Finite Group Databasebeta

← #37543237

Subgroups — #37543237

Order 23168.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2C2×SD16×D181C211yesmaximal, characteristic, complemented
3#26041842C211yesmaximal, characteristic
4#26041755C211yesmaximal, characteristic, complemented
5#26041502C211yesmaximal, characteristic, complemented
6#26041501C211yesmaximal, characteristic, complemented
7#26041500C211yesmaximal, characteristic, complemented
8#26041499C211yesmaximal, characteristic, complemented
9#26041478C211yesmaximal, characteristic, complemented
10#26041477C211yesmaximal, characteristic, complemented
11#26041476C211yesmaximal, characteristic, complemented
12#26041304C211yesmaximal, characteristic, complemented
13#26041282C211yesmaximal, characteristic, complemented
14#26041272C211yesmaximal, characteristic, complemented
15#460×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153845—24no—
20C2×Dic724C2211yescharacteristic, complemented
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic
23C2×#15668437C2211yescharacteristic
24C2×#15668437C2211yescharacteristic, complemented
25C2×#15668436C2211yescharacteristic
26SD16×D181—24no—
27C2×#15668432C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153812C2211yescharacteristic
30#19153804C2211yescharacteristic
31#19153803C2211yescharacteristic
32#19153788—21no—
33#19153788—21no—
34#19153784—21no—
35#19153784—21no—
36Q8×Dic181C2211yescharacteristic, complemented
37C4×Dic362—21no—
38C4×Dic362C2211yescharacteristic, complemented
39#19153781—21no—
40#19153781C2211yescharacteristic, complemented
41#19153774—21no—
42#19153774C2211yescharacteristic
43#19153773—21no—
44#19153773C2211yescharacteristic
45C4⋊#13089678C2211yescharacteristic, complemented
46#19153745C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153738C2211yescharacteristic
51#19153737C2211yescharacteristic
52#19153737C2211yescharacteristic, complemented
53#19153736C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic, complemented
56#127×C181C2211yescharacteristic
57C4×Q8×C181C2211yescharacteristic, complemented
58C4⋊C8×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic
60#102×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668448—22no—
63#15668448—22no—
64C2×Dic362—21no—
65C2×Dic362C2311yescharacteristic
66C2×Dic362C2311yescharacteristic
67Q8×D181—22no—
68Q8×D181D414yescomplemented
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74D4×D181—22no—
75D4×D181—22no—
76C22×Dic181—21no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79Dic724—22no—
80#15668439—22no—
81#15668438—22no—
82#15668437—22no—
83#15668437—22no—
84#15668436—22no—
85#15668432—22no—
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C8×D181—22no—
89C4⋊Dic181—21no—
90C4⋊Dic181C2311yescharacteristic
91#15668428—21no—
92#15668428—21no—
93#15668428D411yescharacteristic
94#15668428D411yescharacteristic
95C22⋊Dic181—21no—
96C22⋊Dic181—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426D411yescharacteristic
101#15668426D411yescharacteristic
102C4×Dic181—21no—
103C4×Dic181—21no—
104C4×Dic181—21no—
105C4×Dic181—21no—
106C4×Dic181C2311yescharacteristic
107C2×Q8×C181C2311yescharacteristic
108C2×D4×C181—21no—
109C2×D4×C181—21no—
110C2×D4×C181C2311yescharacteristic
111C2×D4×C181C2311yescharacteristic
112SD16×C181—22no—
113SD16×C181—22no—
114C2×C8×C181C2311yescharacteristic
115C2×C8×C181C2311yescharacteristic
116C4⋊C4×C181—21no—
117C4⋊C4×C181C2311yescharacteristic
118C42×C181—21no—
119C42×C181C2311yescharacteristic
120C22×D181—21no—
121C22×D181—42no—
122C22×D181—81no—
123C22×D181—81no—
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—24no—
127Dic362—21no—
128Dic362—21no—
129Dic362C2×D412yes—
130D724—21no—
131D724—21no—
132C22⋊D181—42no—
133C22⋊D181—42no—
134C22⋊D181—81no—
135C22⋊D181—81no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—42no—
140C2×Dic181C2×D411yescharacteristic
141C2×Dic181C2×D411yescharacteristic
142C2×Dic181C2×D411yescharacteristic
143C4×D181—24no—
144C4×D181C2×D412yes—
145C4×D181#4112yes—
146#13089678—21no—
147#13089678—21no—
148C23×C181—21no—
149C23×C181—21no—
150Q8×C181—21no—
151Q8×C181C2×D412yes—
152D4×C181—21no—
153D4×C181—21no—
154D4×C181—21no—
155D4×C181—21no—
156D4×C181—81no—
157D4×C181—81no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181C2×D411yescharacteristic
161C2×C4×C181C2×D411yescharacteristic
162C2×C4×C181C2411yescharacteristic
163C8×C181—21no—
164C8×C181—21no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181—41no—
170C22×C181—41no—
171C22×C181C22×D411yescharacteristic
172C2×D181—21no—
173C2×D181—21no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—84no—
178C2×D181#12112yes—
179C4×C181—22no—
180C4×C181—21no—
181C4×C181—21no—
182C4×C181—41no—
183C4×C181C22×D411yescharacteristic
184C4×C181C2×#4111yescharacteristic
185Dic181—22no—
186Dic181—21no—
187Dic181—21no—
188Dic181—41no—
189Dic181—41no—
190Dic181#12112yes—
191C2×C181—81no—
192C2×C181—81no—
193C2×C181C2×#12111yescharacteristic
194C2×C181#57711yescharacteristic
195C2×C181#57711yescharacteristic
196D181—22no—
197D181—81no—
198C181#304311yescharacteristic, complemented, Sylow
199#3043—1811nomaximal, Sylow
200C2×#137—1811no—
201C22×SD16—1811no—
202#535—1811no—
203#519—1811no—
204#464—1811no—
205#464—1811no—
206#464—1811no—
207#464—1811no—
208#460—1811no—
209#460—1811no—
210#460—1811no—
211#460D18111yescharacteristic, complemented
212#424—1811no—
213#418—1811no—
214#416—1811no—
215C2×#41—1811no—
216C22×Q8—1811no—
217C22×D4—1811no—
218#137—3624no—
219C2×Q16—1811no—
220C2×Q16—1811no—
221C2×SD16—3624no—
222C2×SD16—1811no—
223C2×SD16—1811no—
224C2×SD16—1811no—
225C2×SD16—1811no—
226C2×SD16C2×D18111yescharacteristic
227C2×SD16C2×D18111yescharacteristic, complemented
228C2×M16—1811no—
229C22×C8—1811no—
230#127—1811no—
231#127C2×D18111yescharacteristic
232#124—1811no—
233#124—1811no—
234#121—3621no—
235#121—3621no—
236#120—3621no—
237#120—3621no—
238C4×Q8—3621no—
239C4×Q8—3621no—
240C4×Q8—1811no—
241C4×Q8—1811no—
242C4×Q8—1811no—
243C4×Q8C2×D18111yescharacteristic, complemented
244#117—3621no—
245#117—3621no—
246#117—1811no—
247#117—1811no—
248C4⋊C8—1811no—
249C4⋊C8—1811no—
250C4⋊C8—1811no—
251C4⋊C8C2×D18111yescharacteristic, complemented
252#103—1811no—
253#103—1811no—
254#102—1811no—
255#102—1811no—
256#102—1811no—
257#102—1811no—
258#102C2×D18111yescharacteristic
259#102C2×D18111yescharacteristic, complemented
260C24—3621no—
261#41—3622no—
262#41—3622no—
263C2×Q8—3621no—
264C2×Q8—3622no—
265C2×Q8—1814no—
266C2×Q8—1811no—
267C2×Q8—1811no—
268C2×Q8C22×D18111yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—3621no—
272C2×D4—3622no—
273C2×D4—3622no—
274C2×D4—3621no—
275C2×D4—3621no—
276C2×D4—3621no—
277C2×D4—1811no—
278C2×D4C22×D18111yescharacteristic
279C2×D4C22×D18111yescharacteristic
280C22×C4—3621no—
281C22×C4—3621no—
282C22×C4—1811no—
283Q16—3622no—
284Q16—3622no—
285SD16—22no—
286SD16—22no—
287SD16—3622no—
288SD16—3622no—
289SD16—3622no—
290SD16—3622no—
291M16—3622no—
292C2×C8—3622no—
293C2×C8—1811no—
294C2×C8—1811no—
295C2×C8C22×D18111yescharacteristic
296C2×C8C22×D18111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—3621no—
299C4⋊C4—3621no—
300C4⋊C4—3621no—
301C4⋊C4—1811no—
302C4⋊C4—1811no—
303C4⋊C4—1811no—
304C4⋊C4C22×D18111yescharacteristic
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4—1811no—
311C22⋊C4—1811no—
312C42—21no—
313C42—3621no—
314C42—3621no—
315C42—3621no—
316C42—3621no—
317C42—1811no—
318C42C22×D18111yescharacteristic
319C23—21no—
320C23—21no—
321C23—3621no—
322C23—14481no—
323C23—14481no—
324C23—1811no—
325C23—7242no—
326Q8—21no—
327Q8—3624no—
328Q8—3621no—
329Q8—3621no—
330Q8—3621no—
331Q8—1812no—
332Q8D4×D18112yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—3621no—
340D4—3621no—
341D4—14481no—
342D4—14481no—
343D4—7242no—
344D4—7242no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—3624no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—3621no—
351C2×C4—1811no—
352C2×C4—1811no—
353C2×C4—1811no—
354C2×C4—1812no—
355C2×C4—1812no—
356C2×C4—7242no—
357C2×C4D4×D18111yescharacteristic
358C2×C4D4×D18111yescharacteristic
359C2×C4C23×D18111yescharacteristic
360C8—21no—
361C8—21no—
362C8—3621no—
363C8—3621no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3621no—
371C22—3621no—
372C22—14484no—
373C22—1812no—
374C22—7241no—
375C22—7241no—
376C22—7241no—
377C22C2×D4×D18111yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—3622no—
383C4—3621no—
384C4—3621no—
385C4—1812no—
386C4—7241no—
387C4—7241no—
388C4C2×D4×D18111yescharacteristic
389C4#41×D18111yescharacteristic
390C2—81no—
391C2—81no—
392C2—3622no—
393C2—14481no—
394C2#121×D18111yescharacteristic
395C2#2604208511yescharacteristic
396C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#460×C181C2
socleC22×C181C22×D4
prime radical#26041842C2
hypercenter#460D181
hypocenterC181#3043
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×#19153845C2semidirect
2. C2×SD16×D181C2semidirect
4. #26041755C2semidirect
5. #26041502C2semidirect
6. #26041501C2semidirect
7. #26041500C2semidirect
8. #26041499C2semidirect
9. #26041478C2semidirect
10. #26041477C2semidirect
11. #26041476C2semidirect
12. #26041304C2semidirect
13. #26041282C2semidirect
14. #26041272C2semidirect
15. #460×C181C2semidirect
20. C2×Dic724C22semidirect
21. C2×#15668439C22semidirect
24. C2×#15668437C22semidirect
36. Q8×Dic181C22semidirect
38. C4×Dic362C22semidirect
40. #19153781C22semidirect
45. C4⋊#13089678C22semidirect
46. #19153745C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153739C22semidirect
52. #19153737C22semidirect
55. C2×SD16×C181C22semidirect
57. C4×Q8×C181C22semidirect
58. C4⋊C8×C181C22semidirect
60. #102×C181C22semidirect
68. Q8×D181D4semidirect
198. C181#3043semidirect
211. #460D181semidirect
227. C2×SD16C2×D181semidirect
243. C4×Q8C2×D181semidirect
251. C4⋊C8C2×D181semidirect
259. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
C2×SD16×D1812
#260418422
#260417552
#260415022
#260415012
#260415002
#260414992
#260414782
#260414772
#260414762
#260413042
#260412822
#260412722
#460×C1812
#3043181