Finite Group Databasebeta

← C4×#15689049 #26138102

Subgroups — C4×#15689049 #26138102

Order 12032.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689049C211yesmaximal, characteristic
2C2×C4×(C2×D47)⋊C4C212yesmaximal
3C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
4#20241656C211yesmaximal, characteristic, complemented
5#20241649C212yesmaximal, complemented
6#20241647C211yesmaximal, characteristic, complemented
7#20241636C212yesmaximal
8#20241630C211yesmaximal, characteristic
9C4×C47⋊#95C212yesmaximal, complemented
10C4×C47⋊#95C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×(C2×D47)⋊C4C2212yes—
14C22×(C2×D47)⋊C4C2211yescharacteristic
15C22×(C2×D47)⋊C4C2211yescharacteristic
16C22×(C2×D47)⋊C4C2212yes—
17C22×C4×Dic47C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19#15689049C418yesdirect product factor
20C4×(C2×D47)⋊C4—28no—
21C4×(C2×D47)⋊C4—24no—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C22×C42×C47C2212yes—
36C22×C42×C47C2211yescharacteristic
37C4×C4⋊C4×C47—24no—
38C2×#95×C47C2212yes—
39C24×D47C2311yescharacteristic
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—24no—
48C22×C4×D47—24no—
49C22×C4×D47D414yes—
50C2×(C2×D47)⋊C4—24no—
51C2×(C2×D47)⋊C4—24no—
52C2×(C2×D47)⋊C4—22no—
53C2×(C2×D47)⋊C4—24no—
54C2×(C2×D47)⋊C4—22no—
55C2×(C2×D47)⋊C4—22no—
56C2×(C2×D47)⋊C4—24no—
57C2×(C2×D47)⋊C4—22no—
58C2×(C2×D47)⋊C4—24no—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4C2×C418yes—
61C2×(C2×D47)⋊C4C2×C414yes—
62C2×C4×Dic47—24no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47—24no—
65C2×C4×Dic47—24no—
66C2×C4×Dic47—24no—
67C47⋊#95—22no—
68C47⋊#95—22no—
69C47⋊#95—24no—
70C47⋊#95—22no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95C2×C418yescomplemented
77C47⋊#95C2×C414yescomplemented
78C23×C4×C47C2312yes—
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47C2×C414yescomplemented
88C2×C42×C47—24no—
89C2×C42×C47—24no—
90C2×C42×C47—24no—
91C2×C42×C47—24no—
92C2×C42×C47D414yes—
93#95×C47—24no—
94C24×C47C2411yescharacteristic
95C23×D47—21no—
96C23×D47—21no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—21no—
100C23×D47—21no—
101C23×D47—22no—
102C23×D47—22no—
103C23×D47C22×C412yes—
104C23×D47C2×D412yes—
105C23×D47#4112yes—
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×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47C22×C414yes—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47#4114yes—
131C22×C4×C47#4112yes—
132C22×C4×C47#4112yes—
133C22×C4×C47#4114yes—
134C22×C4×C47#4114yes—
135C22×Dic47—21no—
136C22×Dic47—22no—
137C22×Dic47—21no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—22no—
141C22×Dic47—22no—
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—21no—
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47C22×C414yes—
160C22×Dic47C22×C412yes—
161C22×Dic47C2×D412yes—
162C22×Dic47#4112yes—
163C22×Dic47#4114yes—
164C2×C4×D47—28no—
165C2×C4×D47—216no—
166C2×C4×D47—44no—
167C2×C4×D47—44no—
168C2×C4×D47—48no—
169C4⋊C4×C47—28no—
170C4⋊C4×C47—28no—
171C42×C47—28no—
172C42×C47—48no—
173(C2×D47)⋊C4—28no—
174(C2×D47)⋊C4—216no—
175(C2×D47)⋊C4—216no—
176(C2×D47)⋊C4—28no—
177C4×Dic47—48no—
178C4×Dic47—44no—
179C23×C47C23×C411yescharacteristic
180C23×C47C22×D411yescharacteristic
181C23×C47C22×D411yescharacteristic
182C23×C47C2×#4111yescharacteristic
183C23×C47C2×#4111yescharacteristic
184C23×C47C2×#4112yes—
185C23×C47C2×#4111yescharacteristic
186C23×C47C2×#4111yescharacteristic
187C23×C47C2×#4112yes—
188C23×C47C2×#4112yes—
189C23×C47C2×#4112yes—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—24no—
193C22×D47—24no—
194C22×D47—22no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—22no—
198C22×D47—24no—
199C22×D47—22no—
200C22×D47—24no—
201C22×D47—22no—
202C22×D47—41no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—41no—
212C22×D47C4×D414yes—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—22no—
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×C47C4×D414yes—
237C2×C4×C47#12114yes—
238C2×C4×C47#12314yes—
239C2×C4×C47#12318yes—
240C2×C4×C47#12414yes—
241C2×C4×C47#12618yes—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—24no—
249C2×Dic47—22no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—41no—
257C2×Dic47—42no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—41no—
270C2×Dic47—42no—
271C4×D47—416no—
272C4×#397—471nomaximal, Sylow
273C22×C47C2×C4×D411yescharacteristic
274C22×C47C2×C4×D411yescharacteristic
275C22×C47C4×#4111yescharacteristic
276C22×C47C4×#4112yes—
277C22×C47C4×#4112yes—
278C22×C47C2×#12111yescharacteristic
279C22×C47C2×#12311yescharacteristic
280C22×C47C2×#12312yes—
281C22×C47#52511yescharacteristic
282C22×C47#52512yes—
283C22×C47#52511yescharacteristic
284C22×C47C2×#12411yescharacteristic
285C22×C47C2×#12612yes—
286C22×C47#52912yes—
287C22×C47#52912yes—
288C22×C47#52912yes—
289C22×C47#52911yescharacteristic
290C22×C47#52912yes—
291C22×C47#52912yes—
292C22×C47#52912yes—
293C22×C47#52912yes—
294C22×C47#53211yescharacteristic
295C22×C47#53211yescharacteristic
296C2×D47—24no—
297C2×D47—28no—
298C2×D47—44no—
299C2×D47—42no—
300C2×D47—42no—
301C2×D47—42no—
302C2×D47—44no—
303C2×D47—42no—
304C2×D47—44no—
305C2×D47—42no—
306C2×D47—42no—
307C4×C47—24no—
308C4×C47—44no—
309C4×C47—42no—
310C4×C47—44no—
311C4×C47#39718yescomplemented
312Dic47—42no—
313Dic47—44no—
314Dic47—42no—
315Dic47—44no—
316Dic47—42no—
317C2×#397—471no—
318C2×C4×C4⋊C4D4711yescharacteristic, complemented
319C2×C4×C22⋊C4—471no—
320C2×C4×C22⋊C4—472no—
321#1445—471no—
322#1443—472no—
323#1442—471no—
324#1440—472no—
325#1439—471no—
326C4×#95—471no—
327C4×#95—472no—
328C2×C47C4×#12111yescharacteristic
329C2×C47C4×#12311yescharacteristic
330C2×C47C4×#12312yes—
331C2×C47C4×#12411yescharacteristic
332C2×C47C4×#12612yes—
333C2×C47C2×#39711yescharacteristic
334C2×C47#239211yescharacteristic
335C2×C47#239511yescharacteristic
336C2×C47#239512yes—
337C2×C47#239811yescharacteristic
338C2×C47#240112yes—
339D47—44no—
340C24×C4—471no—
341C22×C4⋊C4C2×D4711yescharacteristic
342C22×C4⋊C4C2×D4711yescharacteristic
343C22×C22⋊C4—471no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4—472no—
346C22×C22⋊C4—471no—
347C22×C42—472no—
348C22×C42—471no—
349C22×C42C2×D4712yes—
350C22×C42C2×D4711yescharacteristic
351#397—478no—
352C4×C4⋊C4—24no—
353C4×C22⋊C4—948no—
354C4×C22⋊C4—944no—
355C2×#95—471no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—471no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—472no—
363C2×#95—471no—
364C2×#95—471no—
365C2×#95—472no—
366C2×#95C2×D4712yes—
367C47C4×#39711yescharacteristic, complemented, Sylow
368C25—471no—
369C23×C4—472no—
370C23×C4—471no—
371C23×C4—472no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—474no—
375C23×C4—944no—
376C23×C4—942no—
377C23×C4—944no—
378C23×C4—942no—
379C23×C4C22×D4712yes—
380C23×C4C22×D4712yes—
381C23×C4C22×D4711yescharacteristic
382C23×C4C22×D4711yescharacteristic
383C23×C4C22×D4711yescharacteristic
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4C4×D4714yescomplemented
389C2×C22⋊C4—478no—
390C2×C22⋊C4—474no—
391C2×C22⋊C4—944no—
392C2×C22⋊C4—942no—
393C2×C22⋊C4—944no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4—944no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C22⋊C4—942no—
401C2×C42—24no—
402C2×C42—24no—
403C2×C42—24no—
404C2×C42—24no—
405C2×C42—944no—
406C2×C42—944no—
407C2×C42—944no—
408C2×C42—944no—
409C2×C42—942no—
410C2×C42C22⋊D4714yes—
411#95—24no—
412#95—478no—
413#95—474no—
414#95—944no—
415#95—944no—
416#95—944no—
417#95—944no—
418#95—944no—
419#95—944no—
420#95—942no—
421#95—942no—
422#95—942no—
423C24—472no—
424C24—472no—
425C24—472no—
426C24—941no—
427C24—941no—
428C24—942no—
429C24—941no—
430C24—941no—
431C24—942no—
432C24—942no—
433C24—942no—
434C24C23×D4711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—474no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—474no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—941no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—948no—
483C22×C4—9416no—
484C22×C4—1884no—
485C22×C4—1884no—
486C22×C4—1888no—
487C22×C4C2×C4×D4714yes—
488C22×C4C2×C4×D4712yes—
489C22×C4#1062535014yes—
490C22×C4#1062535012yes—
491C22×C4#1062535014yes—
492C22×C4D4×D4712yes—
493C22×C4#1062535212yes—
494C22×C4#1062535414yes—
495C22×C4C2×C22⋊D4712yes—
496C22×C4C2×C22⋊D4712yes—
497C22×C4C2×C22⋊D4712yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C22⋊C4—9416no—
501C22⋊C4—948no—
502C22⋊C4—9416no—
503C22⋊C4—948no—
504C42—28no—
505C42—48no—
506C42—1888no—
507C42—1884no—
508C23—474no—
509C23—942no—
510C23—942no—
511C23—944no—
512C23—944no—
513C23—942no—
514C23—944no—
515C23—942no—
516C23—944no—
517C23—944no—
518C23—942no—
519C23—944no—
520C23—942no—
521C23—1881no—
522C23—1881no—
523C23—1881no—
524C23—1882no—
525C23—1882no—
526C23—1882no—
527C23—1882no—
528C23—1882no—
529C23—1882no—
530C23—1881no—
531C23C22×C4×D4711yescharacteristic
532C23C2×D4×D4711yescharacteristic
533C23C22×C22⋊D4711yescharacteristic
534C23#41×D4711yescharacteristic
535C23#41×D4711yescharacteristic
536C23#41×D4712yes—
537C23C2×#1062535211yescharacteristic
538C23C2×#1062535011yescharacteristic
539C23C2×#1062535012yes—
540C23C2×#1062535012yes—
541C23C2×#1062535412yes—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—944no—
566C2×C4—942no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—942no—
571C2×C4—944no—
572C2×C4—942no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—1881no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1881no—
593C2×C4—1882no—
594C2×C4—18816no—
595C2×C4C4×C22⋊D4714yes—
596C2×C4#1309399514yes—
597C2×C4#1309400714yes—
598C2×C4#1309400918yes—
599C2×C4#1309401314yes—
600C2×C4#1309401918yes—
601C22—944no—
602C22—948no—
603C22—1882no—
604C22—1882no—
605C22—1882no—
606C22—1884no—
607C22—1882no—
608C22—1882no—
609C22—1884no—
610C22—1882no—
611C22—1884no—
612C22C4×D4×D4711yescharacteristic
613C22C2×C4×C22⋊D4711yescharacteristic
614C22C4×#1062535211yescharacteristic
615C22C4×#1062535012yes—
616C22C4×#1062535412yes—
617C22C2×#1309399511yescharacteristic
618C22C2×#1309400711yescharacteristic
619C22C2×#1309400912yes—
620C22#1568963011yescharacteristic
621C22#1568963212yes—
622C22#1568963411yescharacteristic
623C22C2×#1309401311yescharacteristic
624C22C2×#1309401912yes—
625C22#1568965112yes—
626C22#1568965112yes—
627C22#1568965312yes—
628C22#1568965511yescharacteristic
629C22#1568965712yes—
630C22#1568966012yes—
631C22#1568966012yes—
632C22#1568966012yes—
633C22#1568966911yescharacteristic
634C22#1568967011yescharacteristic
635C4—24no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—1882no—
640C4—1884no—
641C4—1882no—
642C4—1884no—
643C4—1882no—
644C4#1568904918yesdirect product factor
645C2—1884no—
646C2C4×#1309399511yescharacteristic
647C2C4×#1309400711yescharacteristic
648C2C4×#1309400912yes—
649C2C4×#1309401311yescharacteristic
650C2C4×#1309401912yes—
651C2C2×#1568904911yescharacteristic
652C2#2024662611yescharacteristic
653C2#2024664511yescharacteristic
654C2#2024665112yes—
655C2#2024667111yescharacteristic
656C2#2024668912yes—
subgroupG/H
centerC23×C4C22×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C4×C4⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×C4×C4⋊C4D47
hypocenterC47C4×#397
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #20241656C2semidirect
5. #20241649C2semidirect
6. #20241647C2semidirect
9. C4×C47⋊#95C2semidirect
10. C4×C47⋊#95C2semidirect
11. C2×C4×C4⋊C4×C47C2semidirect
19. #15689049C4direct (N × H)
76. C47⋊#95C2×C4semidirect
77. C47⋊#95C2×C4semidirect
87. C2×C4⋊C4×C47C2×C4semidirect
311. C4×C47#397semidirect
318. C2×C4×C4⋊C4D47semidirect
367. C47C4×#397semidirect
388. C2×C4⋊C4C4×D47semidirect
644. C4#15689049direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156890492
C2×C4×(C2×D47)⋊C42
#202416562
#202416492
#202416472
#202416362
#202416302
C4×C47⋊#952
C2×C4×C4⋊C4×C472
C4×#39747