Finite Group Databasebeta

← #37543198

Subgroups — #37543198

Order 23168.

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

normal subgroups only

This group has 584 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×#19153843C212yesmaximal, complemented
2#127×D181C211yesmaximal, characteristic, complemented
3C4×#15668450C211yesmaximal, characteristic, complemented
4#26041479C211yesmaximal, characteristic, complemented
5#26041477C212yesmaximal, complemented
6#26041475C211yesmaximal, characteristic, complemented
7#26041473C212yesmaximal, complemented
8#26041471C211yesmaximal, characteristic, complemented
9#26041301C211yesmaximal, characteristic, complemented
10#26041282C212yesmaximal, complemented
11#460×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2211yescharacteristic
13C2×D4×D181—21no—
14C2×D4×D181—21no—
15C2×D4×D181C2212yes—
16#19153843—28no—
17C2×#15668436C2212yescomplemented
18C2×D1448C2212yescomplemented
19C2×#15668434C2212yescomplemented
20C2×#15668432C2212yes—
21#19153813C2211yescharacteristic
22#19153812—21no—
23#19153812—21no—
24#19153812C2212yes—
25Q8×Dic181C2211yescharacteristic, complemented
26C4×D724—21no—
27C4×D724C2211yescharacteristic, complemented
28#19153778—21no—
29#19153778C2211yescharacteristic, complemented
30#19153774—21no—
31#19153774C2211yescharacteristic
32C42×D181—21no—
33C42×D181C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153745C2212yescomplemented
36#19153738C2212yescomplemented
37#19153737C2212yescomplemented
38#19153736C2212yescomplemented
39C2×SD16×C181C2212yescomplemented
40#127×C181C2211yescharacteristic
41C4×Q8×C181C2211yescharacteristic, complemented
42C4⋊C8×C181C2211yescharacteristic, complemented
43#102×C181C2212yescomplemented
44C23×D181—22no—
45#15668450—22no—
46#15668450D414yescomplemented
47C2×D724—21no—
48C2×D724—21no—
49C2×D724—21no—
50C2×D724C2312yes—
51C2×D724C2311yescharacteristic
52C2×C22⋊D181—22no—
53C2×C22⋊D181—22no—
54C2×C22⋊D181—22no—
55D4×D181—24no—
56D4×D181—24no—
57D4×D181—44no—
58D4×D181—44no—
59C2×C4×D181—21no—
60C2×C4×D181—21no—
61C2×C4×D181—21no—
62C2×C4×D181D411yescharacteristic
63C2×C4×D181D411yescharacteristic
64C2×C4×D181C2311yescharacteristic
65#15668436—24no—
66D1448—24no—
67#15668434—24no—
68#15668432—24no—
69C2×#13089678C2312yes—
70C4⋊Dic181—21no—
71C4⋊Dic181C2311yescharacteristic
72#15668426—21no—
73#15668426—21no—
74#15668426—21no—
75C4×Dic181—21no—
76C4×Dic181—21no—
77C4×Dic181D411yescharacteristic
78C4×Dic181D411yescharacteristic
79C4×Dic181C2311yescharacteristic
80C2×Q8×C181C2311yescharacteristic
81C2×D4×C181—21no—
82C2×D4×C181—21no—
83C2×D4×C181C2312yes—
84SD16×C181—24no—
85C2×C8×C181C2312yes—
86C4⋊C4×C181—21no—
87C4⋊C4×C181C2311yescharacteristic
88C42×C181—21no—
89C42×C181C2311yescharacteristic
90C22×D181—21no—
91C22×D181—22no—
92C22×D181—44no—
93C22×D181—44no—
94C22×D181—44no—
95C22×D181C2×D411yescharacteristic
96D724—24no—
97D724—22no—
98D724—21no—
99D724—22no—
100D724—42no—
101D724—42no—
102D724C2×D412yescomplemented
103C22⋊D181—44no—
104C22⋊D181—44no—
105C22⋊D181—44no—
106C2×Dic181—21no—
107C2×Dic181C2×D411yescharacteristic
108C2×Dic181C2×D411yescharacteristic
109C2×Dic181C2×D411yescharacteristic
110C4×D181—22no—
111C4×D181—24no—
112C4×D181—22no—
113C4×D181—42no—
114C4×D181—42no—
115C4×D181C2×D412yes—
116C4×D181#4112yes—
117#13089678—22no—
118C23×C181—22no—
119Q8×C181—21no—
120Q8×C181C2×D412yescomplemented
121D4×C181—22no—
122D4×C181—22no—
123D4×C181—42no—
124D4×C181—42no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181C2×D411yescharacteristic
128C2×C4×C181C2×D411yescharacteristic
129C2×C4×C181C2411yescharacteristic
130C8×C181—22no—
131C22×C181—42no—
132C22×C181—42no—
133C22×C181—42no—
134C22×C181C22×D411yescharacteristic
135C2×D181—21no—
136C2×D181—21no—
137C2×D181—22no—
138C2×D181—41no—
139C2×D181—41no—
140C2×D181—42no—
141C2×D181—42no—
142C2×D181—42no—
143C2×D181—88no—
144C2×D181#12112yes—
145C4×C181—22no—
146C4×C181—21no—
147C4×C181—41no—
148C4×C181AGL(1,ℤ/8)12yescomplemented
149C4×C181C22×D411yescharacteristic
150C4×C181C2×#4111yescharacteristic
151Dic181—21no—
152Dic181—41no—
153Dic181#12112yes—
154Dic181AGL(1,ℤ/8)12yescomplemented
155C2×C181—82no—
156C2×C181C2×#12111yescharacteristic
157C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
159D181—22no—
160D181—42no—
161D181—82no—
162C181#303811yescharacteristic, complemented, Sylow
163#3038—1811nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1812no—
165C2×#127—1811no—
166C4×#41—1811no—
167#460—1812no—
168#460—1811no—
169#460D18111yescharacteristic, complemented
170#459—1811no—
171#459—1811no—
172#459—1812no—
173#423—1811no—
174#418—1812no—
175C2×#41—1811no—
176C22×D4—3621no—
177C22×D4—3621no—
178C22×D4—1812no—
179AGL(1,ℤ/8)—3628no—
180C2×SD16—1812no—
181C2×SD16C2×D18112yescomplemented
182C2×D8—1812no—
183C2×D8—1812no—
184C2×M16—1812no—
185#127—3621no—
186#127—3621no—
187#127—1812no—
188#127—1811no—
189#127C2×D18111yescharacteristic
190C4×Q8—1811no—
191C4×Q8C2×D18111yescharacteristic, complemented
192C4×D4—3621no—
193C4×D4—3621no—
194C4×D4—1811no—
195C4×D4—1811no—
196#117—3621no—
197#117—1811no—
198C2×C42—3621no—
199C2×C42—1811no—
200C4⋊C8—1812no—
201C4⋊C8—1811no—
202C4⋊C8C2×D18111yescharacteristic, complemented
203#102—1812no—
204#102—1812no—
205#102—1812no—
206#102C2×D18112yescomplemented
207C24—3622no—
208#41—3622no—
209#41—1814no—
210C2×Q8C22×D18111yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—3621no—
214C2×D4—3624no—
215C2×D4—3624no—
216C2×D4—3622no—
217C2×D4—3622no—
218C2×D4—3622no—
219C2×D4—3621no—
220C2×D4—3621no—
221C2×D4—1812no—
222C2×D4—1811no—
223C2×D4—7244no—
224C2×D4—7244no—
225C2×D4C22×D18112yes—
226C22×C4—3621no—
227C22×C4—3621no—
228C22×C4—3621no—
229C22×C4—1811no—
230C22×C4—1811no—
231C22×C4—1811no—
232SD16—24no—
233SD16—3624no—
234D8—3624no—
235D8—3624no—
236M16—3624no—
237C2×C8—1812no—
238C2×C8C22×D18112yes—
239C4⋊C4—21no—
240C4⋊C4—3621no—
241C4⋊C4—1811no—
242C4⋊C4C22×D18111yescharacteristic
243C22⋊C4—3621no—
244C22⋊C4—3621no—
245C22⋊C4—3621no—
246C42—21no—
247C42—3621no—
248C42—3621no—
249C42—1811no—
250C42—1811no—
251C42—1811no—
252C42C22×D18111yescharacteristic
253C23—22no—
254C23—3621no—
255C23—3622no—
256C23—1811no—
257C23—7244no—
258C23—7244no—
259C23—7244no—
260Q8—21no—
261Q8D4×D18112yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3624no—
267D4—3622no—
268D4—3621no—
269D4—3622no—
270D4—1812no—
271D4—7242no—
272D4—7244no—
273D4—7244no—
274D4—7244no—
275D4—7242no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3624no—
279C2×C4—3622no—
280C2×C4—3622no—
281C2×C4—3621no—
282C2×C4—1811no—
283C2×C4—1811no—
284C2×C4—1811no—
285C2×C4—1812no—
286C2×C4—1812no—
287C2×C4—7242no—
288C2×C4—7242no—
289C2×C4D4×D18111yescharacteristic
290C2×C4D4×D18111yescharacteristic
291C2×C4C23×D18111yescharacteristic
292C8—22no—
293C8—3622no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—3621no—
298C22—3621no—
299C22—3622no—
300C22—14488no—
301C22—1812no—
302C22—7242no—
303C22—7242no—
304C22—7242no—
305C22—7241no—
306C22—7241no—
307C22C2×D4×D18111yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—3621no—
312C4—1812no—
313C4—1812no—
314C4—7241no—
315C4#1915384312yescomplemented
316C4C2×D4×D18111yescharacteristic
317C4#41×D18111yescharacteristic
318C2—82no—
319C2—3622no—
320C2—14482no—
321C2—7242no—
322C2#121×D18111yescharacteristic
323C2AGL(1,ℤ/8)×D18111yescharacteristic
324C2C2×#1915384311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#460×C181C2
socleC22×C181C22×D4
hypercenter#460D181
hypocenterC181#3038
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19153843C2semidirect
2. #127×D181C2semidirect
3. C4×#15668450C2semidirect
4. #26041479C2semidirect
5. #26041477C2semidirect
6. #26041475C2semidirect
7. #26041473C2semidirect
8. #26041471C2semidirect
9. #26041301C2semidirect
10. #26041282C2semidirect
11. #460×C181C2semidirect
17. C2×#15668436C22semidirect
18. C2×D1448C22semidirect
19. C2×#15668434C22semidirect
25. Q8×Dic181C22semidirect
27. C4×D724C22semidirect
29. #19153778C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153745C22semidirect
36. #19153738C22semidirect
37. #19153737C22semidirect
38. #19153736C22semidirect
39. C2×SD16×C181C22semidirect
41. C4×Q8×C181C22semidirect
42. C4⋊C8×C181C22semidirect
43. #102×C181C22semidirect
46. #15668450D4semidirect
102. D724C2×D4semidirect
120. Q8×C181C2×D4semidirect
148. C4×C181AGL(1,ℤ/8)semidirect
154. Dic181AGL(1,ℤ/8)semidirect
162. C181#3038semidirect
169. #460D181semidirect
181. C2×SD16C2×D181semidirect
191. C4×Q8C2×D181semidirect
202. C4⋊C8C2×D181semidirect
206. #102C2×D181semidirect
261. Q8D4×D181semidirect
315. C4#19153843semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
#127×D1812
C4×#156684502
#260414792
#260414772
#260414752
#260414732
#260414712
#260413012
#260412822
#460×C1812
#3038181