Finite Group Databasebeta

← #26139039

Subgroups — #26139039

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688982C211yesmaximal, characteristic
2C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
3C2×#95×D47C211yesmaximal, characteristic
4#20241646C212yesmaximal, complemented
5#20241643C212yesmaximal, complemented
6#20241615C211yesmaximal, characteristic, complemented
7#20241614C211yesmaximal, characteristic, complemented
8#20241614C211yesmaximal, characteristic, complemented
9(C2×D47)⋊#95C212yesmaximal
10(C2×D47)⋊#95C212yesmaximal
11C47×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D47C2212yes—
13C23×C4×D47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×(C2×D47)⋊C4C2212yes—
16C22×(C2×D47)⋊C4C2211yescharacteristic
17C22×(C2×D47)⋊C4C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19#15688982—24no—
20C4×(C2×D47)⋊C4C418yescomplemented
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2211yescharacteristic
31C2×C47⋊#95C2211yescharacteristic
32#95×D47—24no—
33C22×C42×C47C2211yescharacteristic
34C2×#95×C47C2211yescharacteristic
35C2×#95×C47C2212yes—
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C24×D47C2311yescharacteristic
39C23×Dic47C2312yes—
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2312yes—
43C23×Dic47C2311yescharacteristic
44C22×C4×D47—24no—
45C22×C4×D47—24no—
46C22×C4×D47—24no—
47C22×C4×D47—24no—
48C22×C4×D47—24no—
49C22×C4×D47—42no—
50C22×C4×D47—42no—
51C22×C4×D47—42no—
52C22×C4×D47—42no—
53C22×C4×D47C2×C414yes—
54C22×C4×D47D414yes—
55C22×C4×D47Q814yes—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—42no—
58C2×Dic47⋊C4—42no—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—22no—
63C2×(C2×D47)⋊C4—24no—
64C2×(C2×D47)⋊C4—24no—
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4C2×C414yes—
67C2×(C2×D47)⋊C4C2×C414yes—
68C2×C4×Dic47—42no—
69C2×C4×Dic47C2×C414yescomplemented
70C47⋊#95—22no—
71C47⋊#95—22no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—42no—
77C47⋊#95—42no—
78C47⋊#95—42no—
79C47⋊#95C2×C414yescomplemented
80C47⋊#95C2×C414yescomplemented
81C23×C4×C47C2312yes—
82C23×C4×C47C2311yescharacteristic
83C23×C4×C47C2311yescharacteristic
84C23×C4×C47C2312yes—
85C23×C4×C47C2311yescharacteristic
86C2×C42×C47—42no—
87C2×C42×C47C2×C414yescomplemented
88#95×C47—22no—
89#95×C47—22no—
90#95×C47—24no—
91#95×C47—42no—
92C24×C47C2411yescharacteristic
93C23×D47—22no—
94C23×D47—22no—
95C23×D47—22no—
96C23×D47—22no—
97C23×D47C22×C412yes—
98C23×D47C2×D412yes—
99C23×D47C2×Q812yes—
100C23×D47#4112yes—
101C23×D47#4112yes—
102C23×D47#4112yes—
103C23×D47#4112yes—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×Q812yes—
132C22×C4×C47#4114yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47C22×C412yes—
157C22×Dic47C22×C412yes—
158C22×Dic47C22×C412yes—
159C22×Dic47C2×D412yes—
160C22×Dic47C2×Q812yes—
161C22×Dic47#4114yes—
162C2×C4×D47—216no—
163C2×C4×D47—44no—
164C2×C4×D47—44no—
165C2×C4×D47—48no—
166C2×C4×D47—48no—
167C2×C4×D47—48no—
168C2×C4×D47—48no—
169C2×C4×D47—48no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C2×C4×D47—48no—
173C2×C4×D47C4⋊C4116yes—
174C42×C47—44no—
175(C2×D47)⋊C4—28no—
176(C2×D47)⋊C4—28no—
177(C2×D47)⋊C4—48no—
178Dic47⋊C4—48no—
179C4×Dic47—44no—
180C23×C47C23×C411yescharacteristic
181C23×C47C22×D411yescharacteristic
182C23×C47C22×Q811yescharacteristic
183C23×C47C2×#4111yescharacteristic
184C23×C47C2×#4111yescharacteristic
185C23×C47C2×#4111yescharacteristic
186C23×C47C2×#4111yescharacteristic
187C23×C47C2×#4112yes—
188C23×C47C2×#4112yes—
189C23×C47H(5,𝔽2)12yes—
190C23×C47H(5,𝔽2)12yes—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—24no—
202C22×D47—24no—
203C22×D47—24no—
204C22×D47—24no—
205C22×D47—24no—
206C22×D47—24no—
207C22×D47C2×C4⋊C414yes—
208C22×D47#11714yes—
209C22×D47#11714yes—
210C22×D47#12414yes—
211C22×D47#12414yes—
212C22×D47#12514yes—
213C22×D47#12514yes—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47C2×C4⋊C414yes—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47C2×C4⋊C414yes—
272C4×D47—416no—
273C4×D47—432no—
274#13966—471nomaximal, Sylow
275C22×C47C22×C4⋊C411yescharacteristic
276C22×C47C2×#11711yescharacteristic
277C22×C47C2×#11711yescharacteristic
278C22×C47C4×#4111yescharacteristic
279C22×C47C4×#4111yescharacteristic
280C22×C47#51811yescharacteristic
281C22×C47#51811yescharacteristic
282C22×C47#52512yes—
283C22×C47C2×#12411yescharacteristic
284C22×C47C2×#12411yescharacteristic
285C22×C47C2×#12511yescharacteristic
286C22×C47C2×#12511yescharacteristic
287C22×C47#53312yes—
288C22×C47#53412yes—
289C22×C47#54312yes—
290C22×C47#55112yes—
291C22×C47#55112yes—
292C22×C47#55112yes—
293C22×C47#55112yes—
294C22×C47#55312yes—
295C22×C47#55312yes—
296C22×C47#55312yes—
297C22×C47#55312yes—
298C2×D47—28no—
299C2×D47—28no—
300C2×D47—28no—
301C2×D47—24no—
302C2×D47—28no—
303C2×D47—24no—
304C2×D47—24no—
305C2×D47—24no—
306C2×D47—24no—
307C2×D47—24no—
308C2×D47#38218yes—
309C4×C47—24no—
310C4×C47—44no—
311C4×C47—42no—
312C4×C47—42no—
313C4×C47—82no—
314Dic47—24no—
315Dic47—44no—
316Dic47—42no—
317Dic47—42no—
318Dic47—82no—
319C2×#380—471no—
320C2×C4×C22⋊C4—471no—
321C22×#95—471no—
322#1442—472no—
323#1442—472no—
324#1434—471no—
325#1434—471no—
326#1434—471no—
327#1434D4711yescharacteristic, complemented
328C22⋊#95—472no—
329C22⋊#95—472no—
330C2×C47C2×#38211yescharacteristic
331C2×C47#229611yescharacteristic
332C2×C47#231511yescharacteristic
333C2×C47#235311yescharacteristic
334C2×C47#235311yescharacteristic
335C2×C47#235311yescharacteristic
336C2×C47#235311yescharacteristic
337C2×C47#256112yes—
338C2×C47#256912yes—
339C2×C47#256912yes—
340C2×C47#259112yes—
341D47—28no—
342C24×C4—471no—
343C24×C4—472no—
344C22×C4⋊C4—472no—
345C22×C22⋊C4—471no—
346C22×C22⋊C4—471no—
347C22×C22⋊C4—472no—
348C22×C42—471no—
349C22×C42C2×D4711yescharacteristic
350#380—944no—
351C4×C22⋊C4—478no—
352C2×#95—471no—
353C2×#95—471no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95—471no—
358C2×#95—471no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—472no—
363C2×#95—944no—
364C2×#95C2×D4712yes—
365C2×#95C2×D4711yescharacteristic
366C2×#95C2×D4711yescharacteristic
367C2×#95C2×D4712yes—
368C47#1396611yescharacteristic, complemented, Sylow
369C25—471no—
370C23×C4—471no—
371C23×C4—474no—
372C23×C4—474no—
373C23×C4—474no—
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—472no—
377C23×C4—472no—
378C23×C4—944no—
379C23×C4—944no—
380C23×C4—944no—
381C23×C4—944no—
382C23×C4—944no—
383C23×C4—1882no—
384C23×C4—1882no—
385C23×C4—1882no—
386C23×C4—1882no—
387C23×C4C22×D4712yes—
388C23×C4C22×D4711yescharacteristic
389C23×C4C22×D4711yescharacteristic
390C23×C4C22×D4712yes—
391C23×C4C22×D4711yescharacteristic
392C2×C4⋊C4—944no—
393C2×C4⋊C4—1882no—
394C2×C4⋊C4—1882no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—474no—
397C2×C22⋊C4—942no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C22⋊C4—942no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—944no—
404C2×C42—42no—
405C2×C42—474no—
406C2×C42—1882no—
407C2×C42C4×D4714yescomplemented
408#95—24no—
409#95—22no—
410#95—22no—
411#95—42no—
412#95—474no—
413#95—474no—
414#95—944no—
415#95—944no—
416#95—944no—
417#95—944no—
418#95—942no—
419#95—942no—
420#95—1882no—
421#95—1882no—
422#95—1882no—
423C24—472no—
424C24—472no—
425C24—472no—
426C24—472no—
427C24—472no—
428C24—472no—
429C24—472no—
430C24—942no—
431C24—942no—
432C24—942no—
433C24—942no—
434C24C23×D4711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—4716no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—474no—
465C22×C4—9416no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—1888no—
490C22×C4—1884no—
491C22×C4—1884no—
492C22×C4—1888no—
493C22×C4—1888no—
494C22×C4—1888no—
495C22×C4—1888no—
496C22×C4—1888no—
497C22×C4—1888no—
498C22×C4—1888no—
499C22×C4C2×C4×D4712yes—
500C22×C4C2×C4×D4712yes—
501C22×C4C2×C4×D4712yes—
502C22×C4#1062535014yes—
503C22×C4D4×D4712yes—
504C22×C4Q8×D4712yes—
505C4⋊C4—1888no—
506C22⋊C4—948no—
507C22⋊C4—948no—
508C22⋊C4—1888no—
509C42—44no—
510C42—1884no—
511C23—474no—
512C23—474no—
513C23—474no—
514C23—474no—
515C23—474no—
516C23—474no—
517C23—474no—
518C23—944no—
519C23—944no—
520C23—944no—
521C23—944no—
522C23—944no—
523C23—944no—
524C23—944no—
525C23—944no—
526C23—944no—
527C23—942no—
528C23—942no—
529C23—942no—
530C23—942no—
531C23—944no—
532C23—944no—
533C23—944no—
534C23C22×C4×D4711yescharacteristic
535C23C2×D4×D4711yescharacteristic
536C23C2×Q8×D4711yescharacteristic
537C23#41×D4711yescharacteristic
538C23#41×D4711yescharacteristic
539C23#41×D4712yes—
540C23#41×D4711yescharacteristic
541C23#41×D4711yescharacteristic
542C23C2×#1062535012yes—
543C23#1309406912yes—
544C23#1309407012yes—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—474no—
574C2×C4—944no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—944no—
587C2×C4—18816no—
588C2×C4—18832no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4C4⋊C4×D4714yes—
605C22—478no—
606C22—948no—
607C22—948no—
608C22—948no—
609C22—944no—
610C22—944no—
611C22—944no—
612C22—944no—
613C22—944no—
614C22—948no—
615C22—944no—
616C22C2×C4⋊C4×D4711yescharacteristic
617C22#117×D4711yescharacteristic
618C22#117×D4711yescharacteristic
619C22#1568959111yescharacteristic
620C22C4×#1062535011yescharacteristic
621C22#1568959611yescharacteristic
622C22#1568959711yescharacteristic
623C22#1568963212yes—
624C22#124×D4711yescharacteristic
625C22#124×D4711yescharacteristic
626C22#125×D4711yescharacteristic
627C22#125×D4711yescharacteristic
628C22#1568967712yes—
629C22#1568968112yes—
630C22#1568974812yes—
631C22#1568979712yes—
632C22#1568979812yes—
633C22#1568980112yes—
634C22#1568980212yes—
635C22#1568981712yes—
636C22#1568981812yes—
637C22#1568982312yes—
638C22#1568982412yes—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—944no—
645C4—3762no—
646C4—1882no—
647C4—1882no—
648C4—1884no—
649C2—948no—
650C2#382×D4711yescharacteristic
651C2#2024586811yescharacteristic
652C2#2024600111yescharacteristic
653C2#2024632111yescharacteristic
654C2#2024632511yescharacteristic
655C2#2024632911yescharacteristic
656C2#2024633111yescharacteristic
657C2#2024854412yes—
658C2#2024861012yes—
659C2#2024861112yes—
660C2#2024886612yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1434C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1434D47
hypocenterC47#13966
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #20241646C2semidirect
5. #20241643C2semidirect
6. #20241615C2semidirect
7. #20241614C2semidirect
8. #20241614C2semidirect
11. C47×#1434C2semidirect
20. C4×(C2×D47)⋊C4C4semidirect
69. C2×C4×Dic47C2×C4semidirect
79. C47⋊#95C2×C4semidirect
80. C47⋊#95C2×C4semidirect
87. C2×C42×C47C2×C4semidirect
327. #1434D47semidirect
368. C47#13966semidirect
407. C2×C42C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889822
C2×C4×(C2×D47)⋊C42
C2×#95×D472
#202416462
#202416432
#202416152
#202416142
(C2×D47)⋊#952
C47×#14342
#1396647