Finite Group Databasebeta

← #37545337

Subgroups — #37545337

Order 23168.

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

normal subgroups only

This group has 556 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#26041950C211yesmaximal, characteristic, complemented
3#26041655C211yesmaximal, characteristic, complemented
4#26041530C212yesmaximal, complemented
5#26041476C211yesmaximal, characteristic, complemented
6#26041474C211yesmaximal, characteristic, complemented
7#26041422C212yesmaximal, complemented
8#26041413C212yesmaximal, complemented
9C4×#15668437C211yesmaximal, characteristic, complemented
10#26041357C211yesmaximal, characteristic, complemented
11D42×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2212yescomplemented
13#19153842—28no—
14C2×#15668437C2211yescharacteristic
15C2×#15668437C2212yescomplemented
16C2×#15668434C2212yes—
17C2×#15668434C2212yescomplemented
18C2×#15668434C2212yescomplemented
19C2×#15668433C2212yescomplemented
20#19153813C2211yescharacteristic, complemented
21#19153812—21no—
22#19153790C2212yescomplemented
23#19153789—22no—
24C4×Dic362C2211yescharacteristic, complemented
25#19153778—21no—
26#19153747C2211yescharacteristic, complemented
27C4⋊#13089678C2211yescharacteristic, complemented
28#19153739C2211yescharacteristic, complemented
29#19153737C2211yescharacteristic
30#19153737C2212yescomplemented
31#19153736C2212yescomplemented
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×Dic362C2311yescharacteristic
45C2×D724—21no—
46C2×D724—21no—
47C2×D724C2312yes—
48C2×C22⋊D181—22no—
49C2×C22⋊D181—22no—
50C2×C4×D181—22no—
51#15668437—24no—
52#15668437D414yescomplemented
53#15668434—24no—
54#15668434—24no—
55#15668434—42no—
56#15668433—24no—
57C2×#13089678C2311yescharacteristic
58C2×#13089678C2312yes—
59C2×#13089678C2311yescharacteristic
60C4⋊Dic181C2311yescharacteristic
61#15668428—21no—
62#15668426—22no—
63C4×Dic181—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—22no—
84Dic362—21no—
85Dic362C2×D412yescomplemented
86D724—22no—
87D724—22no—
88D724—42no—
89D724—81no—
90C22⋊D181—44no—
91C22⋊D181—82no—
92C2×Dic181—21no—
93C2×Dic181—21no—
94C4×D181—44no—
95#13089678—22no—
96#13089678—21no—
97#13089678C2×D412yescomplemented
98C23×C181—21no—
99C23×C181—22no—
100C23×C181—44no—
101C23×C181—44no—
102C23×C181—82no—
103C23×C181C2×D412yes—
104D4×C181—22no—
105D4×C181—22no—
106D4×C181—21no—
107D4×C181—22no—
108D4×C181—44no—
109D4×C181—44no—
110D4×C181—42no—
111D4×C181—81no—
112D4×C181C2×D412yes—
113C2×C4×C181—22no—
114C2×C4×C181—22no—
115C2×C4×C181—21no—
116C2×C4×C181—42no—
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2×D412yes—
119C2×C4×C181C2×D411yescharacteristic
120C2×C4×C181C2411yescharacteristic
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—22no—
125C22×C181—41no—
126C22×C181—41no—
127C22×C181—42no—
128C22×C181—42no—
129C22×C181—42no—
130C22×C181—88no—
131C22×C181C22×D411yescharacteristic
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—42no—
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—42no—
144C2×C181—42no—
145C2×C181—82no—
146C2×C181D4211yescharacteristic
147C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C181#57611yescharacteristic
149D181—82no—
150C181#327411yescharacteristic, complemented, Sylow
151#3274—1811nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1812no—
153#550—1811no—
154D42D18111yescharacteristic, complemented
155#493—1811no—
156#469—1812no—
157#460—1811no—
158#459—1811no—
159#449—1812no—
160#447—1812no—
161C4×SD16—1811no—
162#436—1811no—
163C2×#41—1812no—
164C22×D4—21no—
165C22×D4C2×D18112yes—
166AGL(1,ℤ/8)—3628no—
167C2×SD16—1812no—
168C2×SD16—1811no—
169C2×D8—1812no—
170C2×D8—1812no—
171C2×D8—1812no—
172C2×M16—1812no—
173#127—3621no—
174#127—1811no—
175#127C2×D18111yescharacteristic
176#121—21no—
177#121—3622no—
178#121—1812no—
179#121C2×D18112yes—
180#120—22no—
181C4×Q8—1811no—
182C4×D4—3621no—
183C4×D4C2×D18111yescharacteristic
184C4×D4C2×D18111yescharacteristic
185#106—1811no—
186C4⋊C8—1811no—
187#103—1811no—
188#102—1811no—
189#102—1812no—
190#102—1812no—
191C22⋊C8—1812no—
192C4×C8—1811no—
193C24—22no—
194#41—3624no—
195#41—7242no—
196C2×Q8—1811no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—3622no—
205C2×D4—3622no—
206C2×D4—3621no—
207C2×D4—3621no—
208C2×D4—1812no—
209C2×D4C22⋊D18111yescharacteristic
210C2×D4C22×D18112yes—
211C2×D4C22×D18111yescharacteristic
212C22×C4—21no—
213C22×C4—3622no—
214C22×C4C22×D18112yes—
215SD16—3624no—
216SD16—1814no—
217D8—3624no—
218D8—3624no—
219D8—7242no—
220M16—3624no—
221C2×C8—1811no—
222C2×C8—1812no—
223C2×C8—1811no—
224C4⋊C4—3621no—
225C4⋊C4—1811no—
226C4⋊C4C22⋊D18111yescharacteristic
227C4⋊C4C22×D18111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—3622no—
231C22⋊C4C22⋊D18112yes—
232C42—3621no—
233C42C22×D18111yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—3622no—
240C23C2×C22⋊D18112yes—
241Q8—3621no—
242Q8—1812no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—3622no—
252D4—3622no—
253D4—14481no—
254D4—14482no—
255D4—7244no—
256D4—7242no—
257D4D4×D18112yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3621no—
263C2×C4—3621no—
264C2×C4—7244no—
265C2×C4C2×C22⋊D18111yescharacteristic
266C2×C4C2×C22⋊D18112yes—
267C2×C4C2×C22⋊D18111yescharacteristic
268C2×C4C23×D18111yescharacteristic
269C8—3622no—
270C8—3621no—
271C8—1812no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—7242no—
283C22—7242no—
284C22—7242no—
285C22C22×C22⋊D18111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—3622no—
290C4—7241no—
291C4#1915384212yescomplemented
292C4C2×D4×D18111yescharacteristic
293C4#1915386111yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—14482no—
298C2D4×C22⋊D18111yescharacteristic
299C2C2×#1915384211yescharacteristic
300C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD42×C181C2
socleC22×C181C22×D4
hypercenterD42D181
hypocenterC181#3274
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. #26041950C2semidirect
3. #26041655C2semidirect
4. #26041530C2semidirect
5. #26041476C2semidirect
6. #26041474C2semidirect
7. #26041422C2semidirect
8. #26041413C2semidirect
9. C4×#15668437C2semidirect
10. #26041357C2semidirect
11. D42×C181C2semidirect
12. C2×#15668449C22semidirect
15. C2×#15668437C22semidirect
17. C2×#15668434C22semidirect
18. C2×#15668434C22semidirect
19. C2×#15668433C22semidirect
20. #19153813C22semidirect
22. #19153790C22semidirect
24. C4×Dic362C22semidirect
26. #19153747C22semidirect
27. C4⋊#13089678C22semidirect
28. #19153739C22semidirect
30. #19153737C22semidirect
31. #19153736C22semidirect
32. C22⋊#13089678C22semidirect
33. C4×#13089678C22semidirect
52. #15668437D4semidirect
85. Dic362C2×D4semidirect
97. #13089678C2×D4semidirect
138. C4×C181AGL(1,ℤ/8)semidirect
150. C181#3274semidirect
154. D42D181semidirect
291. C4#19153842semidirect

Maximal subgroups

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

subgroupindex
C2×#191538422
#260419502
#260416552
#260415302
#260414762
#260414742
#260414222
#260414132
C4×#156684372
#260413572
D42×C1812
#3274181