q8—3622no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—3621no—
250D4—1812no—
251D4—7244no—
252D4D4×D18112yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3621no—
258C2×C4—3622no—
259C2×C4—7244no—
260C2×C4—7242no—
261C2×C4C2×C22⋊D18111yescharacteristic
262C2×C4C2×C22⋊D18112yes—
263C2×C4C2×C22⋊D18111yescharacteristic
264C2×C4C23×D18111yescharacteristic
265C8—3622no—
266C8—3621no—
267C8—1812no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3622no—
279C22—7241no—
280C22—7241no—
281C22C22×C22⋊D18111yescharacteristic
282C4—22no—
283C4—21no—
284C4—41no—
285C4—7242no—
286C4—7241no—
287C4#1915384212yescomplemented
288C4C2×D4×D18111yescharacteristic
289C4#1915386111yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—7242no—
294C2D4×C22⋊D18111yescharacteristic
295C2C2×#1915384211yescharacteristic
296C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD42×C181C2
socleC22×C181C22×D4
hypercenterD42D181
hypocenterC181#3272
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19153842C2semidirect
2. #26041934C2semidirect
3. #26041651C2semidirect
4. #26041523C2semidirect
5. #26041484C2semidirect
6. #26041471C2semidirect
7. #26041430C2semidirect
8. #26041422C2semidirect
9. C4×#15668434C2semidirect
10. #26041357C2semidirect
11. D42×C181C2semidirect
12. C2×#15668449C22semidirect
15. C2×#15668437C22semidirect
16. C2×#15668437C22semidirect
18. C2×#15668434C22semidirect
19. C2×#15668433C22semidirect
20. #19153816C22semidirect
22. #19153795C22semidirect
24. C4×D724C22semidirect
26. #19153749C22semidirect
27. C4⋊#13089678C22semidirect
28. #19153739C22semidirect
30. #19153737C22semidirect
31. #19153736C22semidirect
32. C22⋊#13089678C22semidirect
33. C4×#13089678C22semidirect
53. #15668434D4semidirect
87. D724C2×D4semidirect
95. #13089678C2×D4semidirect
136. C4×C181AGL(1,ℤ/8)semidirect
148. C181#3272semidirect
152. D42D181semidirect
287. C4#19153842semidirect

Maximal subgroups

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

subgroupindex
C2×#191538422
#260419342
#260416512
#260415232
#260414842
#260414712
#260414302
#260414222
C4×#156684342
#260413572
D42×C1812
#3272181