Finite Group Databasebeta

← #26146987

Subgroups — #26146987

Order 12032.

1434 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689048C211yesmaximal, characteristic
2C2×#15688995C212yesmaximal
3#20241651C212yesmaximal, complemented
4#20241649C212yesmaximal, complemented
5#20241646C212yesmaximal, complemented
6#20241640C212yesmaximal, complemented
7#20241635C212yesmaximal
8#20241610C211yesmaximal, characteristic
9C47×#1439C211yesmaximal, characteristic
10C24×Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4C2212yes—
13C22×Dic47⋊C4C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C22⋊Dic47C2211yescharacteristic
16#15689048C418yescomplemented
17#15688995—28no—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2212yes—
28C22×C4⋊C4×C47C2211yescharacteristic
29C22×C22⋊C4×C47C2212yes—
30C22×C22⋊C4×C47C2211yescharacteristic
31C2×#95×C47C2212yes—
32C2×#95×C47C2211yescharacteristic
33C23×Dic47—42no—
34C23×Dic47—42no—
35C23×Dic47—41no—
36C23×Dic47—41no—
37C23×Dic47D414yes—
38C23×Dic47C2312yes—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C2×C4⋊Dic47—41no—
43C2×C4⋊Dic47—41no—
44C2×C4⋊Dic47C2×C414yescomplemented
45C2×Dic47⋊C4—24no—
46C2×Dic47⋊C4—24no—
47C2×Dic47⋊C4—42no—
48C2×Dic47⋊C4—42no—
49C2×Dic47⋊C4—42no—
50C2×Dic47⋊C4—42no—
51C2×C22⋊Dic47—24no—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—22no—
54C2×C22⋊Dic47—41no—
55C2×C22⋊Dic47—41no—
56C2×C22⋊Dic47C2×C418yes—
57C47⋊#95—24no—
58C47⋊#95—24no—
59C47⋊#95—42no—
60C47⋊#95—42no—
61C47⋊#95—42no—
62C47⋊#95—42no—
63C47⋊#95—42no—
64C47⋊#95—42no—
65C47⋊#95C2×C414yescomplemented
66C47⋊#95C2×C418yescomplemented
67C25×C47C2311yescharacteristic
68C23×C4×C47C2312yes—
69C23×C4×C47C2312yes—
70C23×C4×C47C2311yescharacteristic
71C23×C4×C47C2311yescharacteristic
72C2×C4⋊C4×C47—41no—
73C2×C4⋊C4×C47—42no—
74C2×C22⋊C4×C47—24no—
75C2×C22⋊C4×C47—41no—
76C2×C22⋊C4×C47—41no—
77C2×C22⋊C4×C47—42no—
78C2×C22⋊C4×C47—42no—
79C2×C22⋊C4×C47C2×C414yes—
80#95×C47—41no—
81#95×C47—42no—
82C24×C47—21no—
83C24×C47—21no—
84C24×C47—22no—
85C24×C47—22no—
86C24×C47—21no—
87C24×C47—22no—
88C24×C47—22no—
89C24×C47—21no—
90C24×C47C22×C412yes—
91C24×C47C2×D412yes—
92C24×C47#4112yes—
93C24×C47C2411yescharacteristic
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C22×C412yes—
118C22×Dic47—22no—
119C22×Dic47—22no—
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—44no—
144C22×Dic47—48no—
145C22×Dic47—44no—
146C22×Dic47—48no—
147C22×Dic47—44no—
148C22×Dic47C22×C414yes—
149C22×Dic47C22×C414yes—
150C22×Dic47C2×D414yes—
151C4⋊C4×C47—44no—
152C22⋊C4×C47—44no—
153C22⋊C4×C47—48no—
154C22⋊Dic47—216no—
155C22⋊Dic47—44no—
156C4⋊Dic47—44no—
157Dic47⋊C4—48no—
158Dic47⋊C4—48no—
159C23×C47—22no—
160C23×C47—24no—
161C23×C47—24no—
162C23×C47—22no—
163C23×C47—24no—
164C23×C47—22no—
165C23×C47—24no—
166C23×C47—22no—
167C23×C47—22no—
168C23×C47—24no—
169C23×C47—24no—
170C23×C47—22no—
171C23×C47—41no—
172C23×C47—42no—
173C23×C47—41no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—41no—
177C23×C47—42no—
178C23×C47—42no—
179C23×C47—41no—
180C23×C47—42no—
181C23×C47C4×D414yes—
182C23×C47C23×C411yescharacteristic
183C23×C47C22×D411yescharacteristic
184C23×C47C2×#4111yescharacteristic
185C23×C47H(5,𝔽2)11yescharacteristic
186C23×C47H(5,𝔽2)11yescharacteristic
187C23×C47H(5,𝔽2)12yes—
188C23×C47H(5,𝔽2)12yes—
189C23×C47H(5,𝔽2)11yescharacteristic
190C23×C472−1+412yes—
191C23×C472−1+412yes—
192C23×C472−1+411yescharacteristic
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×Dic47—24no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—24no—
221C2×Dic47—24no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—416no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246#14627—471nomaximal, Sylow
247C22×C47—28no—
248C22×C47—24no—
249C22×C47—42no—
250C22×C47—44no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—44no—
256C22×C47—42no—
257C22×C47—44no—
258C22×C47C2×C4×D411yescharacteristic
259C22×C47#51811yescharacteristic
260C22×C47#52012yes—
261C22×C47#52012yes—
262C22×C47#52011yescharacteristic
263C22×C47#53411yescharacteristic
264C22×C47#53412yes—
265C22×C47#53612yes—
266C22×C47#53711yescharacteristic
267C22×C47#53811yescharacteristic
268C22×C47#53912yes—
269C22×C47#53912yes—
270C22×C47#53911yescharacteristic
271C22×C47#56011yescharacteristic
272C22×C47#56012yes—
273C22×C47H(3,𝔽4)11yescharacteristic
274C22×C47#56212yes—
275C22×C47#56212yes—
276C22×C47#56311yescharacteristic
277C22×C47#56312yes—
278C22×C47#56312yes—
279C22×C47#56311yescharacteristic
280C22×C47#56312yes—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—82no—
284C4×C47—82no—
285Dic47—44no—
286Dic47—44no—
287Dic47—44no—
288Dic47—82no—
289C2×#397—471no—
290C2×#385—472no—
291#1444—472no—
292#1443—472no—
293#1442—472no—
294#1441—472no—
295#1440—472no—
296#1439D4711yescharacteristic
297C22⋊#95—471no—
298C2×C47—44no—
299C2×C47#232212yes—
300C2×C47#232511yescharacteristic
301C2×C47#237611yescharacteristic
302C2×C47#237911yescharacteristic
303C2×C47#238312yes—
304C2×C47#281211yescharacteristic
305C2×C47#281412yes—
306C2×C47#281711yescharacteristic
307C2×C47#282712yes—
308C2×C47#283411yescharacteristic
309C2×C47#283511yescharacteristic
310C24×C4—471no—
311C22×C4⋊C4—471no—
312C22×C4⋊C4—472no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C22⋊C4—472no—
316C22×C22⋊C4—471no—
317C22×C22⋊C4C2×D4712yes—
318C22×C22⋊C4C2×D4711yescharacteristic
319#397—478no—
320#385—948no—
321C2×#95—471no—
322C2×#95—472no—
323C2×#95—472no—
324C2×#95—472no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95—472no—
330C2×#95—472no—
331C2×#95C2×D4712yes—
332C2×#95C2×D4711yescharacteristic
333C47#1462711yescharacteristic, complemented, Sylow
334C25C22×D4711yescharacteristic
335C23×C4—472no—
336C23×C4—472no—
337C23×C4—472no—
338C23×C4—474no—
339C23×C4—472no—
340C23×C4—1882no—
341C23×C4—1882no—
342C23×C4—1881no—
343C23×C4—1881no—
344C23×C4C22×D4712yes—
345C23×C4C22×D4712yes—
346C23×C4C22×D4711yescharacteristic
347C23×C4C22×D4711yescharacteristic
348C2×C4⋊C4—42no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—474no—
351C2×C4⋊C4—944no—
352C2×C4⋊C4—944no—
353C2×C4⋊C4—1882no—
354C2×C4⋊C4—1882no—
355C2×C4⋊C4—1882no—
356C2×C4⋊C4—1882no—
357C2×C4⋊C4—1881no—
358C2×C4⋊C4—1881no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—478no—
365C2×C22⋊C4—944no—
366C2×C22⋊C4—944no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—1881no—
369C2×C22⋊C4—1881no—
370C2×C22⋊C4C4×D4714yes—
371#95—42no—
372#95—41no—
373#95—474no—
374#95—478no—
375#95—944no—
376#95—944no—
377#95—1882no—
378#95—1882no—
379#95—1882no—
380#95—1882no—
381#95—1882no—
382#95—1882no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24C2×C4×D4712yes—
392C24D4×D4712yes—
393C24#1062535412yes—
394C24C23×D4711yescharacteristic
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—474no—
418C22×C4—474no—
419C22×C4—474no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—1884no—
446C22×C4—1888no—
447C22×C4—1884no—
448C22×C4—1888no—
449C22×C4—1884no—
450C22×C4C2×C4×D4712yes—
451C22×C4C2×C4×D4712yes—
452C4⋊C4—44no—
453C4⋊C4—1884no—
454C4⋊C4—1888no—
455C4⋊C4—1888no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—9416no—
459C22⋊C4—1884no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23#1309398614yes—
483C23C22×C4×D4711yescharacteristic
484C23C2×D4×D4711yescharacteristic
485C23C2×#1062535411yescharacteristic
486C23#1309406911yescharacteristic
487C23#1309406911yescharacteristic
488C23#1309406912yes—
489C23#1309406912yes—
490C23#1309407011yescharacteristic
491C23#1309407112yes—
492C23#1309407112yes—
493C23#1309407111yescharacteristic
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—944no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—18816no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C22—28no—
548C22—24no—
549C22—42no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22C2×#1309398611yescharacteristic
559C22#1568959611yescharacteristic
560C22#1568960212yes—
561C22#1568960212yes—
562C22#1568960311yescharacteristic
563C22#1568968111yescharacteristic
564C22#1568968312yes—
565C22#1568969512yes—
566C22#1568970011yescharacteristic
567C22#1568971311yescharacteristic
568C22#1568972212yes—
569C22#1568972212yes—
570C22#1568972211yescharacteristic
571C22#1568986011yescharacteristic
572C22#1568986112yes—
573C22#1568986611yescharacteristic
574C22#1568986812yes—
575C22#1568986812yes—
576C22#1568987311yescharacteristic
577C22#1568987312yes—
578C22#1568987512yes—
579C22#1568987612yes—
580C22#1568987611yescharacteristic
581C4—42no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—3762no—
586C4—1884no—
587C4—1884no—
588C4—1884no—
589C2—44no—
590C2#2024605612yes—
591C2#2024608811yescharacteristic
592C2#2024653811yescharacteristic
593C2#2024655811yescharacteristic
594C2#2024657512yes—
595C2#2025138211yescharacteristic
596C2#2025140512yes—
597C2#2025142611yescharacteristic
598C2#2025150212yes—
599C2#2025157411yescharacteristic
600C2#2025157911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1439C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1439D47
hypocenterC47#14627
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #20241651C2semidirect
4. #20241649C2semidirect
5. #20241646C2semidirect
6. #20241640C2semidirect
16. #15689048C4semidirect
44. C2×C4⋊Dic47C2×C4semidirect
65. C47⋊#95C2×C4semidirect
66. C47⋊#95C2×C4semidirect
333. C47#14627semidirect

Maximal subgroups

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

subgroupindex
C2×#156890482
C2×#156889952
#202416512
#202416492
#202416462
#202416402
#202416352
#202416102
C47×#14392
#1462747