Finite Group Databasebeta

← C47×#62005 #26101703

Subgroups — C47×#62005 #26101703

Order 12032.

1836 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3517C211yesmaximal, characteristic
2Q8×#41×C47C211yesmaximal, characteristic, complemented
3C47×#3395C211yesmaximal, characteristic, complemented
4C47×#3392C211yesmaximal, characteristic, complemented
5C47×#3363C211yesmaximal, characteristic, complemented
6C47×#3362C211yesmaximal, characteristic, complemented
7C47×#3333C213yesmaximal, complemented
8C47×#3331C213yesmaximal, complemented
9C47×#3278C213yesmaximal, complemented
10C47×#3276C213yesmaximal, complemented
11C47×#3252C213yesmaximal, complemented
12C47×#3210C213yesmaximal, complemented
13C47×#3108C213yesmaximal, complemented
14C47×#3037C213yesmaximal, complemented
15C47×#2991C211yesmaximal, characteristic, complemented
16C2×#137×C47C2216yes—
17Q82×C47C2211yescharacteristic
18C47×#557—21no—
19C47×#557C2211yescharacteristic
20C47×#557C2211yescharacteristic
21C47×#554—23no—
22C47×#554C2213yes—
23C47×#553—23no—
24C47×#550—21no—
25C47×#550C2213yes—
26C47×#550C2211yescharacteristic
27D4×Q8×C47C2213yes—
28C47×#547C2213yes—
29C47×#540—23no—
30C47×#533—23no—
31C47×#533C2213yes—
32C47×#532C2213yes—
33C47×#529C2213yes—
34C47×#520—21no—
35C4×#41×C47—21no—
36C4×#41×C47C2211yescharacteristic
37C2×C4×Q8×C47C2213yes—
38C47×#494C2213yes—
39C47×#492C2213yes—
40C47×#487C2213yes—
41C47×#486C2213yes—
42C47×#483C2213yes—
43C47×#480C2213yes—
44C47×#479C2211yescharacteristic
45C47×#479C2213yes—
46C47×#477C2213yes—
47C47×#476C2213yes—
48C47×#470C2216yes—
49C47×#468C2216yes—
50C47×#464C2213yes—
51C47×#464C2216yes—
52C47×#464C2213yes—
53C47×#462C2213yes—
54C47×#462C2213yes—
55C47×#462C2213yes—
56C47×#461C2213yes—
57C47×#460C2213yes—
58C47×#460C2213yes—
59C47×#452C2216yes—
60C47×#452C2216yes—
61C47×#451C2213yes—
62C47×#448C2213yes—
63C8×Q8×C47C2211yescharacteristic
64C8×Q8×C47C2211yescharacteristic
65C4×Q16×C47C2213yes—
66C4×Q16×C47C2213yes—
67C4×SD16×C47C2213yes—
68C4×SD16×C47C2213yes—
69C47×#436C2213yes—
70C47×#436C2213yes—
71C47×#431C2213yes—
72C47×#423C2213yes—
73C47×#417C2216yes—
74C2×#41×C47—21no—
75C2×#41×C47D411yescharacteristic
76C2×#41×C47C2316yes—
77C22×Q8×C47C2313yes—
78#137×C47—412no—
79C2×Q16×C47C2313yes—
80C2×Q16×C47C2313yes—
81C2×Q16×C47C2316yes—
82C2×SD16×C47C2313yes—
83C2×SD16×C47C2313yes—
84C2×SD16×C47C2316yes—
85C2×M16×C47C2316yes—
86C4⋊Q8×C47—23no—
87C4⋊Q8×C47—23no—
88C4⋊Q8×C47C2313yes—
89C4⋊Q8×C47C2313yes—
90C4⋊Q8×C47C2313yes—
91#127×C47—23no—
92#127×C47C2313yes—
93#126×C47—23no—
94#125×C47—23no—
95#125×C47—23no—
96#125×C47—23no—
97#125×C47—23no—
98#125×C47C2311yescharacteristic
99#124×C47C2313yes—
100#123×C47—23no—
101#123×C47—26no—
102C22⋊Q8×C47—23no—
103C22⋊Q8×C47—26no—
104C22⋊Q8×C47—26no—
105C22⋊Q8×C47C2313yes—
106C22⋊Q8×C47C2316yes—
107#121×C47—23no—
108#121×C47—23no—
109#121×C47—23no—
110#121×C47—23no—
111#121×C47—26no—
112#121×C47C2313yes—
113C4×Q8×C47—21no—
114C4×Q8×C47—21no—
115C4×Q8×C47—21no—
116C4×Q8×C47—23no—
117C4×Q8×C47—23no—
118C4×Q8×C47—23no—
119C4×Q8×C47D411yescharacteristic
120C4×Q8×C47C2313yes—
121C4×Q8×C47C2313yes—
122C4×Q8×C47C2313yes—
123C4×Q8×C47C2311yescharacteristic
124C4×Q8×C47C2313yes—
125C4×Q8×C47C2311yescharacteristic
126C4×D4×C47—23no—
127C4×D4×C47—23no—
128C4×D4×C47—23no—
129C4×D4×C47—23no—
130C4×D4×C47—23no—
131C4×D4×C47—23no—
132C4×D4×C47D413yes—
133C4×D4×C47C2313yes—
134C4×D4×C47C2313yes—
135#117×C47—23no—
136#117×C47—23no—
137#117×C47D413yes—
138C2×C4⋊C4×C47—23no—
139C2×C4⋊C4×C47—23no—
140C2×C4⋊C4×C47C2313yes—
141C2×C42×C47—23no—
142C2×C42×C47C2313yes—
143#107×C47C2313yes—
144#107×C47C2311yescharacteristic
145#106×C47C2313yes—
146#106×C47C2311yescharacteristic
147C4⋊C8×C47C2313yes—
148C4⋊C8×C47C2313yes—
149C4⋊C8×C47C2313yes—
150C4⋊C8×C47C2311yescharacteristic
151#103×C47C2311yescharacteristic
152#103×C47C2313yes—
153#103×C47C2313yes—
154#103×C47C2316yes—
155#103×C47C2313yes—
156#103×C47C2313yes—
157#103×C47C2311yescharacteristic
158#102×C47C2313yes—
159#102×C47C2316yes—
160#102×C47C2313yes—
161C22⋊C8×C47C2313yes—
162C22⋊C8×C47C2313yes—
163C4×C8×C47C2313yes—
164C4×C8×C47C2313yes—
165#41×C47—24no—
166#41×C47—46no—
167#41×C47—46no—
168#41×C47—44no—
169C2×Q8×C47—23no—
170C2×Q8×C47—23no—
171C2×Q8×C47—26no—
172C2×Q8×C47—43no—
173C2×Q8×C47C2×D411yescharacteristic
174C2×Q8×C47C2413yes—
175C2×Q8×C47C2413yes—
176C2×Q8×C47C2411yescharacteristic
177C2×D4×C47—23no—
178C2×D4×C47—26no—
179C2×D4×C47—23no—
180C2×D4×C47—23no—
181C2×D4×C47C2×D413yes—
182C2×D4×C47C2413yes—
183C22×C4×C47—23no—
184C22×C4×C47—23no—
185C22×C4×C47—23no—
186C22×C4×C47—26no—
187C22×C4×C47C2×D413yes—
188C22×C4×C47C2×D413yes—
189C22×C4×C47C2413yes—
190Q16×C47—26no—
191Q16×C47—43no—
192Q16×C47—46no—
193SD16×C47—26no—
194SD16×C47—43no—
195SD16×C47—46no—
196M16×C47—46no—
197C2×C8×C47C2411yescharacteristic
198C2×C8×C47C2413yes—
199C2×C8×C47C2413yes—
200C2×C8×C47C2411yescharacteristic
201C4⋊C4×C47—21no—
202C4⋊C4×C47—23no—
203C4⋊C4×C47—23no—
204C4⋊C4×C47—23no—
205C4⋊C4×C47—23no—
206C4⋊C4×C47—23no—
207C4⋊C4×C47—23no—
208C4⋊C4×C47—21no—
209C4⋊C4×C47—23no—
210C4⋊C4×C47—23no—
211C4⋊C4×C47—21no—
212C4⋊C4×C47—23no—
213C4⋊C4×C47—23no—
214C4⋊C4×C47—26no—
215C4⋊C4×C47—23no—
216C4⋊C4×C47C2×D413yes—
217C4⋊C4×C47C2×D413yes—
218C4⋊C4×C47C2411yescharacteristic
219C4⋊C4×C47C2411yescharacteristic
220C4⋊C4×C47C2411yescharacteristic
221C4⋊C4×C47C2413yes—
222C22⋊C4×C47—26no—
223C22⋊C4×C47—23no—
224C22⋊C4×C47—23no—
225C22⋊C4×C47—23no—
226C22⋊C4×C47—23no—
227C22⋊C4×C47—23no—
228C22⋊C4×C47C2×D416yes—
229C42×C47—21no—
230C42×C47—23no—
231C42×C47—23no—
232C42×C47—23no—
233C42×C47—23no—
234C42×C47C2×D413yes—
235C42×C47C2×D413yes—
236C42×C47C2411yescharacteristic
237C42×C47C2413yes—
238C23×C47—23no—
239C23×C47C22×D413yes—
240Q8×C47—21no—
241Q8×C47—23no—
242Q8×C47—23no—
243Q8×C47—23no—
244Q8×C47—23no—
245Q8×C47—46no—
246Q8×C47—83no—
247Q8×C47#13714yescomplemented
248Q8×C47H(5,𝔽2)12yes—
249D4×C47—26no—
250D4×C47—23no—
251D4×C47—23no—
252D4×C47—46no—
253D4×C47—83no—
254D4×C47—83no—
255D4×C47—86no—
256C2×C4×C47—21no—
257C2×C4×C47—26no—
258C2×C4×C47—23no—
259C2×C4×C47—23no—
260C2×C4×C47—21no—
261C2×C4×C47—21no—
262C2×C4×C47—23no—
263C2×C4×C47—23no—
264C2×C4×C47—23no—
265C2×C4×C47—21no—
266C2×C4×C47—43no—
267C2×C4×C47—46no—
268C2×C4×C47—46no—
269C2×C4×C47—86no—
270C2×C4×C47—83no—
271C2×C4×C47C22×D411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C22×D413yes—
274C2×C4×C47C22×D413yes—
275C2×C4×C47C22×D413yes—
276C2×C4×C47C2511yescharacteristic
277C8×C47—23no—
278C8×C47—23no—
279C8×C47—21no—
280C8×C47—21no—
281#62005C4711yesmaximal, characteristic, direct product factor, Sylow
282C22×C47—23no—
283C22×C47—23no—
284C22×C47—23no—
285C22×C47—43no—
286C22×C47—43no—
287C22×C47—43no—
288C22×C47C23×D411yescharacteristic
289C4×C47—21no—
290C4×C47—23no—
291C4×C47—23no—
292C4×C47—22no—
293C4×C47—21no—
294C4×C47—41no—
295C4×C47—43no—
296C4×C47—41no—
297C4×C47—43no—
298C4×C47—43no—
299C4×C47—41no—
300C4×C47C2×#13716yes—
301C4×C47C2×H(5,𝔽2)11yescharacteristic
302C4×C47#58511yescharacteristic
303#3517C2×C4711yescharacteristic
304Q8×#41C2×C4711yescharacteristic, complemented
305#3395C2×C4711yescharacteristic, complemented
306#3392C2×C4711yescharacteristic, complemented
307#3363C2×C4711yescharacteristic, complemented
308#3362C2×C4711yescharacteristic, complemented
309#3333C2×C4713yescomplemented
310#3331C2×C4713yescomplemented
311#3278C2×C4713yescomplemented
312#3276C2×C4713yescomplemented
313#3252C2×C4713yescomplemented
314#3210C2×C4713yescomplemented
315#3108C2×C4713yescomplemented
316#3037C2×C4713yescomplemented
317#2991C2×C4711yescharacteristic, complemented
318C2×C47—43no—
319C2×C47—83no—
320C2×C47#348811yescharacteristic
321C2×C47C22×#13711yescharacteristic
322C2×C47#358411yescharacteristic
323C2×#137C22×C4716yes—
324Q82C22×C4711yescharacteristic
325#557—21no—
326#557C22×C4711yescharacteristic
327#557C22×C4711yescharacteristic
328#554—23no—
329#554C22×C4713yes—
330#553—23no—
331#550—21no—
332#550C22×C4713yes—
333#550C22×C4711yescharacteristic
334D4×Q8C22×C4713yes—
335#547C22×C4713yes—
336#540—23no—
337#533—23no—
338#533C22×C4713yes—
339#532C22×C4713yes—
340#529C22×C4713yes—
341#520—21no—
342C4×#41—21no—
343C4×#41C22×C4711yescharacteristic
344C2×C4×Q8C22×C4713yes—
345#494C22×C4713yes—
346#492C22×C4713yes—
347#487C22×C4713yes—
348#486C22×C4713yes—
349#483C22×C4713yes—
350#480C22×C4713yes—
351#479C22×C4711yescharacteristic
352#479C22×C4713yes—
353#477C22×C4713yes—
354#476C22×C4713yes—
355#470C22×C4716yes—
356#468C22×C4716yes—
357#464C22×C4713yes—
358#464C22×C4713yes—
359#464C22×C4716yes—
360#462C22×C4713yes—
361#462C22×C4713yes—
362#462C22×C4713yes—
363#461C22×C4713yes—
364#460C22×C4713yes—
365#460C22×C4713yes—
366#452C22×C4716yes—
367#452C22×C4716yes—
368#451C22×C4713yes—
369#448C22×C4713yes—
370C8×Q8C22×C4711yescharacteristic
371C8×Q8C22×C4711yescharacteristic
372C4×Q16C22×C4713yes—
373C4×Q16C22×C4713yes—
374C4×SD16C22×C4713yes—
375C4×SD16C22×C4713yes—
376#436C22×C4713yes—
377#436C22×C4713yes—
378#431C22×C4713yes—
379#423C22×C4713yes—
380#417C22×C4716yes—
381C47#6200511yescharacteristic, direct product factor, Sylow
382C2×#41—21no—
383C2×#41D4×C4711yescharacteristic
384C2×#41C23×C4716yes—
385C22×Q8C23×C4713yes—
386#137—412no—
387C2×Q16C23×C4713yes—
388C2×Q16C23×C4716yes—
389C2×Q16C23×C4713yes—
390C2×SD16C23×C4713yes—
391C2×SD16C23×C4716yes—
392C2×SD16C23×C4713yes—
393C2×M16C23×C4716yes—
394C4⋊Q8—23no—
395C4⋊Q8—23no—
396C4⋊Q8C23×C4713yes—
397C4⋊Q8C23×C4713yes—
398C4⋊Q8C23×C4713yes—
399#127—23no—
400#127C23×C4713yes—
401#126—23no—
402#125—23no—
403#125—23no—
404#125—23no—
405#125—23no—
406#125C23×C4711yescharacteristic
407#124C23×C4713yes—
408#123—23no—
409#123—26no—
410C22⋊Q8—23no—
411C22⋊Q8—26no—
412C22⋊Q8—26no—
413C22⋊Q8C23×C4713yes—
414C22⋊Q8C23×C4716yes—
415#121—23no—
416#121—23no—
417#121—23no—
418#121—23no—
419#121—26no—
420#121C23×C4713yes—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—23no—
425C4×Q8—23no—
426C4×Q8—23no—
427C4×Q8D4×C4711yescharacteristic
428C4×Q8C23×C4711yescharacteristic
429C4×Q8C23×C4713yes—
430C4×Q8C23×C4713yes—
431C4×Q8C23×C4713yes—
432C4×Q8C23×C4713yes—
433C4×Q8C23×C4711yescharacteristic
434C4×D4—23no—
435C4×D4—23no—
436C4×D4—23no—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—23no—
440C4×D4D4×C4713yes—
441C4×D4C23×C4713yes—
442C4×D4C23×C4713yes—
443#117—23no—
444#117—23no—
445#117D4×C4713yes—
446C2×C4⋊C4—23no—
447C2×C4⋊C4—23no—
448C2×C4⋊C4C23×C4713yes—
449C2×C42—23no—
450C2×C42C23×C4713yes—
451#107C23×C4711yescharacteristic
452#107C23×C4713yes—
453#106C23×C4711yescharacteristic
454#106C23×C4713yes—
455C4⋊C8C23×C4713yes—
456C4⋊C8C23×C4713yes—
457C4⋊C8C23×C4713yes—
458C4⋊C8C23×C4711yescharacteristic
459#103C23×C4713yes—
460#103C23×C4716yes—
461#103C23×C4711yescharacteristic
462#103C23×C4713yes—
463#103C23×C4711yescharacteristic
464#103C23×C4713yes—
465#103C23×C4713yes—
466#102C23×C4713yes—
467#102C23×C4716yes—
468#102C23×C4713yes—
469C22⋊C8C23×C4713yes—
470C22⋊C8C23×C4713yes—
471C4×C8C23×C4713yes—
472C4×C8C23×C4713yes—
473#41—24no—
474#41—46no—
475#41—46no—
476#41—44no—
477C2×Q8—23no—
478C2×Q8—23no—
479C2×Q8—26no—
480C2×Q8—43no—
481C2×Q8C2×D4×C4711yescharacteristic
482C2×Q8C24×C4713yes—
483C2×Q8C24×C4713yes—
484C2×Q8C24×C4711yescharacteristic
485C2×D4—23no—
486C2×D4—26no—
487C2×D4—23no—
488C2×D4—23no—
489C2×D4C2×D4×C4713yes—
490C2×D4C24×C4713yes—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—23no—
494C22×C4—26no—
495C22×C4C2×D4×C4713yes—
496C22×C4C2×D4×C4713yes—
497C22×C4C24×C4713yes—
498Q16—26no—
499Q16—43no—
500Q16—46no—
501SD16—26no—
502SD16—43no—
503SD16—46no—
504M16—46no—
505C2×C8C24×C4711yescharacteristic
506C2×C8C24×C4711yescharacteristic
507C2×C8C24×C4713yes—
508C2×C8C24×C4713yes—
509C4⋊C4—21no—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4—23no—
513C4⋊C4—23no—
514C4⋊C4—23no—
515C4⋊C4—23no—
516C4⋊C4—21no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—21no—
520C4⋊C4—23no—
521C4⋊C4—23no—
522C4⋊C4—26no—
523C4⋊C4—23no—
524C4⋊C4C2×D4×C4713yes—
525C4⋊C4C2×D4×C4713yes—
526C4⋊C4C24×C4711yescharacteristic
527C4⋊C4C24×C4713yes—
528C4⋊C4C24×C4711yescharacteristic
529C4⋊C4C24×C4711yescharacteristic
530C22⋊C4—26no—
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—23no—
534C22⋊C4—23no—
535C22⋊C4—23no—
536C22⋊C4C2×D4×C4716yes—
537C42—21no—
538C42—23no—
539C42—23no—
540C42—23no—
541C42—23no—
542C42C2×D4×C4713yes—
543C42C2×D4×C4713yes—
544C42C24×C4713yes—
545C42C24×C4711yescharacteristic
546C23—23no—
547C23C22×D4×C4713yes—
548Q8—21no—
549Q8—23no—
550Q8—23no—
551Q8—23no—
552Q8—23no—
553Q8—46no—
554Q8—83no—
555Q8#137×C4714yescomplemented
556Q8H(5,𝔽2)×C4712yes—
557D4—26no—
558D4—23no—
559D4—23no—
560D4—46no—
561D4—83no—
562D4—83no—
563D4—86no—
564C2×C4—21no—
565C2×C4—26no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—23no—
573C2×C4—21no—
574C2×C4—43no—
575C2×C4—46no—
576C2×C4—46no—
577C2×C4—86no—
578C2×C4—83no—
579C2×C4C22×D4×C4711yescharacteristic
580C2×C4C22×D4×C4711yescharacteristic
581C2×C4C22×D4×C4713yes—
582C2×C4C22×D4×C4713yes—
583C2×C4C22×D4×C4713yes—
584C2×C4C25×C4711yescharacteristic
585C8—23no—
586C8—23no—
587C8—21no—
588C8—21no—
589C22—23no—
590C22—23no—
591C22—23no—
592C22—43no—
593C22—43no—
594C22—43no—
595C22C23×D4×C4711yescharacteristic
596C4—21no—
597C4—21no—
598C4—23no—
599C4—23no—
600C4—22no—
601C4—41no—
602C4—41no—
603C4—43no—
604C4—41no—
605C4—43no—
606C4—43no—
607C4C2×#137×C4716yes—
608C4C2×H(5,𝔽2)×C4711yescharacteristic
609C4C47×#58511yescharacteristic
610C2—43no—
611C2—83no—
612C2C47×#348811yescharacteristic
613C2C22×#137×C4711yescharacteristic
614C2C47×#358411yescharacteristic
subgroupG/H
center, socleC22×C47C23×D4
commutator, Frattini, cosocleC2×C4C25×C47
prime radicalC47×#3517C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C4×#41×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×#41×C47C2semidirect
3. C47×#3395C2semidirect
4. C47×#3392C2semidirect
5. C47×#3363C2semidirect
6. C47×#3362C2semidirect
7. C47×#3333C2semidirect
8. C47×#3331C2semidirect
9. C47×#3278C2semidirect
10. C47×#3276C2semidirect
11. C47×#3252C2semidirect
12. C47×#3210C2semidirect
13. C47×#3108C2semidirect
14. C47×#3037C2semidirect
15. C47×#2991C2semidirect
247. Q8×C47#137semidirect
281. #62005C47direct (N × H)
304. Q8×#41C2×C47semidirect
305. #3395C2×C47semidirect
306. #3392C2×C47semidirect
307. #3363C2×C47semidirect
308. #3362C2×C47semidirect
309. #3333C2×C47semidirect
310. #3331C2×C47semidirect
311. #3278C2×C47semidirect
312. #3276C2×C47semidirect
313. #3252C2×C47semidirect
314. #3210C2×C47semidirect
315. #3108C2×C47semidirect
316. #3037C2×C47semidirect
317. #2991C2×C47semidirect
381. C47#62005direct (N × H)
555. Q8#137×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#35172
Q8×#41×C472
C47×#33952
C47×#33922
C47×#33632
C47×#33622
C47×#33332
C47×#33312
C47×#32782
C47×#32762
C47×#32522
C47×#32102
C47×#31082
C47×#30372
C47×#29912
#6200547