Finite Group Databasebeta

← C47×#15166 #26054864

Subgroups — C47×#15166 #26054864

Order 12032.

1310 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C47×#398C212yesmaximal
2C2×C47×#389C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C47C211yesmaximal, characteristic
4C47×#1444C211yesmaximal, characteristic
5C47×#1442C211yesmaximal, characteristic
6C47×#1441C212yesmaximal
7C47×#1441C211yesmaximal, characteristic
8C47×#1441C211yesmaximal, characteristic
9C47×#1436C212yesmaximal
10C47×C4⋊#95C211yesmaximal, characteristic
11C47×#1434C212yesmaximal
12C22×C4⋊C4×C47C2212yes—
13C22×C4⋊C4×C47C2211yescharacteristic
14C22×C4⋊C4×C47C2212yes—
15C22×C4⋊C4×C47C2212yes—
16C22×C4⋊C4×C47C2211yescharacteristic
17C22×C4⋊C4×C47C2211yescharacteristic
18C22×C4⋊C4×C47C2211yescharacteristic
19C22×C42×C47C2211yescharacteristic
20C22×C42×C47C2211yescharacteristic
21C22×C42×C47C2211yescharacteristic
22C47×#398—28no—
23C47×#389—24no—
24C4×C4⋊C4×C47—24no—
25C2×#95×C47C2211yescharacteristic
26C2×#95×C47C2211yescharacteristic
27C2×#95×C47C2212yes—
28C2×#95×C47C2212yes—
29C2×#95×C47C2212yes—
30C2×#95×C47C2212yes—
31C2×#95×C47C2212yes—
32C2×#95×C47C2212yes—
33C2×#95×C47C2212yes—
34C2×#95×C47C2211yescharacteristic
35C2×#95×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2212yes—
38C23×C4×C47C2312yes—
39C23×C4×C47C2312yes—
40C23×C4×C47C2311yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C23×C4×C47C2312yes—
44C23×C4×C47C2312yes—
45C23×C4×C47C2311yescharacteristic
46C23×C4×C47C2311yescharacteristic
47C23×C4×C47C2311yescharacteristic
48C23×C4×C47C2311yescharacteristic
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47—22no—
51C2×C4⋊C4×C47—22no—
52C2×C4⋊C4×C47—24no—
53C2×C4⋊C4×C47—22no—
54C2×C4⋊C4×C47—22no—
55C2×C4⋊C4×C47—22no—
56C2×C4⋊C4×C47—24no—
57C2×C4⋊C4×C47—24no—
58C2×C4⋊C4×C47—24no—
59C2×C4⋊C4×C47—24no—
60C2×C4⋊C4×C47—24no—
61C2×C4⋊C4×C47—24no—
62C2×C4⋊C4×C47—24no—
63C2×C4⋊C4×C47—42no—
64C2×C4⋊C4×C47—42no—
65C2×C4⋊C4×C47—42no—
66C2×C4⋊C4×C47D414yes—
67C2×C4⋊C4×C47Q814yes—
68C2×C42×C47—22no—
69C2×C42×C47—22no—
70C2×C42×C47—22no—
71C2×C42×C47—42no—
72C2×C42×C47—42no—
73C2×C42×C47D414yes—
74C2×C42×C47Q814yes—
75#95×C47—22no—
76#95×C47—22no—
77#95×C47—24no—
78#95×C47—24no—
79#95×C47—24no—
80#95×C47—24no—
81#95×C47—24no—
82#95×C47—22no—
83#95×C47—24no—
84#95×C47—24no—
85#95×C47—24no—
86#95×C47—22no—
87#95×C47—42no—
88C24×C47C2411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
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×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×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×Q812yes—
144C22×C4×C47C2×Q812yes—
145C22×C4×C47C2×Q812yes—
146C22×C4×C47C2×Q812yes—
147C22×C4×C47C2×Q812yes—
148C22×C4×C47C2×Q812yes—
149C22×C4×C47#4114yes—
150C22×C4×C47#4114yes—
151C22×C4×C47#4112yes—
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C4⋊C4×C47—28no—
155C4⋊C4×C47—28no—
156C4⋊C4×C47—28no—
157C4⋊C4×C47—48no—
158C4⋊C4×C47—48no—
159C4⋊C4×C47—44no—
160C4⋊C4×C47—48no—
161C42×C47—28no—
162C42×C47—44no—
163C42×C47—44no—
164C23×C47C22×D411yescharacteristic
165C23×C47C22×D411yescharacteristic
166C23×C47C22×Q811yescharacteristic
167C23×C47C22×Q811yescharacteristic
168C23×C47C2×#4111yescharacteristic
169C23×C47C2×#4111yescharacteristic
170C23×C47C2×#4111yescharacteristic
171C23×C47C2×#4112yes—
172C23×C47C2×#4112yes—
173C23×C47H(5,𝔽2)12yes—
174C23×C472−1+412yes—
175C2×C4×C47—22no—
176C2×C4×C47—22no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—24no—
188C2×C4×C47—24no—
189C2×C4×C47—24no—
190C2×C4×C47—24no—
191C2×C4×C47—24no—
192C2×C4×C47—24no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—42no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
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#12114yes—
237C2×C4×C47C22⋊Q814yes—
238C2×C4×C47C22⋊Q814yes—
239C2×C4×C47C4⋊Q814yes—
240#15166C4711yesmaximal, characteristic, direct product factor, Sylow
241C22×C47C2×#12111yescharacteristic
242C22×C47C2×C22⋊Q811yescharacteristic
243C22×C47C2×C22⋊Q811yescharacteristic
244C22×C47#52511yescharacteristic
245C22×C47#52912yes—
246C22×C47C2×C4⋊Q811yescharacteristic
247C22×C47#53211yescharacteristic
248C22×C47#53211yescharacteristic
249C22×C47#53311yescharacteristic
250C22×C47#53311yescharacteristic
251C22×C47#53312yes—
252C22×C47#53411yescharacteristic
253C22×C47#53611yescharacteristic
254C22×C47#53912yes—
255C22×C47#54412yes—
256C22×C47#54612yes—
257C22×C47#54812yes—
258C22×C47#55212yes—
259C22×C47#55212yes—
260C22×C47#55312yes—
261C22×C47#55312yes—
262C22×C47#55412yes—
263C22×C47#55712yes—
264C4×C47—24no—
265C4×C47—24no—
266C4×C47—24no—
267C4×C47—24no—
268C4×C47—42no—
269C4×C47—42no—
270C4×C47—44no—
271C4×C47—42no—
272C4×C47—44no—
273C4×C47—82no—
274C4×C47—82no—
275C2×#398C2×C4712yes—
276C2×#389C2×C4711yescharacteristic
277C2×C4×C4⋊C4C2×C4711yescharacteristic
278#1444C2×C4711yescharacteristic
279#1442C2×C4711yescharacteristic
280#1441C2×C4712yes—
281#1441C2×C4711yescharacteristic
282#1441C2×C4711yescharacteristic
283#1436C2×C4712yes—
284C4⋊#95C2×C4711yescharacteristic
285#1434C2×C4712yes—
286C2×C47#239611yescharacteristic
287C2×C47#253611yescharacteristic
288C2×C47#254311yescharacteristic
289C2×C47#255012yes—
290C2×C47#255812yes—
291C2×C47#256511yescharacteristic
292C2×C47#257911yescharacteristic
293C2×C47#259012yes—
294C2×C47#260612yes—
295C2×C47#262711yescharacteristic
296C2×C47#285711yescharacteristic
297C22×C4⋊C4C22×C4712yes—
298C22×C4⋊C4C22×C4711yescharacteristic
299C22×C4⋊C4C22×C4712yes—
300C22×C4⋊C4C22×C4712yes—
301C22×C4⋊C4C22×C4711yescharacteristic
302C22×C4⋊C4C22×C4711yescharacteristic
303C22×C4⋊C4C22×C4711yescharacteristic
304C22×C42C22×C4711yescharacteristic
305C22×C42C22×C4711yescharacteristic
306C22×C42C22×C4711yescharacteristic
307#398—28no—
308#389—24no—
309C4×C4⋊C4—24no—
310C2×#95C22×C4712yes—
311C2×#95C22×C4712yes—
312C2×#95C22×C4712yes—
313C2×#95C22×C4712yes—
314C2×#95C22×C4712yes—
315C2×#95C22×C4712yes—
316C2×#95C22×C4712yes—
317C2×#95C22×C4711yescharacteristic
318C2×#95C22×C4711yescharacteristic
319C2×#95C22×C4712yes—
320C2×#95C22×C4711yescharacteristic
321C2×#95C22×C4711yescharacteristic
322C2×#95C22×C4712yes—
323C47#1516611yescharacteristic, direct product factor, Sylow
324C23×C4C23×C4712yes—
325C23×C4C23×C4712yes—
326C23×C4C23×C4711yescharacteristic
327C23×C4C23×C4711yescharacteristic
328C23×C4C23×C4711yescharacteristic
329C23×C4C23×C4712yes—
330C23×C4C23×C4712yes—
331C23×C4C23×C4711yescharacteristic
332C23×C4C23×C4711yescharacteristic
333C23×C4C23×C4711yescharacteristic
334C23×C4C23×C4711yescharacteristic
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4D4×C4714yes—
353C2×C4⋊C4Q8×C4714yes—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42D4×C4714yes—
360C2×C42Q8×C4714yes—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—42no—
374C24C24×C4711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
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—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4C2×D4×C4712yes—
424C22×C4C2×D4×C4712yes—
425C22×C4C2×D4×C4712yes—
426C22×C4C2×D4×C4712yes—
427C22×C4C2×D4×C4712yes—
428C22×C4C2×D4×C4712yes—
429C22×C4C2×Q8×C4712yes—
430C22×C4C2×Q8×C4712yes—
431C22×C4C2×Q8×C4712yes—
432C22×C4C2×Q8×C4712yes—
433C22×C4C2×Q8×C4712yes—
434C22×C4C2×Q8×C4712yes—
435C22×C4#41×C4714yes—
436C22×C4#41×C4714yes—
437C22×C4#41×C4712yes—
438C22×C4#41×C4712yes—
439C22×C4#41×C4712yes—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C42—28no—
448C42—44no—
449C42—44no—
450C23C22×D4×C4711yescharacteristic
451C23C22×D4×C4711yescharacteristic
452C23C22×Q8×C4711yescharacteristic
453C23C22×Q8×C4711yescharacteristic
454C23C2×#41×C4711yescharacteristic
455C23C2×#41×C4711yescharacteristic
456C23C2×#41×C4711yescharacteristic
457C23C2×#41×C4712yes—
458C23C2×#41×C4712yes—
459C23H(5,𝔽2)×C4712yes—
460C232−1+4×C4712yes—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
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—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4#121×C4714yes—
523C2×C4C22⋊Q8×C4714yes—
524C2×C4C22⋊Q8×C4714yes—
525C2×C4C4⋊Q8×C4714yes—
526C22C2×#121×C4711yescharacteristic
527C22C2×C22⋊Q8×C4711yescharacteristic
528C22C2×C22⋊Q8×C4711yescharacteristic
529C22C47×#52511yescharacteristic
530C22C47×#52912yes—
531C22C2×C4⋊Q8×C4711yescharacteristic
532C22C47×#53211yescharacteristic
533C22C47×#53211yescharacteristic
534C22C47×#53312yes—
535C22C47×#53311yescharacteristic
536C22C47×#53311yescharacteristic
537C22C47×#53411yescharacteristic
538C22C47×#53611yescharacteristic
539C22C47×#53912yes—
540C22C47×#54412yes—
541C22C47×#54612yes—
542C22C47×#54812yes—
543C22C47×#55212yes—
544C22C47×#55212yes—
545C22C47×#55312yes—
546C22C47×#55312yes—
547C22C47×#55412yes—
548C22C47×#55712yes—
549C4—24no—
550C4—24no—
551C4—24no—
552C4—24no—
553C4—44no—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C2C47×#239611yescharacteristic
561C2C47×#253611yescharacteristic
562C2C47×#254311yescharacteristic
563C2C47×#255012yes—
564C2C47×#255812yes—
565C2C47×#256511yescharacteristic
566C2C47×#257911yescharacteristic
567C2C47×#259012yes—
568C2C47×#260612yes—
569C2C47×#262711yescharacteristic
570C2C47×#285711yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C47C24
commutator, Frattini, cosocleC24C24×C47
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C47 ◁ G

Complements

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

NH ≅ G/Ntype
240. #15166C47direct (N × H)
323. C47#15166direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C47×#3982
C2×C47×#3892
C2×C4×C4⋊C4×C472
C47×#14442
C47×#14422
C47×#14412
C47×#14362
C47×C4⋊#952
C47×#14342
#1516647