Finite Group Databasebeta

← C41×#23901 #23800102

Subgroups — C41×#23901 #23800102

Order 10496.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3484C211yesmaximal, characteristic
2C22×#123×C41C211yesmaximal, characteristic, complemented
3C2×C41×#458C212yesmaximal, complemented
4C41×#2849C212yesmaximal, complemented
5C41×#2022C214yesmaximal, complemented
6C41×#1898C212yesmaximal, complemented
7C41×#1890C212yesmaximal, complemented
8C41×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C41C2211yescharacteristic
10C23×D4×C41C2211yescharacteristic
11C24×C4×C41C2211yescharacteristic, complemented
12C41×#546—24no—
13D42×C41—24no—
14C41×#535—21no—
15C41×#534—21no—
16C41×#534C2212yes—
17C41×#525—21no—
18C2×#123×C41—21no—
19C2×#123×C41—21no—
20C2×#123×C41—28no—
21C2×#123×C41C2212yes—
22C2×#123×C41C2212yescomplemented
23C2×#123×C41C2214yescomplemented
24C2×#120×C41—21no—
25C2×#120×C41C2212yes—
26C41×#518C2211yescharacteristic
27C22×C4⋊C4×C41—21no—
28C22×C22⋊C4×C41—21no—
29C22×C22⋊C4×C41C2211yescharacteristic, complemented
30C41×#458—28no—
31C2×#99×C41C2212yes—
32C2×#99×C41C2214yescomplemented
33C2×#99×C41C2212yescomplemented
34C41×#386—24no—
35C41×#385—24no—
36C41×#381C2214yescomplemented
37C41×#379C2212yescomplemented
38C41×#342C2214yescomplemented
39C25×C41D411yescharacteristic
40C25×C41C2311yescharacteristic
41H(5,𝔽2)×C41—24no—
42H(5,𝔽2)×C41—42no—
43C2×#41×C41—21no—
44C2×#41×C41—22no—
45C22×D4×C41—21no—
46C22×D4×C41—22no—
47C22×D4×C41—24no—
48C22×D4×C41—22no—
49C22×D4×C41—24no—
50C22×D4×C41—42no—
51C22×D4×C41D411yescharacteristic
52C22×D4×C41D411yescharacteristic
53C22×D4×C41D414yes—
54C22×D4×C41C2312yes—
55C22×D4×C41C2311yescharacteristic
56C22×D4×C41C2311yescharacteristic
57C23×C4×C41—21no—
58C23×C4×C41—21no—
59C23×C4×C41—21no—
60C23×C4×C41—24no—
61C23×C4×C41—24no—
62C23×C4×C41—42no—
63C23×C4×C41D411yescharacteristic
64C23×C4×C41C2312yes—
65#127×C41—22no—
66#124×C41—22no—
67#123×C41—28no—
68#123×C41—28no—
69#123×C41—22no—
70#123×C41—24no—
71#123×C41—22no—
72#123×C41—22no—
73#123×C41—24no—
74#123×C41—24no—
75#123×C41—28no—
76#123×C41—44no—
77#123×C41D418yescomplemented
78C22⋊Q8×C41—22no—
79#121×C41—22no—
80#121×C41—24no—
81#121×C41—24no—
82#121×C41—24no—
83#120×C41—24no—
84#120×C41—22no—
85#120×C41—22no—
86#120×C41—22no—
87#120×C41—22no—
88#120×C41—24no—
89#120×C41—22no—
90C4×D4×C41—24no—
91C4×D4×C41—24no—
92#117×C41—21no—
93C2×C4⋊C4×C41—28no—
94C2×C4⋊C4×C41—22no—
95C2×C4⋊C4×C41—22no—
96C2×C4⋊C4×C41—24no—
97C2×C22⋊C4×C41—28no—
98C2×C22⋊C4×C41—22no—
99C2×C22⋊C4×C41—22no—
100C2×C22⋊C4×C41—22no—
101C2×C22⋊C4×C41—21no—
102C2×C22⋊C4×C41—21no—
103C2×C22⋊C4×C41—24no—
104C2×C22⋊C4×C41—24no—
105C2×C22⋊C4×C41—24no—
106C2×C22⋊C4×C41D414yes—
107C2×C22⋊C4×C41C2314yes—
108C2×C22⋊C4×C41C2312yes—
109C2×C22⋊C4×C41C2312yes—
110#99×C41—24no—
111#99×C41—28no—
112#99×C41—42no—
113#95×C41—28no—
114C24×C41—22no—
115C24×C41—22no—
116C24×C41—21no—
117C24×C41—22no—
118C24×C41—22no—
119C24×C41—44no—
120C24×C41—41no—
121C24×C41—44no—
122C24×C41C2×D411yescharacteristic
123C24×C41C2×D414yes—
124C24×C41C2×D411yescharacteristic
125C24×C41C2×D412yes—
126C24×C41C2×D411yescharacteristic
127C24×C41C2×D411yescharacteristic
128C24×C41C2411yescharacteristic
129#41×C41—48no—
130#41×C41—42no—
131#41×C41—42no—
132C2×Q8×C41—21no—
133C2×D4×C41—22no—
134C2×D4×C41—22no—
135C2×D4×C41—22no—
136C2×D4×C41—21no—
137C2×D4×C41—24no—
138C2×D4×C41—22no—
139C2×D4×C41—22no—
140C2×D4×C41—22no—
141C2×D4×C41—24no—
142C2×D4×C41—22no—
143C2×D4×C41—22no—
144C2×D4×C41—22no—
145C2×D4×C41—22no—
146C2×D4×C41—22no—
147C2×D4×C41—24no—
148C2×D4×C41—24no—
149C2×D4×C41—24no—
150C2×D4×C41—22no—
151C2×D4×C41—41no—
152C2×D4×C41—41no—
153C2×D4×C41—416no—
154C2×D4×C41—48no—
155C2×D4×C41—42no—
156C2×D4×C41—42no—
157C2×D4×C41—48no—
158C2×D4×C41—84no—
159C2×D4×C41C2×D414yes—
160C22×C4×C41—24no—
161C22×C4×C41—24no—
162C22×C4×C41—21no—
163C22×C4×C41—22no—
164C22×C4×C41—24no—
165C22×C4×C41—22no—
166C22×C4×C41—24no—
167C22×C4×C41—22no—
168C22×C4×C41—22no—
169C22×C4×C41—22no—
170C22×C4×C41—24no—
171C22×C4×C41—22no—
172C22×C4×C41—22no—
173C22×C4×C41—24no—
174C22×C4×C41—41no—
175C22×C4×C41—416no—
176C22×C4×C41—44no—
177C22×C4×C41—42no—
178C22×C4×C41—42no—
179C22×C4×C41—42no—
180C22×C4×C41—42no—
181C22×C4×C41C2×D412yes—
182C22×C4×C41C2×D411yescharacteristic
183C22×C4×C41C2×D411yescharacteristic
184C22×C4×C41C2×D414yes—
185C22×C4×C41C2×D414yescomplemented
186C4⋊C4×C41—28no—
187C4⋊C4×C41—22no—
188C4⋊C4×C41—24no—
189C4⋊C4×C41—28no—
190C4⋊C4×C41—48no—
191C22⋊C4×C41—24no—
192C22⋊C4×C41—28no—
193C22⋊C4×C41—22no—
194C22⋊C4×C41—24no—
195C22⋊C4×C41—24no—
196C22⋊C4×C41—28no—
197C22⋊C4×C41—24no—
198C22⋊C4×C41—24no—
199C22⋊C4×C41—22no—
200C22⋊C4×C41—22no—
201C22⋊C4×C41—22no—
202C22⋊C4×C41—24no—
203C22⋊C4×C41—24no—
204C22⋊C4×C41—48no—
205C22⋊C4×C41—48no—
206C22⋊C4×C41C2×D414yescomplemented
207C42×C41—22no—
208C23×C41—21no—
209C23×C41—21no—
210C23×C41—24no—
211C23×C41—21no—
212C23×C41—21no—
213C23×C41—21no—
214C23×C41—28no—
215C23×C41—28no—
216C23×C41—21no—
217C23×C41—21no—
218C23×C41—21no—
219C23×C41—22no—
220C23×C41—21no—
221C23×C41—24no—
222C23×C41—44no—
223C23×C41—42no—
224C23×C41—42no—
225C23×C41—44no—
226C23×C41—48no—
227C23×C41—48no—
228C23×C41—42no—
229C23×C41—42no—
230C23×C41—48no—
231C23×C41—42no—
232C23×C41—44no—
233C23×C41—44no—
234C23×C41—82no—
235C23×C41—84no—
236C23×C41—82no—
237C23×C41#12012yes—
238C23×C41#12012yes—
239C23×C41#12012yes—
240C23×C41C22×D411yescharacteristic
241C23×C41C22×D412yes—
242C23×C41C22×D412yes—
243C23×C41H(5,𝔽2)12yes—
244Q8×C41—42no—
245D4×C41—28no—
246D4×C41—44no—
247D4×C41—42no—
248D4×C41—44no—
249D4×C41—44no—
250D4×C41—42no—
251D4×C41—42no—
252D4×C41—44no—
253D4×C41—44no—
254D4×C41—48no—
255D4×C41—84no—
256D4×C41—84no—
257C2×C4×C41—21no—
258C2×C4×C41—21no—
259C2×C4×C41—24no—
260C2×C4×C41—24no—
261C2×C4×C41—21no—
262C2×C4×C41—24no—
263C2×C4×C41—22no—
264C2×C4×C41—21no—
265C2×C4×C41—21no—
266C2×C4×C41—28no—
267C2×C4×C41—24no—
268C2×C4×C41—44no—
269C2×C4×C41—44no—
270C2×C4×C41—48no—
271C2×C4×C41—44no—
272C2×C4×C41—48no—
273C2×C4×C41—48no—
274C2×C4×C41—42no—
275C2×C4×C41—42no—
276C2×C4×C41—42no—
277C2×C4×C41—44no—
278C2×C4×C41—44no—
279C2×C4×C41—48no—
280C2×C4×C41—44no—
281C2×C4×C41—82no—
282C2×C4×C41#12012yes—
283#23901C4111yesmaximal, characteristic, direct product factor, Sylow
284C22×C41—24no—
285C22×C41—24no—
286C22×C41—22no—
287C22×C41—24no—
288C22×C41—22no—
289C22×C41—22no—
290C22×C41—22no—
291C22×C41—22no—
292C22×C41—24no—
293C22×C41—42no—
294C22×C41—42no—
295C22×C41—416no—
296C22×C41—42no—
297C22×C41—42no—
298C22×C41—42no—
299C22×C41—42no—
300C22×C41—42no—
301C22×C41—42no—
302C22×C41—41no—
303C22×C41—44no—
304C22×C41—42no—
305C22×C41—81no—
306C22×C41—88no—
307C22×C41—84no—
308C22×C41—81no—
309C22×C41—162no—
310C22×C41#45814yescomplemented
311C22×C41C2×#12011yescharacteristic
312C22×C41C2×#12011yescharacteristic
313C22×C41#53411yescharacteristic
314C22×C41D4214yes—
315C4×C41—22no—
316C4×C41—44no—
317C4×C41—44no—
318C4×C41—41no—
319C4×C41—42no—
320C4×C41—44no—
321C4×C41—84no—
322#3484C2×C4111yescharacteristic
323C22×#123C2×C4111yescharacteristic, complemented
324C2×#458C2×C4112yescomplemented
325#2849C2×C4112yescomplemented
326#2022C2×C4114yescomplemented
327#1898C2×C4112yescomplemented
328#1890C2×C4112yescomplemented
329#1781C2×C4111yescharacteristic, complemented
330C2×C41—21no—
331C2×C41—24no—
332C2×C41—21no—
333C2×C41—24no—
334C2×C41—42no—
335C2×C41—42no—
336C2×C41—81no—
337C2×C41—82no—
338C2×C41#240311yescharacteristic
339C2×C41C2×#45812yes—
340C2×H(5,𝔽2)C22×C4111yescharacteristic
341C23×D4C22×C4111yescharacteristic
342C24×C4C22×C4111yescharacteristic, complemented
343#546—24no—
344D42—24no—
345#535—21no—
346#534—21no—
347#534C22×C4112yes—
348#525—21no—
349C2×#123—21no—
350C2×#123—21no—
351C2×#123—28no—
352C2×#123C22×C4112yes—
353C2×#123C22×C4112yescomplemented
354C2×#123C22×C4114yescomplemented
355C2×#120—21no—
356C2×#120C22×C4112yes—
357#518C22×C4111yescharacteristic
358C22×C4⋊C4—21no—
359C22×C22⋊C4—21no—
360C22×C22⋊C4C22×C4111yescharacteristic, complemented
361#458—28no—
362C2×#99C22×C4112yes—
363C2×#99C22×C4114yescomplemented
364C2×#99C22×C4112yescomplemented
365#386—24no—
366#385—24no—
367#381C22×C4114yescomplemented
368#379C22×C4112yescomplemented
369#342C22×C4114yescomplemented
370C41#2390111yescharacteristic, direct product factor, Sylow
371C25D4×C4111yescharacteristic
372C25C23×C4111yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C4111yescharacteristic
384C22×D4D4×C4111yescharacteristic
385C22×D4D4×C4114yes—
386C22×D4C23×C4112yes—
387C22×D4C23×C4111yescharacteristic
388C22×D4C23×C4111yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C4111yescharacteristic
396C23×C4C23×C4112yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C4118yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C4114yes—
439C2×C22⋊C4C23×C4114yes—
440C2×C22⋊C4C23×C4112yes—
441C2×C22⋊C4C23×C4112yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C4111yescharacteristic
455C24C2×D4×C4114yes—
456C24C2×D4×C4111yescharacteristic
457C24C2×D4×C4112yes—
458C24C2×D4×C4111yescharacteristic
459C24C2×D4×C4111yescharacteristic
460C24C24×C4111yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C4114yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C4112yes—
514C22×C4C2×D4×C4111yescharacteristic
515C22×C4C2×D4×C4111yescharacteristic
516C22×C4C2×D4×C4114yes—
517C22×C4C2×D4×C4114yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C4114yescomplemented
539C42—22no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—28no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C4112yes—
570C23#120×C4112yes—
571C23#120×C4112yes—
572C23C22×D4×C4111yescharacteristic
573C23C22×D4×C4112yes—
574C23C22×D4×C4112yes—
575C23H(5,𝔽2)×C4112yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—44no—
586D4—48no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C4112yes—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C41×#45814yescomplemented
642C22C2×#120×C4111yescharacteristic
643C22C2×#120×C4111yescharacteristic
644C22C41×#53411yescharacteristic
645C22D42×C4114yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—21no—
656C2—24no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C41×#240311yescharacteristic
662C2C2×C41×#45812yes—
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC24C24×C41
prime radicalC41×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C25×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C41C2semidirect
3. C2×C41×#458C2semidirect
4. C41×#2849C2semidirect
5. C41×#2022C2semidirect
6. C41×#1898C2semidirect
7. C41×#1890C2semidirect
8. C41×#1781C2semidirect
11. C24×C4×C41C22semidirect
22. C2×#123×C41C22semidirect
23. C2×#123×C41C22semidirect
29. C22×C22⋊C4×C41C22semidirect
32. C2×#99×C41C22semidirect
33. C2×#99×C41C22semidirect
36. C41×#381C22semidirect
37. C41×#379C22semidirect
38. C41×#342C22semidirect
77. #123×C41D4semidirect
185. C22×C4×C41C2×D4semidirect
206. C22⋊C4×C41C2×D4semidirect
283. #23901C41direct (N × H)
310. C22×C41#458semidirect
323. C22×#123C2×C41semidirect
324. C2×#458C2×C41semidirect
325. #2849C2×C41semidirect
326. #2022C2×C41semidirect
327. #1898C2×C41semidirect
328. #1890C2×C41semidirect
329. #1781C2×C41semidirect
342. C24×C4C22×C41semidirect
353. C2×#123C22×C41semidirect
354. C2×#123C22×C41semidirect
360. C22×C22⋊C4C22×C41semidirect
363. C2×#99C22×C41semidirect
364. C2×#99C22×C41semidirect
367. #381C22×C41semidirect
368. #379C22×C41semidirect
369. #342C22×C41semidirect
370. C41#23901direct (N × H)
409. #123D4×C41semidirect
517. C22×C4C2×D4×C41semidirect
538. C22⋊C4C2×D4×C41semidirect
641. C22C41×#458semidirect

Maximal subgroups

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

subgroupindex
C41×#34842
C22×#123×C412
C2×C41×#4582
C41×#28492
C41×#20222
C41×#18982
C41×#18902
C41×#17812
#2390141