Finite Group Databasebeta

← #26184041

Subgroups — #26184041

Order 12032.

1270 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257152C211yesmaximal, characteristic
2#20246328C211yesmaximal, characteristic, complemented
3#20241957C212yesmaximal, complemented
4#20241936C212yesmaximal, complemented
5#20241918C212yesmaximal, complemented
6#20241905C212yesmaximal, complemented
7#20241897C212yesmaximal, complemented
8C2×#15688858C212yesmaximal, complemented
9C47×#2973C211yesmaximal, characteristic, complemented
10C2×#13094069—21no—
11#15689777C414yescomplemented
12#15689768—24no—
13D4×D188C414yescomplemented
14#15689704—21no—
15#15689685—21no—
16#15689685—21no—
17#15689673C2212yes—
18C2×#13093998C2212yes—
19C2×#13093998C2212yes—
20#15689016—21no—
21#15689009—21no—
22#15689003—21no—
23#15688991C2211yescharacteristic, complemented
24#15688986C2211yescharacteristic, complemented
25#15688983C2211yescharacteristic, complemented
26C22⋊C4×Dic47—21no—
27C4×C22⋊Dic47C2211yescharacteristic, complemented
28#15688969C2211yescharacteristic, complemented
29C2×C47⋊#95C2211yescharacteristic, complemented
30#15688858C2214yescomplemented
31#15688858C2214yescomplemented
32#15688858C2214yescomplemented
33#15688858C2214yescomplemented
34C2×C4×D4×C47C2211yescharacteristic
35C47×#435—24no—
36C47×#432C2212yescomplemented
37C47×#407C2212yescomplemented
38C2×C22⋊C8×C47C2212yescomplemented
39#13094069—44no—
40#13094069—44no—
41C2×#10625350—22no—
42C2×#10625352—24no—
43C22×D188C2×C412yes—
44C22×D188C2×C412yescomplemented
45C22×C22⋊D47—22no—
46C22×C22⋊D47—22no—
47C2×D4×D47—24no—
48C23×Dic47—21no—
49#13094024C2×C412yescomplemented
50#13094021C2×C412yescomplemented
51#13094012—22no—
52#13094008—24no—
53#13094008—24no—
54#13094004—24no—
55#13093999—24no—
56#13093998—22no—
57#13093998—22no—
58#13093998—22no—
59#13093998—22no—
60#13093998—22no—
61#13093998C2×C414yes—
62#13093998C2×C414yescomplemented
63#13093998C2314yes—
64#13093994—24no—
65#13093994—24no—
66C4×D188—22no—
67C4×D188C2×C414yescomplemented
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2×C412yes—
70C2×C4⋊Dic47C2×C412yescomplemented
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—21no—
73C2×C22⋊Dic47—21no—
74C2×(C2×D47)⋊C4—22no—
75C2×(C2×D47)⋊C4—22no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78C47⋊#95—21no—
79C47⋊#95—21no—
80C47⋊#95—21no—
81C47⋊#95—21no—
82C47⋊#95—21no—
83C47⋊#95C2311yescharacteristic
84C47⋊#95C2311yescharacteristic
85C47⋊#95C2311yescharacteristic
86C47⋊#95C2311yescharacteristic
87C22×D4×C47D411yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2311yescharacteristic
90C2×M16×C47—22no—
91C22×C8×C47—22no—
92C4×D4×C47C2×C414yes—
93C4×D4×C47D414yes—
94C2×C4⋊C4×C47D411yescharacteristic
95C2×C22⋊C4×C47D411yescharacteristic
96C2×C22⋊C4×C47D411yescharacteristic
97C2×C42×C47C2311yescharacteristic
98C4⋊C8×C47—22no—
99C22⋊C8×C47—22no—
100C22⋊C8×C47—22no—
101C22⋊C8×C47C2314yes—
102#97×C47—22no—
103C24×C47C2×D411yescharacteristic
104C24×C47C2×D411yescharacteristic
105C23×D47—22no—
106C23×D47—22no—
107C2×D4×C47—22no—
108C2×D4×C47—22no—
109C2×D4×C47C22⋊C414yes—
110C22×C4×C47—81no—
111C22×C4×C47—81no—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2411yescharacteristic
121C2×C22⋊D47—24no—
122C2×C22⋊D47—24no—
123C2×C22⋊D47—24no—
124C2×C22⋊D47—44no—
125C2×C22⋊D47—44no—
126C2×C22⋊D47—44no—
127C2×C22⋊D47—44no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—42no—
135C22×Dic47—42no—
136C22×Dic47—81no—
137#10625352—416no—
138D4×D47—416no—
139#10625350—44no—
140#10625350—44no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—82no—
146C2×D188—82no—
147C2×D188C22×C414yes—
148C2×C4×D47—24no—
149C2×Dic94—22no—
150M16×C47—82no—
151C2×C8×C47—24no—
152C2×C8×C47—82no—
153C4⋊C4×C47C22⋊C414yes—
154C22⋊C4×C47C22⋊C414yes—
155C22⋊C4×C47C22⋊C414yes—
156C42×C47C22×C412yes—
157C42×C47C2×D412yes—
158C22⋊Dic47—42no—
159C22⋊Dic47—42no—
160C22⋊Dic47—81no—
161(C2×D47)⋊C4—24no—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—24no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—81no—
169C4⋊Dic47C22×C414yes—
170C4×Dic47—42no—
171C4×Dic47—81no—
172C23×C47—21no—
173C23×C47—21no—
174C23×C47—41no—
175C23×C47—41no—
176C23×C47—41no—
177C23×C47—41no—
178C23×C47C2×C22⋊C412yes—
179C23×C47C2×C22⋊C412yes—
180C23×C47AGL(1,ℤ/8)12yes—
181C23×C47AGL(1,ℤ/8)12yes—
182C23×C47C22×D411yescharacteristic
183C22×D47—24no—
184C22×D47—44no—
185C22×D47—44no—
186C22×D47—82no—
187C22×D47—82no—
188C22×D47—82no—
189C22×D47—82no—
190D4×C47—28no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47C2×C22⋊C412yes—
202C2×C4×C47C2×C22⋊C412yes—
203C2×C4×C47C2×C22⋊C412yes—
204C2×C4×C47C2×C22⋊C412yes—
205C2×C4×C47C23×C411yescharacteristic
206C2×C4×C47C22×D411yescharacteristic
207C22⋊D47—48no—
208C22⋊D47—44no—
209C22⋊D47—44no—
210C22⋊D47—48no—
211C2×Dic47—22no—
212C2×Dic47—24no—
213C2×Dic47—41no—
214C2×Dic47—44no—
215C2×Dic47—41no—
216C2×Dic47—44no—
217C2×Dic47—41no—
218C2×Dic47—41no—
219C2×Dic47—41no—
220C2×Dic47—41no—
221C2×Dic47—41no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—84no—
227D188—44no—
228D188—44no—
229D188—82no—
230D188—82no—
231D188—82no—
232C4×D47—48no—
233Dic94—44no—
234C8×C47—44no—
235#17620—471nomaximal, Sylow
236C22×C47—21no—
237C22×C47—22no—
238C22×C47—21no—
239C22×C47—22no—
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—42no—
244C22×C47—42no—
245C22×C47—81no—
246C22×C47—81no—
247C22×C47#41814yes—
248C22×C47C22×C22⋊C411yescharacteristic
249C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
251C2×D47—44no—
252C2×D47—44no—
253C2×D47—44no—
254C2×D47—164no—
255C2×D47—164no—
256C4×C47—21no—
257C4×C47—21no—
258C4×C47—22no—
259C4×C47—22no—
260C4×C47—22no—
261C4×C47—22no—
262Dic47—42no—
263Dic47—44no—
264Dic47—81no—
265Dic47—81no—
266Dic47—81no—
267#3496—471no—
268#2973D4711yescharacteristic, complemented
269#2353—471no—
270#1519—472no—
271#1514—472no—
272#1510—472no—
273#1507—472no—
274#1505—472no—
275C2×#327—472no—
276C2×C47—21no—
277C2×C47—21no—
278C2×C47—24no—
279C2×C47—42no—
280C2×C47#288311yescharacteristic
281C2×C47C2×#41811yescharacteristic
282C2×C47#290111yescharacteristic
283D47—84no—
284C2×H(5,𝔽2)—941no—
285#547—474no—
286#546—944no—
287D42—474no—
288#537—941no—
289#534—941no—
290#534—941no—
291#532—472no—
292C2×#121—472no—
293C2×#121—472no—
294C2×C4×D4C2×D4711yescharacteristic
295#435—24no—
296#432C2×D4712yescomplemented
297#407C2×D4712yescomplemented
298C2×C22⋊C8C2×D4712yescomplemented
299#388—941no—
300#387—941no—
301#386—941no—
302#383—471no—
303#381—471no—
304#380—471no—
305C4×C22⋊C4—471no—
306C4×C22⋊C4—941no—
307#376—471no—
308C2×#95—471no—
309#327—474no—
310#327—474no—
311#327—474no—
312#327—474no—
313C47#1762011yescharacteristic, complemented, Sylow
314H(5,𝔽2)—1884no—
315H(5,𝔽2)—1884no—
316C2×#41—944no—
317C2×#41—942no—
318C22×D4—472no—
319C22×D4—472no—
320C22×D4—942no—
321C22×D4—942no—
322C22×D4—944no—
323C22×D4D18811yescharacteristic
324C23×C4—941no—
325C23×C4C22×D4711yescharacteristic
326C23×C4C22×D4711yescharacteristic
327C2×M16—22no—
328C22×C8—22no—
329C4⋊Q8—472no—
330#127—472no—
331#124—942no—
332#123—944no—
333#123—944no—
334C22⋊Q8—944no—
335#121—474no—
336#121—474no—
337#121—474no—
338#121—942no—
339#121—942no—
340#121—942no—
341#121—942no—
342#121—942no—
343#121—944no—
344#120—944no—
345#120—944no—
346C4×D4—474no—
347C4×D4—942no—
348C4×D4C4×D4714yes—
349C4×D4C22⋊D4714yes—
350C2×C4⋊C4—472no—
351C2×C4⋊C4—472no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4D18811yescharacteristic
354C2×C22⋊C4—942no—
355C2×C22⋊C4—942no—
356C2×C22⋊C4—941no—
357C2×C22⋊C4—941no—
358C2×C22⋊C4—941no—
359C2×C22⋊C4D18811yescharacteristic
360C2×C22⋊C4D18811yescharacteristic
361C2×C42—941no—
362C2×C42—941no—
363C2×C42C22×D4711yescharacteristic
364C4⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8C22×D4714yes—
368#97—22no—
369#95—471no—
370#95—471no—
371#95—471no—
372#95—471no—
373#95—941no—
374#95—941no—
375#95—941no—
376#95—941no—
377#95—941no—
378C24—942no—
379C24—942no—
380C24C2×D18811yescharacteristic
381C24C2×D18811yescharacteristic
382#41—1884no—
383#41—1884no—
384#41—18816no—
385C2×Q8—942no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—474no—
389C2×D4—944no—
390C2×D4—944no—
391C2×D4—944no—
392C2×D4—942no—
393C2×D4—942no—
394C2×D4—942no—
395C2×D4—942no—
396C2×D4—3762no—
397C2×D4—3762no—
398C2×D4—18816no—
399C2×D4—1884no—
400C2×D4—1884no—
401C2×D4—1884no—
402C2×D4—1884no—
403C2×D4(C2×D47)⋊C414yes—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—941no—
409C22×C4—944no—
410C22×C4—941no—
411C22×C4—941no—
412C22×C4—941no—
413C22×C4—3761no—
414C22×C4—1882no—
415C22×C4—1882no—
416C22×C4C2×C4×D4712yes—
417C22×C4C2×C4×D4712yes—
418C22×C4C2×D18811yescharacteristic
419C22×C4C2×D18811yescharacteristic
420C22×C4C2×D18811yescharacteristic
421C22×C4C2×D18811yescharacteristic
422C22×C4C2×C22⋊D4712yes—
423C22×C4C2×C22⋊D4712yes—
424C22×C4C23×D4711yescharacteristic
425M16—82no—
426C2×C8—24no—
427C2×C8—82no—
428C4⋊C4—474no—
429C4⋊C4—942no—
430C4⋊C4—942no—
431C4⋊C4—942no—
432C4⋊C4—3761no—
433C4⋊C4(C2×D47)⋊C414yes—
434C22⋊C4—944no—
435C22⋊C4—944no—
436C22⋊C4—944no—
437C22⋊C4—944no—
438C22⋊C4—3761no—
439C22⋊C4—1882no—
440C22⋊C4—1882no—
441C22⋊C4(C2×D47)⋊C414yes—
442C22⋊C4(C2×D47)⋊C414yes—
443C42—3761no—
444C42—1882no—
445C42C2×C4×D4712yes—
446C42C2×C22⋊D4712yes—
447C23—21no—
448C23—21no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—944no—
454C23—3762no—
455C23—3762no—
456C23—3762no—
457C23—3762no—
458C23—1884no—
459C23—1884no—
460C23C2×(C2×D47)⋊C412yes—
461C23C2×(C2×D47)⋊C412yes—
462C23#1309405212yes—
463C23#1309405212yes—
464C23C22×D18811yescharacteristic
465Q8—1884no—
466D4—28no—
467D4—3762no—
468D4—3762no—
469D4—3762no—
470D4—1884no—
471D4—1884no—
472D4—1888no—
473D4—1888no—
474D4—1884no—
475D4—1884no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—942no—
487C2×C4—944no—
488C2×C4—3764no—
489C2×C4—1888no—
490C2×C4—1881no—
491C2×C4—1881no—
492C2×C4—1881no—
493C2×C4—1881no—
494C2×C4—1881no—
495C2×C4—1881no—
496C2×C4—1881no—
497C2×C4—1881no—
498C2×C4—1881no—
499C2×C4—1881no—
500C2×C4—1884no—
501C2×C4—1884no—
502C2×C4—1881no—
503C2×C4C2×(C2×D47)⋊C412yes—
504C2×C4C2×(C2×D47)⋊C412yes—
505C2×C4C2×(C2×D47)⋊C412yes—
506C2×C4C2×(C2×D47)⋊C412yes—
507C2×C4C22×C4×D4711yescharacteristic
508C2×C4C22×C22⋊D4711yescharacteristic
509C8—44no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—1884no—
522C22—1884no—
523C22—1884no—
524C22—7524no—
525C22—7524no—
526C22#1568912914yes—
527C22C22×(C2×D47)⋊C411yescharacteristic
528C22C2×#1309405211yescharacteristic
529C22C2×#1309405211yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—3761no—
537C4—3761no—
538C4—3761no—
539C4—1882no—
540C4—1884no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—42no—
545C2—3764no—
546C2#2025184811yescharacteristic
547C2C2×#1568912911yescharacteristic
548C2#2025193311yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2973C2
socleC22×C47C22×C22⋊C4
prime radical#20257152C2
cosocleC22×C4×C47C24
hypercenter#2973D47
hypocenterC47#17620
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20246328C2semidirect
3. #20241957C2semidirect
4. #20241936C2semidirect
5. #20241918C2semidirect
6. #20241905C2semidirect
7. #20241897C2semidirect
8. C2×#15688858C2semidirect
9. C47×#2973C2semidirect
11. #15689777C4semidirect
13. D4×D188C4semidirect
23. #15688991C22semidirect
24. #15688986C22semidirect
25. #15688983C22semidirect
27. C4×C22⋊Dic47C22semidirect
28. #15688969C22semidirect
29. C2×C47⋊#95C22semidirect
30. #15688858C22semidirect
31. #15688858C22semidirect
32. #15688858C22semidirect
33. #15688858C22semidirect
36. C47×#432C22semidirect
37. C47×#407C22semidirect
38. C2×C22⋊C8×C47C22semidirect
44. C22×D188C2×C4semidirect
49. #13094024C2×C4semidirect
50. #13094021C2×C4semidirect
62. #13093998C2×C4semidirect
67. C4×D188C2×C4semidirect
70. C2×C4⋊Dic47C2×C4semidirect
268. #2973D47semidirect
296. #432C2×D47semidirect
297. #407C2×D47semidirect
298. C2×C22⋊C8C2×D47semidirect
313. C47#17620semidirect

Maximal subgroups

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

subgroupindex
#202571522
#202463282
#202419572
#202419362
#202419182
#202419052
#202418972
C2×#156888582
C47×#29732
#1762047