Finite Group Databasebeta

← #37545343

Subgroups — #37545343

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×#19153844C212yesmaximal, complemented
2#26041950C211yesmaximal, characteristic, complemented
3D4×D724C211yesmaximal, characteristic, complemented
4#26041656C211yesmaximal, characteristic, complemented
5#26041532C212yesmaximal, complemented
6#26041480C211yesmaximal, characteristic, complemented
7#26041475C211yesmaximal, characteristic, complemented
8#26041426C212yesmaximal, complemented
9#26041415C212yesmaximal, complemented
10C4×#15668438C211yesmaximal, characteristic, complemented
11#436×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2212yescomplemented
13C22×D724C2212yes—
14C2×D4×D181—21no—
15#19153844—28no—
16C2×#15668438C2211yescharacteristic
17C2×#15668438C2212yescomplemented
18C2×D1448C2212yes—
19C2×D1448C2212yescomplemented
20C2×D1448C2212yescomplemented
21#19153813C2211yescharacteristic
22#19153813C2211yescharacteristic, complemented
23#19153812—21no—
24#19153791—21no—
25#19153790C2212yes—
26#19153790C2212yescomplemented
27#19153789—22no—
28#19153786—22no—
29C4×Dic362C2211yescharacteristic, complemented
30C4×D724C2211yescharacteristic
31#19153778—21no—
32#19153748C2211yescharacteristic, complemented
33#19153741C2211yescharacteristic, complemented
34#19153738C2211yescharacteristic
35#19153738C2212yescomplemented
36#19153738C2212yescomplemented
37C2×M16×C181C2212yescomplemented
38C4×D4×C181C2211yescharacteristic
39C4⋊C8×C181C2211yescharacteristic, complemented
40C22⋊C8×C181C2212yescomplemented
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43#15668449—24no—
44#15668449—42no—
45C2×Dic362C2311yescharacteristic
46C2×D724—24no—
47C2×D724—21no—
48C2×D724—22no—
49C2×D724—21no—
50C2×D724—21no—
51C2×D724—21no—
52C2×D724—42no—
53C2×D724C2312yes—
54C2×D724C2312yes—
55C2×D724C2311yescharacteristic
56C2×C22⋊D181—22no—
57C2×C22⋊D181—22no—
58C2×C22⋊D181—22no—
59D4×D181—44no—
60C2×C4×D181—22no—
61C2×C4×D181—21no—
62#15668438—24no—
63#15668438D414yescomplemented
64D1448—24no—
65D1448—24no—
66D1448—42no—
67C4⋊Dic181C2311yescharacteristic
68C4⋊Dic181C2311yescharacteristic
69#15668428—21no—
70#15668426—22no—
71#15668426—21no—
72#15668426—22no—
73C4×Dic181—21no—
74C2×D4×C181D411yescharacteristic
75C22×C4×C181C2312yes—
76M16×C181—24no—
77C2×C8×C181C2311yescharacteristic
78C2×C8×C181C2312yes—
79C2×C8×C181C2311yescharacteristic
80C4⋊C4×C181D411yescharacteristic
81C22⋊C4×C181D412yes—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—22no—
86C22×D181—44no—
87C22×D181—44no—
88C22×D181—82no—
89Dic362—21no—
90Dic362C2×D412yescomplemented
91D724—22no—
92D724—22no—
93D724—21no—
94D724—22no—
95D724—22no—
96D724—44no—
97D724—42no—
98D724—42no—
99D724—81no—
100D724—81no—
101D724C2×D412yes—
102C22⋊D181—44no—
103C22⋊D181—44no—
104C22⋊D181—82no—
105C2×Dic181—21no—
106C2×Dic181—21no—
107C2×Dic181—21no—
108C4×D181—44no—
109C4×D181—42no—
110C23×C181C2×D412yes—
111D4×C181—22no—
112C2×C4×C181—22no—
113C2×C4×C181—22no—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D412yes—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2411yescharacteristic
118C8×C181—22no—
119C8×C181—21no—
120C8×C181C2×D412yescomplemented
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—22no—
124C22×C181C22×D411yescharacteristic
125C2×D181—22no—
126C2×D181—41no—
127C2×D181—41no—
128C2×D181—42no—
129C2×D181—42no—
130C2×D181—42no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—88no—
135C4×C181—22no—
136C4×C181—21no—
137C4×C181AGL(1,ℤ/8)12yescomplemented
138C4×C181C22×D411yescharacteristic
139C4×C181H(5,𝔽2)11yescharacteristic
140Dic181—22no—
141Dic181—41no—
142Dic181—41no—
143C2×C181—42no—
144C2×C181D4211yescharacteristic
145C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C181#57611yescharacteristic
147D181—42no—
148D181—82no—
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#460—1811no—
158#459—1811no—
159#449—1812no—
160#447—1812no—
161C4×SD16—1811no—
162#436D18111yescharacteristic, complemented
163C2×#41—1812no—
164C22×D4—3621no—
165C22×D4—1812no—
166AGL(1,ℤ/8)—3628no—
167C2×SD16—1812no—
168C2×SD16—1811no—
169C2×D8—1812no—
170C2×D8—1812no—
171C2×D8—1812no—
172C2×M16C2×D18112yescomplemented
173#127—3621no—
174#127—1811no—
175#127—1811no—
176#121—3621no—
177#121—3622no—
178#121—1812no—
179#121—1812no—
180#120—3622no—
181C4×Q8—1811no—
182C4×D4—3621no—
183C4×D4—1811no—
184C4×D4C2×D18111yescharacteristic
185#106—1811no—
186C4⋊C8C2×D18111yescharacteristic, complemented
187#103—1811no—
188#102—1811no—
189#102—1812no—
190#102—1812no—
191C22⋊C8C2×D18112yescomplemented
192C4×C8C2×D18111yescharacteristic, complemented
193C24—3622no—
194#41—3624no—
195#41—7242no—
196C2×Q8—1811no—
197C2×D4—3624no—
198C2×D4—3622no—
199C2×D4—3622no—
200C2×D4—3622no—
201C2×D4—3621no—
202C2×D4—3621no—
203C2×D4—3621no—
204C2×D4—3622no—
205C2×D4—3621no—
206C2×D4—1812no—
207C2×D4—1812no—
208C2×D4—1811no—
209C2×D4—7242no—
210C2×D4—7244no—
211C2×D4D72411yescharacteristic
212C22×C4—3622no—
213C22×C4—3621no—
214C22×C4C22×D18112yes—
215SD16—3624no—
216SD16—1814no—
217D8—3624no—
218D8—3624no—
219D8—7242no—
220M16—24no—
221C2×C8C22×D18111yescharacteristic
222C2×C8C22×D18112yes—
223C2×C8C22×D18111yescharacteristic
224C4⋊C4—3621no—
225C4⋊C4—1811no—
226C4⋊C4—1811no—
227C4⋊C4D72411yescharacteristic
228C22⋊C4—3622no—
229C22⋊C4—3621no—
230C22⋊C4—3622no—
231C22⋊C4D72412yes—
232C42—3621no—
233C42C22×D18111yescharacteristic
234C23—3621no—
235C23—3622no—
236C23—3622no—
237C23—14482no—
238C23—7244no—
239C23—7244no—
240C23C2×D72412yes—
241Q8—3621no—
242Q8—1812no—
243D4—22no—
244D4—3622no—
245D4—3622no—
246D4—3621no—
247D4—3622no—
248D4—3622no—
249D4—14481no—
250D4—14481no—
251D4—14482no—
252D4—1812no—
253D4—7244no—
254D4—7244no—
255D4—7244no—
256D4—7242no—
257D4—7242no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3621no—
261C2×C4—3621no—
262C2×C4—3621no—
263C2×C4—7244no—
264C2×C4—7242no—
265C2×C4C2×D72411yescharacteristic
266C2×C4C2×D72412yes—
267C2×C4C2×D72411yescharacteristic
268C2×C4C23×D18111yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D18112yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—3622no—
276C22—14488no—
277C22—7242no—
278C22—7242no—
279C22—7242no—
280C22—7242no—
281C22—7242no—
282C22—7242no—
283C22—7241no—
284C22—7241no—
285C22C22×D72411yescharacteristic
286C4—22no—
287C4—21no—
288C4—3622no—
289C4—7241no—
290C4—7241no—
291C4#1915384412yescomplemented
292C4C2×D4×D18111yescharacteristic
293C4#1915386211yescharacteristic
294C2—42no—
295C2—14482no—
296C2—14482no—
297C2—7242no—
298C2D4×D72411yescharacteristic
299C2C2×#1915384411yescharacteristic
300C2#2604208411yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#436×C181C2
socleC22×C181C22×D4
hypercenter#436D181
hypocenterC181#3274
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. #26041950C2semidirect
3. D4×D724C2semidirect
4. #26041656C2semidirect
5. #26041532C2semidirect
6. #26041480C2semidirect
7. #26041475C2semidirect
8. #26041426C2semidirect
9. #26041415C2semidirect
10. C4×#15668438C2semidirect
11. #436×C181C2semidirect
12. C2×#15668449C22semidirect
17. C2×#15668438C22semidirect
19. C2×D1448C22semidirect
20. C2×D1448C22semidirect
22. #19153813C22semidirect
26. #19153790C22semidirect
29. C4×Dic362C22semidirect
32. #19153748C22semidirect
33. #19153741C22semidirect
35. #19153738C22semidirect
36. #19153738C22semidirect
37. C2×M16×C181C22semidirect
39. C4⋊C8×C181C22semidirect
40. C22⋊C8×C181C22semidirect
41. C4×C8×C181C22semidirect
63. #15668438D4semidirect
90. Dic362C2×D4semidirect
120. C8×C181C2×D4semidirect
137. C4×C181AGL(1,ℤ/8)semidirect
150. C181#3274semidirect
162. #436D181semidirect
172. C2×M16C2×D181semidirect
186. C4⋊C8C2×D181semidirect
191. C22⋊C8C2×D181semidirect
192. C4×C8C2×D181semidirect
271. C8D4×D181semidirect
291. C4#19153844semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
#260419502
D4×D7242
#260416562
#260415322
#260414802
#260414752
#260414262
#260414152
C4×#156684382
#436×C1812
#3274181