Finite Group Databasebeta

← #26423443

Subgroups — #26423443

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251675C212yesmaximal
2#20250583C214yesmaximal, complemented
3#20250293C214yesmaximal, complemented
4#20250046C214yesmaximal, complemented
5#20249854C214yesmaximal, complemented
6#20249502C211yesmaximal, characteristic, complemented
7#20249358C211yesmaximal, characteristic, complemented
8#20247628C214yesmaximal
9#20247523C214yesmaximal, complemented
10#20246121C212yesmaximal, complemented
11C2×C47×#380C211yesmaximal, characteristic
12C2×#13094017C2214yes—
13C2×#13094014C2214yescomplemented
14C2×#13094003C2212yes—
15C2×#13094003C2212yes—
16C2×#13094003C2212yes—
17#15689065C2214yescomplemented
18#15689061C2214yescomplemented
19#15689057C2214yescomplemented
20#15689056C2214yescomplemented
21#15689055C2214yescomplemented
22#15689054C2214yescomplemented
23#15689053C2214yescomplemented
24#15689051C2214yes—
25#15689051C2214yes—
26#15689048C2214yes—
27#15689046C2214yes—
28#15689044C2214yescomplemented
29#15689027C2214yescomplemented
30#15689020C2214yescomplemented
31#15689018C2214yescomplemented
32#15689017C2214yes—
33#15689016C2214yes—
34#15689011C2214yes—
35#15689010C2214yescomplemented
36#15689009C2214yescomplemented
37#15689006C2214yescomplemented
38#15689004C2214yescomplemented
39#15688996C2214yes—
40#15688996C2214yes—
41#15688996C2214yes—
42#15688996C2214yes—
43#15688996C2214yes—
44#15688983C2212yes—
45#15688983C2212yes—
46#15688983C2212yes—
47#15688983C2214yes—
48#15688979C2214yes—
49C4×C22⋊Dic47C2214yes—
50C24×C4×C47C2211yescharacteristic
51C22×C4⋊C4×C47C2212yes—
52C22×C22⋊C4×C47C2212yes—
53C47×#380C2214yes—
54C47×#380C2214yes—
55C2×#95×C47C2211yescharacteristic
56C2×#95×C47C2211yescharacteristic
57C22×Dic94C2312yes—
58#13094017—48no—
59#13094014—48no—
60#13094003—28no—
61#13094003—44no—
62#13094003—44no—
63C2×C4⋊Dic47C2312yes—
64C2×C4⋊Dic47C2312yes—
65C2×C4⋊Dic47C2312yes—
66C2×C4⋊Dic47C2314yes—
67C2×C4⋊Dic47C2314yes—
68C2×Dic47⋊C4C2314yes—
69C2×Dic47⋊C4C2314yes—
70C2×Dic47⋊C4C2314yes—
71C2×Dic47⋊C4C2314yes—
72C2×Dic47⋊C4C2314yes—
73C2×Dic47⋊C4C2314yes—
74C2×Dic47⋊C4C2314yes—
75C2×C22⋊Dic47C2314yes—
76C2×C22⋊Dic47C2314yes—
77C2×C22⋊Dic47C2314yes—
78C2×C22⋊Dic47C2314yes—
79C2×C22⋊Dic47C2314yes—
80C2×C22⋊Dic47C2314yes—
81C2×C4×Dic47C2314yes—
82C2×C4×Dic47C2314yes—
83C47⋊#95C2314yes—
84C47⋊#95C2314yes—
85C47⋊#95C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2314yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2314yes—
93C47⋊#95C2314yes—
94C25×C47C2311yescharacteristic
95C23×C4×C47—24no—
96C23×C4×C47—44no—
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2312yes—
99C23×C4×C47C2312yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2312yes—
102C23×C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47—44no—
104C2×C4⋊C4×C47C2314yes—
105C2×C4⋊C4×C47C2314yes—
106C2×C22⋊C4×C47—44no—
107C2×C22⋊C4×C47C2314yes—
108C2×C22⋊C4×C47C2314yes—
109#95×C47C2314yes—
110#95×C47C2314yes—
111C24×C47—22no—
112C24×C47—22no—
113C24×C47—22no—
114C24×C47—24no—
115C24×C47—42no—
116C24×C47C2411yescharacteristic
117C24×C47C2411yescharacteristic
118C24×C47C2411yescharacteristic
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—24no—
122C22×C4×C47—24no—
123C22×C4×C47—24no—
124C22×C4×C47—24no—
125C22×C4×C47—24no—
126C22×C4×C47—24no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—416no—
135C22×C4×C47#4114yes—
136C22×C4×C47C2414yes—
137C22×C4×C47C2414yes—
138C22×C4×C47C2412yes—
139C22×C4×C47C2412yes—
140C22×Dic47C2414yes—
141C22×Dic47C2414yes—
142C22×Dic47C2414yes—
143C22×Dic47C2414yes—
144C2×Dic94—24no—
145C2×Dic94—42no—
146C2×Dic94—42no—
147C4⋊C4×C47—28no—
148C4⋊C4×C47—44no—
149C22⋊C4×C47—28no—
150C22⋊C4×C47—44no—
151C22⋊Dic47—28no—
152C22⋊Dic47—28no—
153C22⋊Dic47—28no—
154C22⋊Dic47—28no—
155C22⋊Dic47—44no—
156C22⋊Dic47—44no—
157C4⋊Dic47—24no—
158C4⋊Dic47—42no—
159C4⋊Dic47—42no—
160C4⋊Dic47—44no—
161C4⋊Dic47—44no—
162Dic47⋊C4—28no—
163Dic47⋊C4—28no—
164Dic47⋊C4—28no—
165Dic47⋊C4—44no—
166Dic47⋊C4—44no—
167Dic47⋊C4—44no—
168Dic47⋊C4—44no—
169C4×Dic47—44no—
170C4×Dic47—44no—
171C23×C47—22no—
172C23×C47—21no—
173C23×C47—21no—
174C23×C47—28no—
175C23×C47—21no—
176C23×C47—21no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—21no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—21no—
183C23×C47—48no—
184C23×C47—48no—
185C23×C47—44no—
186C23×C47—44no—
187C23×C47C2×#4112yes—
188C23×C47H(5,𝔽2)12yes—
189C23×C472−1+414yes—
190C23×C472−1+412yes—
191C23×C47C2511yescharacteristic
192C2×C4×C47—28no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—44no—
204C2×C4×C47—48no—
205C2×C4×C47—48no—
206C2×C4×C47—84no—
207C2×C4×C47—84no—
208C2×C4×C47C2×#4114yes—
209C2×C4×C47H(5,𝔽2)14yes—
210C2×Dic47—24no—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—24no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—24no—
221C2×Dic47—24no—
222Dic94—84no—
223#36728—471nomaximal, Sylow
224C22×C47—22no—
225C22×C47—22no—
226C22×C47—24no—
227C22×C47—24no—
228C22×C47—22no—
229C22×C47—22no—
230C22×C47—24no—
231C22×C47—24no—
232C22×C47—416no—
233C22×C47—41no—
234C22×C47—41no—
235C22×C47—41no—
236C22×C47—41no—
237C22×C47—42no—
238C22×C47—42no—
239C22×C47#54114yes—
240C22×C47C22×#4111yescharacteristic
241C22×C47C2×H(5,𝔽2)12yes—
242C22×C47C2×H(5,𝔽2)11yescharacteristic
243C22×C47C2×2−1+412yes—
244C22×C47C2×2−1+411yescharacteristic
245C4×C47—24no—
246C4×C47—24no—
247C4×C47—44no—
248C4×C47—44no—
249Dic47—44no—
250Dic47—44no—
251Dic47—44no—
252Dic47—44no—
253#2848—472no—
254#2747—474no—
255#2722—474no—
256#2704—474no—
257#2690—474no—
258#2659—471no—
259#2646—471no—
260#2480—474no—
261#2473—474no—
262#2329—472no—
263C2×#380D4711yescharacteristic
264C2×C47—24no—
265C2×C47—42no—
266C2×C47—42no—
267C2×C47C2×#54111yescharacteristic
268C2×C47#349011yescharacteristic
269C2×C47#349211yescharacteristic
270C2×C47#353612yes—
271C2×C47#353812yes—
272C24×C4C2×D4711yescharacteristic
273C2×#126—474no—
274C2×#125—474no—
275C2×C22⋊Q8—472no—
276C2×C22⋊Q8—472no—
277C2×C22⋊Q8—472no—
278C22×C4⋊C4C2×D4712yes—
279C22×C22⋊C4C2×D4712yes—
280#400—474no—
281#400—474no—
282#398—474no—
283#398—474no—
284#398—474no—
285#398—474no—
286#398—474no—
287#397—474no—
288#397—474no—
289#397—474no—
290#396—474no—
291#395—474no—
292#391—474no—
293#389—474no—
294#389—474no—
295#388—474no—
296#388—474no—
297#388—474no—
298#387—474no—
299#387—474no—
300#387—474no—
301#387—474no—
302#385—474no—
303#385—474no—
304#385—474no—
305#385—474no—
306#385—474no—
307#380—472no—
308#380—472no—
309#380—472no—
310#380—474no—
311#380C2×D4714yes—
312#380C2×D4714yes—
313#379—474no—
314C4×C22⋊C4—474no—
315C2×#95C2×D4711yescharacteristic
316C2×#95C2×D4711yescharacteristic
317C47#3672811yescharacteristic, complemented, Sylow
318C25C22×D4711yescharacteristic
319C22×Q8—472no—
320C23×C4—24no—
321C23×C4—44no—
322C23×C4C22×D4711yescharacteristic
323C23×C4C22×D4712yes—
324C23×C4C22×D4712yes—
325C23×C4C22×D4712yes—
326C23×C4C22×D4712yes—
327C23×C4C22×D4711yescharacteristic
328#126—1888no—
329#125—1888no—
330C22⋊Q8—948no—
331C22⋊Q8—1884no—
332C22⋊Q8—1884no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—474no—
335C2×C4⋊C4—474no—
336C2×C4⋊C4—474no—
337C2×C4⋊C4—472no—
338C2×C4⋊C4—472no—
339C2×C4⋊C4—472no—
340C2×C4⋊C4—474no—
341C2×C4⋊C4—474no—
342C2×C4⋊C4—474no—
343C2×C4⋊C4—474no—
344C2×C4⋊C4—474no—
345C2×C4⋊C4—474no—
346C2×C4⋊C4C22×D4714yes—
347C2×C4⋊C4C22×D4714yes—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—474no—
350C2×C22⋊C4—474no—
351C2×C22⋊C4—474no—
352C2×C22⋊C4—474no—
353C2×C22⋊C4—474no—
354C2×C22⋊C4—474no—
355C2×C22⋊C4C22×D4714yes—
356C2×C22⋊C4C22×D4714yes—
357C2×C42—474no—
358C2×C42—474no—
359#95—474no—
360#95—474no—
361#95—474no—
362#95—474no—
363#95—474no—
364#95—474no—
365#95—474no—
366#95—474no—
367#95—474no—
368#95—474no—
369#95—474no—
370#95C22×D4714yes—
371#95C22×D4714yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C23×D4711yescharacteristic
378C24C23×D4711yescharacteristic
379C24C23×D4711yescharacteristic
380C2×Q8—944no—
381C2×Q8—1882no—
382C2×Q8—1882no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—474no—
400C22×C4—474no—
401C22×C4—474no—
402C22×C4—474no—
403C22×C4#1062535014yes—
404C22×C4C23×D4714yes—
405C22×C4C23×D4714yes—
406C22×C4C23×D4712yes—
407C22×C4C23×D4712yes—
408C4⋊C4—28no—
409C4⋊C4—44no—
410C4⋊C4—948no—
411C4⋊C4—944no—
412C4⋊C4—948no—
413C4⋊C4—948no—
414C4⋊C4—1884no—
415C4⋊C4—1884no—
416C4⋊C4—1884no—
417C4⋊C4—1884no—
418C4⋊C4—1882no—
419C4⋊C4—1882no—
420C4⋊C4—1884no—
421C4⋊C4—1884no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—948no—
425C22⋊C4—948no—
426C22⋊C4—948no—
427C22⋊C4—948no—
428C22⋊C4—1884no—
429C22⋊C4—1884no—
430C42—1884no—
431C42—1884no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#1062535012yes—
449C23#1309406912yes—
450C23#1309407114yes—
451C23#1309407212yes—
452C23C24×D4711yescharacteristic
453Q8—3764no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—944no—
471C2×C4—944no—
472C2×C4—944no—
473C2×C4—944no—
474C2×C4—944no—
475C2×C4—944no—
476C2×C4—944no—
477C2×C4—944no—
478C2×C4—944no—
479C2×C4—944no—
480C2×C4—944no—
481C2×C4—944no—
482C2×C4C2×#1062535014yes—
483C2×C4#1309406914yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22#1568973514yes—
500C22C22×#1062535011yescharacteristic
501C22C2×#1309406912yes—
502C22C2×#1309406911yescharacteristic
503C22C2×#1309407112yes—
504C22C2×#1309407211yescharacteristic
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—1884no—
510C4—1884no—
511C4—1884no—
512C4—1884no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#1568973511yescharacteristic
517C2#2025707711yescharacteristic
518C2#2025709711yescharacteristic
519C2#2025761012yes—
520C2#2025763312yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC2×C47×#380C2
prime radicalC25×C47C23
hypercenterC2×#380D47
hypocenterC47#36728
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #20250583C2semidirect
3. #20250293C2semidirect
4. #20250046C2semidirect
5. #20249854C2semidirect
6. #20249502C2semidirect
7. #20249358C2semidirect
9. #20247523C2semidirect
10. #20246121C2semidirect
13. C2×#13094014C22semidirect
17. #15689065C22semidirect
18. #15689061C22semidirect
19. #15689057C22semidirect
20. #15689056C22semidirect
21. #15689055C22semidirect
22. #15689054C22semidirect
23. #15689053C22semidirect
28. #15689044C22semidirect
29. #15689027C22semidirect
30. #15689020C22semidirect
31. #15689018C22semidirect
35. #15689010C22semidirect
36. #15689009C22semidirect
37. #15689006C22semidirect
38. #15689004C22semidirect
317. C47#36728semidirect

Maximal subgroups

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

subgroupindex
#202516752
#202505832
#202502932
#202500462
#202498542
#202495022
#202493582
#202476282
#202475232
#202461212
C2×C47×#3802
#3672847