Finite Group Databasebeta

← D47×#1805 #26215739

Subgroups — D47×#1805 #26215739

Order 12032.

1510 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D47C211yesmaximal, characteristic
2C2×C4⋊C8×D47C211yesmaximal, characteristic
3C2×#103×D47C212yesmaximal
4#382×D47C211yesmaximal, characteristic
5#20243404C211yesmaximal, characteristic, complemented
6#20243403C211yesmaximal, characteristic, complemented
7#20243402C212yesmaximal, complemented
8#20243401C212yesmaximal, complemented
9#20243400C211yesmaximal, characteristic, complemented
10#340×D47C212yesmaximal
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C22×C8×D47C2212yes—
14C2×Q8×Dic47C2211yescharacteristic
15C2×C4×Dic94—21no—
16C2×C4×Dic94C2211yescharacteristic
17C2×#13093989—21no—
18C2×#13093989C2211yescharacteristic
19C4×Q8×D47—24no—
20C4×Q8×D47C418yescomplemented
21C2×C4⋊C4×D47—21no—
22C2×C4⋊C4×D47C2212yes—
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×Dic47⋊C8C2212yes—
28C4⋊C8×D47—24no—
29C2×#13093949C2212yes—
30C2×#13093948C2212yes—
31C2×#13093947C2212yes—
32#103×D47—28no—
33#15688989C2211yescharacteristic
34#15688988C2212yes—
35#15688987—21no—
36#15688987—21no—
37#95×D47—21no—
38#95×D47—21no—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4⋊C8×C47C2211yescharacteristic
44C2×#103×C47C2212yes—
45C47×#382C2211yescharacteristic
46C47×#340C2212yes—
47C22×Dic94—21no—
48C22×Dic94C2311yescharacteristic
49C2×Q8×D47—22no—
50C2×Q8×D47—22no—
51C2×Q8×D47—22no—
52C2×Q8×D47C2×C414yes—
53C2×Q8×D47D414yes—
54C2×Q8×D47Q814yes—
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47—22no—
58C22×C4×D47D411yescharacteristic
59C22×C4×D47D411yescharacteristic
60C22×C4×D47C2311yescharacteristic
61C22×C47⋊C8C2312yes—
62C2×C8×D47—24no—
63C2×C8×D47—24no—
64C2×C8×D47—24no—
65Q8×Dic47—22no—
66Q8×Dic47C2×C414yescomplemented
67C4×Dic94—22no—
68C4×Dic94—28no—
69C4×Dic94C2×C414yescomplemented
70#13093989—28no—
71#13093989—22no—
72#13093989C2×C414yescomplemented
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2312yes—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—22no—
80C4⋊C4×D47—28no—
81C4⋊C4×D47—22no—
82C4⋊C4×D47—24no—
83C4⋊C4×D47—24no—
84C4⋊C4×D47C2×C414yes—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C42×D47—28no—
92C42×D47C2×C414yes—
93C42×D47D414yes—
94C4⋊(C47⋊C8)—22no—
95Dic47⋊C8—24no—
96#13093949—24no—
97#13093948—24no—
98#13093947—24no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—22no—
102C47⋊#95—22no—
103C22×Q8×C47C2311yescharacteristic
104C22×C8×C47C2312yes—
105C4×Q8×C47—22no—
106C4×Q8×C47C2×C414yescomplemented
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2312yes—
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112C4⋊C8×C47—22no—
113#103×C47—24no—
114#95×C47—21no—
115#95×C47—21no—
116C23×D47C2×D411yescharacteristic
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47C22×C412yes—
121C2×Q8×C47C2×D412yes—
122C2×Q8×C47C2×Q812yes—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—22no—
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C22×Dic47C2×D411yescharacteristic
135Q8×D47—28no—
136Q8×D47—28no—
137Q8×D47—28no—
138Q8×D47C4⋊C4116yes—
139C2×C4×D47—24no—
140C2×C4×D47—24no—
141C2×C4×D47—24no—
142C2×C4×D47—24no—
143C2×C4×D47—24no—
144C2×C4×D47—24no—
145C2×C4×D47—44no—
146C2×C4×D47—44no—
147C2×C4×D47—44no—
148C2×C4×D47—44no—
149C2×C4×D47C22⋊C414yes—
150C2×C4×D47C22⋊C414yes—
151C2×C4×D47C22×C412yes—
152C2×C4×D47C2×D412yes—
153C2×C4×D47C2×D412yes—
154C2×C4×D47C2×Q812yes—
155C2×C4×D47#4112yes—
156C2×C4×D47#4112yes—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—24no—
161C2×Dic94—24no—
162C2×Dic94—24no—
163C2×Dic94C22×C412yes—
164C2×Dic94C2×D412yes—
165C2×Dic94C2×Q812yes—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C4⋊C4×C47—24no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47C22×C412yes—
174C42×C47—24no—
175C42×C47C22×C412yes—
176C42×C47C2×D412yes—
177C4⋊Dic47—21no—
178C4⋊Dic47—24no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47C22×C412yes—
182Dic47⋊C4—24no—
183Dic47⋊C4—24no—
184Dic47⋊C4—24no—
185Dic47⋊C4—44no—
186C4×Dic47—24no—
187C4×Dic47—24no—
188C4×Dic47—24no—
189C4×Dic47C22⋊C414yes—
190C4×Dic47C22⋊C414yes—
191C4×Dic47C22×C412yes—
192C4×Dic47C2×D412yes—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C8×D47—48no—
197C23×C47C22×D411yescharacteristic
198C22×D47C2×C22⋊C412yes—
199C22×D47#12112yes—
200C22×D47C22⋊Q812yes—
201C22×D47#13514yes—
202C22×D47#13714yes—
203Q8×C47—22no—
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47C2×C4⋊C414yes—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47C2×C22⋊C412yes—
218C2×C4×C47C2×C22⋊C412yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C22×Q811yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47C2×#4111yescharacteristic
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47C2×C22⋊C412yes—
236C2×Dic47C2×C22⋊C412yes—
237C2×Dic47C2×C22⋊C412yes—
238C2×Dic47#12112yes—
239C2×Dic47C22⋊Q812yes—
240C4×D47—216no—
241C4×D47—28no—
242C4×D47—28no—
243C4×D47—48no—
244C4×D47—88no—
245C4×D47C2×C4⋊C414yes—
246C4×D47#11714yes—
247C4×D47#12114yes—
248C4×D47C22⋊Q814yes—
249Dic94—22no—
250Dic94—22no—
251Dic94—22no—
252Dic94—216no—
253Dic94C2×C4⋊C414yes—
254C8×C47—42no—
255C47⋊C8—42no—
256C2×#1805—471nomaximal, Sylow
257C22×C47C22×C22⋊C411yescharacteristic
258C22×C47C2×#12111yescharacteristic
259C22×C47C2×C22⋊Q811yescharacteristic
260C22×C47C2×#13512yes—
261C22×C47C2×#13712yes—
262C2×D47#38014yes—
263C2×D47#41614yes—
264C2×D47#41914yes—
265C2×D47#46418yes—
266C2×D47#47918yes—
267C4×C47—22no—
268C4×C47—22no—
269C4×C47—24no—
270C4×C47—42no—
271C4×C47—82no—
272C4×C47C22×C4⋊C411yescharacteristic
273C4×C47C2×#11711yescharacteristic
274C4×C47C2×#12111yescharacteristic
275C4×C47C2×C22⋊Q811yescharacteristic
276Dic47—22no—
277Dic47—22no—
278Dic47—24no—
279Dic47—42no—
280Dic47—82no—
281Dic47#38014yes—
282C22×C4×Q8—471no—
283C22×C4⋊C8—471no—
284C22×#103—472no—
285C2×#382—471no—
286#1805—471no—
287#1805—472no—
288#1805—471no—
289#1805—472no—
290#1805—471no—
291#1805D4711yescharacteristic, direct product factor
292C2×#340—472no—
293C2×C47C2×#38011yescharacteristic
294C2×C47C2×#41611yescharacteristic
295C2×C47C2×#41911yescharacteristic
296C2×C47C2×#46412yes—
297C2×C47C2×#47912yes—
298D47#180518yesdirect product factor
299C23×Q8—471no—
300C23×C8—472no—
301C2×C4×Q8—471no—
302C2×C4×Q8—478no—
303C2×C4×Q8—471no—
304C2×C4×Q8—471no—
305C2×C4×Q8—941no—
306C2×C4×Q8—941no—
307C2×C4×Q8—944no—
308C2×C4×Q8C2×D4711yescharacteristic
309C22×C4⋊C4—471no—
310C22×C4⋊C4—472no—
311C22×C4⋊C4—941no—
312C22×C42—471no—
313C22×C42—941no—
314C2×C4⋊C8—472no—
315C2×C4⋊C8—471no—
316C2×C4⋊C8—944no—
317C2×C4⋊C8C2×D4711yescharacteristic
318C2×#103—472no—
319C2×#103—472no—
320C2×#103—472no—
321C2×#103—948no—
322C2×#103C2×D4712yes—
323#382—472no—
324#382—471no—
325#382—941no—
326#382—941no—
327#382C2×D4711yescharacteristic
328C2×#95—941no—
329C2×#95—941no—
330#340—472no—
331#340—472no—
332#340—472no—
333#340C2×D4712yes—
334C47C2×#180511yescharacteristic, complemented, Sylow
335C22×Q8—474no—
336C22×Q8—474no—
337C22×Q8—474no—
338C22×Q8—471no—
339C22×Q8—941no—
340C22×Q8—942no—
341C22×Q8—942no—
342C22×Q8—942no—
343C22×Q8C22×D4711yescharacteristic
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—941no—
348C23×C4—941no—
349C23×C4—942no—
350C22×C8—472no—
351C22×C8—944no—
352C22×C8—944no—
353C22×C8—944no—
354C22×C8C22×D4712yes—
355C4×Q8—22no—
356C4×Q8—474no—
357C4×Q8—474no—
358C4×Q8—474no—
359C4×Q8—948no—
360C4×Q8—948no—
361C4×Q8—942no—
362C4×Q8—942no—
363C4×Q8—942no—
364C4×Q8C4×D4714yescomplemented
365C2×C4⋊C4—21no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—948no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4C22×D4712yes—
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C42—21no—
381C2×C42—471no—
382C2×C42—471no—
383C2×C42—471no—
384C2×C42—474no—
385C2×C42—474no—
386C2×C42—948no—
387C2×C42—941no—
388C2×C42—941no—
389C2×C42—941no—
390C2×C42C22×D4711yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—944no—
393C4⋊C8—942no—
394#103—24no—
395#103—944no—
396#103—944no—
397#103—944no—
398#95—21no—
399#95—21no—
400#95—942no—
401#95—942no—
402#95—941no—
403#95—941no—
404C24—471no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—4716no—
409C2×Q8—472no—
410C2×Q8—472no—
411C2×Q8—472no—
412C2×Q8—944no—
413C2×Q8—944no—
414C2×Q8—944no—
415C2×Q8—948no—
416C2×Q8—948no—
417C2×Q8—948no—
418C2×Q8—941no—
419C2×Q8—941no—
420C2×Q8—941no—
421C2×Q8C2×C4×D4712yes—
422C2×Q8D4×D4712yes—
423C2×Q8Q8×D4712yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—471no—
428C22×C4—472no—
429C22×C4—472no—
430C22×C4—472no—
431C22×C4—472no—
432C22×C4—472no—
433C22×C4—472no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—474no—
437C22×C4—474no—
438C22×C4—944no—
439C22×C4—944no—
440C22×C4—944no—
441C22×C4—944no—
442C22×C4—944no—
443C22×C4—944no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—942no—
447C22×C4—1884no—
448C22×C4—1884no—
449C22×C4—1884no—
450C22×C4—1884no—
451C22×C4D4×D4711yescharacteristic
452C22×C4D4×D4711yescharacteristic
453C22×C4C23×D4711yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—942no—
458C2×C8—942no—
459C2×C8—942no—
460C2×C8—1888no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—472no—
466C4⋊C4—944no—
467C4⋊C4—944no—
468C4⋊C4—944no—
469C4⋊C4—944no—
470C4⋊C4—941no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4—1884no—
474C4⋊C4C2×C4×D4712yes—
475C42—24no—
476C42—474no—
477C42—474no—
478C42—472no—
479C42—472no—
480C42—944no—
481C42—944no—
482C42—944no—
483C42C2×C4×D4712yes—
484C42D4×D4712yes—
485C23—472no—
486C23—472no—
487C23—472no—
488C23—474no—
489C23—474no—
490C23C2×D4×D4711yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—22no—
494Q8—474no—
495Q8—9416no—
496Q8—942no—
497Q8—942no—
498Q8—942no—
499Q8C4⋊C4×D4714yes—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—472no—
511C2×C4—472no—
512C2×C4—472no—
513C2×C4—474no—
514C2×C4—474no—
515C2×C4—474no—
516C2×C4—474no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—948no—
520C2×C4—948no—
521C2×C4—9416no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—3768no—
529C2×C4—1888no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4C22⋊C4×D4712yes—
535C2×C4C22⋊C4×D4712yes—
536C2×C4C22×C4×D4711yescharacteristic
537C2×C4C2×D4×D4711yescharacteristic
538C2×C4C2×D4×D4711yescharacteristic
539C2×C4C2×Q8×D4711yescharacteristic
540C2×C4#41×D4711yescharacteristic
541C2×C4#41×D4711yescharacteristic
542C8—42no—
543C8—1882no—
544C22—478no—
545C22—474no—
546C22—478no—
547C22—474no—
548C22—474no—
549C22C2×C22⋊C4×D4711yescharacteristic
550C22#121×D4711yescharacteristic
551C22C22⋊Q8×D4711yescharacteristic
552C22#135×D4712yes—
553C22#137×D4712yes—
554C4—22no—
555C4—22no—
556C4—24no—
557C4—42no—
558C4—82no—
559C4—474no—
560C4—942no—
561C4—942no—
562C4—944no—
563C4—3762no—
564C4—1882no—
565C4C2×C4⋊C4×D4711yescharacteristic
566C4#117×D4711yescharacteristic
567C4#121×D4711yescharacteristic
568C4C22⋊Q8×D4711yescharacteristic
569C2—478no—
570C2#380×D4711yescharacteristic
571C2#416×D4711yescharacteristic
572C2#419×D4711yescharacteristic
573C2#464×D4712yes—
574C2#479×D4712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47C2×#1805
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
5. #20243404C2semidirect
6. #20243403C2semidirect
7. #20243402C2semidirect
8. #20243401C2semidirect
9. #20243400C2semidirect
11. C47×#1805C2semidirect
20. C4×Q8×D47C4semidirect
66. Q8×Dic47C2×C4semidirect
69. C4×Dic94C2×C4semidirect
72. #13093989C2×C4semidirect
106. C4×Q8×C47C2×C4semidirect
291. #1805D47direct (N × H)
298. D47#1805direct (N × H)
334. C47C2×#1805semidirect
364. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D472
C2×C4⋊C8×D472
C2×#103×D472
#382×D472
#202434042
#202434032
#202434022
#202434012
#202434002
#340×D472
C47×#18052
C2×#180547