Finite Group Databasebeta

← C41×#24059 #23800260

Subgroups — C41×#24059 #23800260

Order 10496.

1348 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3485C211yesmaximal, characteristic
2C41×#2623C212yesmaximal, complemented
3C41×#2311C211yesmaximal, characteristic, complemented
4C41×#2044C214yesmaximal, complemented
5C41×#1899C212yesmaximal, complemented
6C41×#1867C212yesmaximal, complemented
7C41×#1790C212yesmaximal, complemented
8C41×#1782C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C41C2211yescharacteristic
10C22×#41×C41C2211yescharacteristic
11C41×#547—22no—
12C41×#547—22no—
13C41×#546—22no—
14C41×#546—22no—
15C41×#536—21no—
16C41×#534—21no—
17C41×#534C2212yes—
18C41×#525—21no—
19C2×#123×C41—21no—
20C2×#123×C41C2212yes—
21C41×#518C2211yescharacteristic
22C2×#117×C41C2211yescharacteristic, complemented
23C22×C4⋊C4×C41C2211yescharacteristic, complemented
24C2×#99×C41C2212yes—
25C2×#99×C41C2214yes—
26C2×#99×C41C2212yescomplemented
27C41×#400—22no—
28C41×#399—22no—
29C41×#398—22no—
30C41×#397—22no—
31C41×#391—21no—
32C41×#390—21no—
33C41×#386C2214yes—
34C41×#386C2212yes—
35C41×#386C2212yescomplemented
36C41×#385C2214yescomplemented
37C41×#381C2212yescomplemented
38C41×#380—21no—
39C41×#380—21no—
40C41×#342C2214yescomplemented
41H(5,𝔽2)×C41—24no—
42H(5,𝔽2)×C41—42no—
43C2×#41×C41—24no—
44C2×#41×C41—21no—
45C2×#41×C41—22no—
46C2×#41×C41—22no—
47C2×#41×C41—22no—
48C2×#41×C41—42no—
49C22×Q8×C41D411yescharacteristic
50C22×D4×C41—21no—
51C22×D4×C41—22no—
52C22×D4×C41D411yescharacteristic
53C22×D4×C41C2312yes—
54C22×D4×C41C2311yescharacteristic
55C22×D4×C41C2311yescharacteristic
56C23×C4×C41D411yescharacteristic
57C23×C4×C41D411yescharacteristic
58C23×C4×C41C2312yes—
59C23×C4×C41C2311yescharacteristic
60C4⋊Q8×C41—22no—
61#124×C41—22no—
62#123×C41—22no—
63#123×C41—22no—
64#123×C41—22no—
65#123×C41—22no—
66#123×C41—22no—
67#123×C41—22no—
68#123×C41—22no—
69#123×C41—22no—
70#123×C41—22no—
71C22⋊Q8×C41—21no—
72C22⋊Q8×C41—21no—
73C22⋊Q8×C41—22no—
74C22⋊Q8×C41—22no—
75#121×C41—21no—
76#121×C41—21no—
77#121×C41—22no—
78#121×C41—22no—
79#121×C41—22no—
80#121×C41—22no—
81#120×C41—22no—
82#120×C41—22no—
83#120×C41—22no—
84C4×D4×C41—22no—
85C4×D4×C41—22no—
86C4×D4×C41—22no—
87C4×D4×C41—22no—
88#117×C41—21no—
89#117×C41—42no—
90C2×C4⋊C4×C41—21no—
91C2×C4⋊C4×C41—21no—
92C2×C4⋊C4×C41—21no—
93C2×C4⋊C4×C41—21no—
94C2×C4⋊C4×C41—22no—
95C2×C4⋊C4×C41—22no—
96C2×C4⋊C4×C41—21no—
97C2×C4⋊C4×C41—24no—
98C2×C4⋊C4×C41D412yes—
99C2×C4⋊C4×C41D412yes—
100C2×C22⋊C4×C41—21no—
101C2×C22⋊C4×C41—21no—
102C2×C22⋊C4×C41—21no—
103C2×C22⋊C4×C41—22no—
104C2×C22⋊C4×C41—22no—
105C2×C22⋊C4×C41—24no—
106C2×C22⋊C4×C41D412yes—
107C2×C22⋊C4×C41D412yes—
108C2×C22⋊C4×C41C2312yes—
109C2×C22⋊C4×C41C2314yes—
110C2×C22⋊C4×C41C2312yes—
111C2×C42×C41—21no—
112#99×C41—24no—
113#99×C41—44no—
114#99×C41—42no—
115#95×C41—22no—
116#95×C41—22no—
117#95×C41—22no—
118#95×C41—22no—
119#95×C41—24no—
120#95×C41—24no—
121C24×C41—21no—
122C24×C41C2×D412yes—
123C24×C41C2×D411yescharacteristic
124C24×C41C2×D411yescharacteristic
125C24×C41C2411yescharacteristic
126#41×C41—48no—
127#41×C41—48no—
128#41×C41—48no—
129#41×C41—42no—
130#41×C41—42no—
131C2×Q8×C41—21no—
132C2×Q8×C41—22no—
133C2×Q8×C41—22no—
134C2×Q8×C41—41no—
135C2×D4×C41—22no—
136C2×D4×C41—22no—
137C2×D4×C41—22no—
138C2×D4×C41—21no—
139C2×D4×C41—22no—
140C2×D4×C41—22no—
141C2×D4×C41—22no—
142C2×D4×C41—22no—
143C2×D4×C41—22no—
144C2×D4×C41—24no—
145C2×D4×C41—22no—
146C2×D4×C41—24no—
147C2×D4×C41—22no—
148C2×D4×C41—22no—
149C2×D4×C41—22no—
150C2×D4×C41—41no—
151C2×D4×C41—48no—
152C2×D4×C41—42no—
153C2×D4×C41—42no—
154C2×D4×C41#4114yes—
155C22×C4×C41—22no—
156C22×C4×C41—22no—
157C22×C4×C41—21no—
158C22×C4×C41—22no—
159C22×C4×C41—22no—
160C22×C4×C41—21no—
161C22×C4×C41—21no—
162C22×C4×C41—21no—
163C22×C4×C41—21no—
164C22×C4×C41—21no—
165C22×C4×C41—21no—
166C22×C4×C41—22no—
167C22×C4×C41—24no—
168C22×C4×C41—22no—
169C22×C4×C41—22no—
170C22×C4×C41—24no—
171C22×C4×C41—41no—
172C22×C4×C41—41no—
173C22×C4×C41—82no—
174C22×C4×C41C2×D412yes—
175C22×C4×C41C2×D411yescharacteristic
176C22×C4×C41C2×D412yes—
177C22×C4×C41C2×D412yes—
178C22×C4×C41C2×D411yescharacteristic
179C22×C4×C41C2×D411yescharacteristic
180C22×C4×C41C2×D411yescharacteristic
181C22×C4×C41C2×D412yes—
182C22×C4×C41C2×D412yes—
183C4⋊C4×C41—22no—
184C4⋊C4×C41—22no—
185C4⋊C4×C41—21no—
186C4⋊C4×C41—21no—
187C4⋊C4×C41—22no—
188C4⋊C4×C41—22no—
189C4⋊C4×C41—22no—
190C4⋊C4×C41—22no—
191C4⋊C4×C41—42no—
192C4⋊C4×C41—41no—
193C4⋊C4×C41—41no—
194C4⋊C4×C41—42no—
195C4⋊C4×C41—84no—
196C22⋊C4×C41—22no—
197C22⋊C4×C41—22no—
198C22⋊C4×C41—22no—
199C22⋊C4×C41—22no—
200C22⋊C4×C41—22no—
201C22⋊C4×C41—22no—
202C22⋊C4×C41—22no—
203C22⋊C4×C41—22no—
204C22⋊C4×C41—22no—
205C22⋊C4×C41—22no—
206C22⋊C4×C41—22no—
207C22⋊C4×C41—24no—
208C22⋊C4×C41—24no—
209C22⋊C4×C41—24no—
210C22⋊C4×C41—82no—
211C22⋊C4×C41—82no—
212C22⋊C4×C41—84no—
213C42×C41—22no—
214C42×C41—42no—
215C23×C41—21no—
216C23×C41—21no—
217C23×C41—22no—
218C23×C41—21no—
219C23×C41—21no—
220C23×C41—21no—
221C23×C41—24no—
222C23×C41—41no—
223C23×C41—41no—
224C23×C41—41no—
225C23×C41—41no—
226C23×C41—41no—
227C23×C41—41no—
228C23×C41—42no—
229C23×C41—42no—
230C23×C41—42no—
231C23×C41—81no—
232C23×C41—81no—
233C23×C41#12312yes—
234C23×C41C22×D411yescharacteristic
235C23×C41C22×D412yes—
236C23×C41C2×#4112yes—
237C23×C41H(5,𝔽2)12yes—
238Q8×C41—44no—
239Q8×C41—42no—
240D4×C41—28no—
241D4×C41—44no—
242D4×C41—42no—
243D4×C41—44no—
244D4×C41—42no—
245D4×C41—42no—
246D4×C41—44no—
247D4×C41—44no—
248D4×C41—82no—
249D4×C41—82no—
250C2×C4×C41—21no—
251C2×C4×C41—21no—
252C2×C4×C41—21no—
253C2×C4×C41—21no—
254C2×C4×C41—21no—
255C2×C4×C41—21no—
256C2×C4×C41—24no—
257C2×C4×C41—24no—
258C2×C4×C41—22no—
259C2×C4×C41—22no—
260C2×C4×C41—21no—
261C2×C4×C41—22no—
262C2×C4×C41—21no—
263C2×C4×C41—21no—
264C2×C4×C41—24no—
265C2×C4×C41—42no—
266C2×C4×C41—42no—
267C2×C4×C41—42no—
268C2×C4×C41—42no—
269C2×C4×C41—44no—
270C2×C4×C41—42no—
271C2×C4×C41—42no—
272C2×C4×C41—42no—
273C2×C4×C41—42no—
274C2×C4×C41—44no—
275C2×C4×C41—44no—
276C2×C4×C41—42no—
277C2×C4×C41—42no—
278C2×C4×C41—44no—
279C2×C4×C41—44no—
280C2×C4×C41—42no—
281C2×C4×C41—42no—
282C2×C4×C41—42no—
283C2×C4×C41—44no—
284C2×C4×C41—81no—
285C2×C4×C41—81no—
286C2×C4×C41#12011yescharacteristic
287C2×C4×C41#12011yescharacteristic
288C2×C4×C41#12012yes—
289C2×C4×C41#12312yes—
290#24059C4111yesmaximal, characteristic, direct product factor, Sylow
291C22×C41—22no—
292C22×C41—21no—
293C22×C41—22no—
294C22×C41—21no—
295C22×C41—22no—
296C22×C41—42no—
297C22×C41—42no—
298C22×C41—42no—
299C22×C41—42no—
300C22×C41—41no—
301C22×C41—41no—
302C22×C41—41no—
303C22×C41—42no—
304C22×C41—41no—
305C22×C41—42no—
306C22×C41—41no—
307C22×C41—42no—
308C22×C41—81no—
309C22×C41—81no—
310C22×C41—84no—
311C22×C41—162no—
312C22×C41C2×#12011yescharacteristic
313C22×C41C2×#12311yescharacteristic
314C22×C41#53411yescharacteristic
315C22×C41#54614yes—
316C4×C41—21no—
317C4×C41—21no—
318C4×C41—22no—
319C4×C41—22no—
320C4×C41—42no—
321C4×C41—41no—
322C4×C41—42no—
323C4×C41—42no—
324C4×C41—82no—
325C4×C41—84no—
326C4×C41—82no—
327#3485C2×C4111yescharacteristic
328#2623C2×C4112yescomplemented
329#2311C2×C4111yescharacteristic, complemented
330#2044C2×C4114yescomplemented
331#1899C2×C4112yescomplemented
332#1867C2×C4112yescomplemented
333#1790C2×C4112yescomplemented
334#1782C2×C4111yescharacteristic, complemented
335C2×C41—21no—
336C2×C41—24no—
337C2×C41—21no—
338C2×C41—42no—
339C2×C41—81no—
340C2×C41—82no—
341C2×C41#240511yescharacteristic
342C2×C41#302711yescharacteristic
343C2×C41#302811yescharacteristic
344C2×H(5,𝔽2)C22×C4111yescharacteristic
345C22×#41C22×C4111yescharacteristic
346#547—22no—
347#547—22no—
348#546—22no—
349#546—22no—
350#536—21no—
351#534—21no—
352#534C22×C4112yes—
353#525—21no—
354C2×#123—21no—
355C2×#123C22×C4112yes—
356#518C22×C4111yescharacteristic
357C2×#117C22×C4111yescharacteristic, complemented
358C22×C4⋊C4C22×C4111yescharacteristic, complemented
359C2×#99C22×C4112yes—
360C2×#99C22×C4114yes—
361C2×#99C22×C4112yescomplemented
362#400—22no—
363#399—22no—
364#398—22no—
365#397—22no—
366#391—21no—
367#390—21no—
368#386C22×C4112yes—
369#386C22×C4114yes—
370#386C22×C4112yescomplemented
371#385C22×C4114yescomplemented
372#381C22×C4112yescomplemented
373#380—21no—
374#380—21no—
375#342C22×C4114yescomplemented
376C41#2405911yescharacteristic, direct product factor, Sylow
377H(5,𝔽2)—24no—
378H(5,𝔽2)—42no—
379C2×#41—24no—
380C2×#41—21no—
381C2×#41—22no—
382C2×#41—22no—
383C2×#41—22no—
384C2×#41—42no—
385C22×Q8D4×C4111yescharacteristic
386C22×D4—21no—
387C22×D4—22no—
388C22×D4D4×C4111yescharacteristic
389C22×D4C23×C4112yes—
390C22×D4C23×C4111yescharacteristic
391C22×D4C23×C4111yescharacteristic
392C23×C4D4×C4111yescharacteristic
393C23×C4D4×C4111yescharacteristic
394C23×C4C23×C4111yescharacteristic
395C23×C4C23×C4112yes—
396C4⋊Q8—22no—
397#124—22no—
398#123—22no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403#123—22no—
404#123—22no—
405#123—22no—
406#123—22no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411#121—21no—
412#121—21no—
413#121—22no—
414#121—22no—
415#121—22no—
416#121—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424#117—21no—
425#117—42no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—24no—
434C2×C4⋊C4D4×C4112yes—
435C2×C4⋊C4D4×C4112yes—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—24no—
442C2×C22⋊C4D4×C4112yes—
443C2×C22⋊C4D4×C4112yes—
444C2×C22⋊C4C23×C4112yes—
445C2×C22⋊C4C23×C4114yes—
446C2×C22⋊C4C23×C4112yes—
447C2×C42—21no—
448#99—24no—
449#99—44no—
450#99—42no—
451#95—22no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—24no—
456#95—24no—
457C24—21no—
458C24C2×D4×C4112yes—
459C24C2×D4×C4111yescharacteristic
460C24C2×D4×C4111yescharacteristic
461C24C24×C4111yescharacteristic
462#41—48no—
463#41—48no—
464#41—48no—
465#41—42no—
466#41—42no—
467C2×Q8—21no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—41no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—24no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—41no—
487C2×D4—48no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4#41×C4114yes—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—82no—
510C22×C4C2×D4×C4112yes—
511C22×C4C2×D4×C4111yescharacteristic
512C22×C4C2×D4×C4112yes—
513C22×C4C2×D4×C4112yes—
514C22×C4C2×D4×C4111yescharacteristic
515C22×C4C2×D4×C4111yescharacteristic
516C22×C4C2×D4×C4111yescharacteristic
517C22×C4C2×D4×C4112yes—
518C22×C4C2×D4×C4112yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—42no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—42no—
531C4⋊C4—84no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—84no—
549C42—22no—
550C42—42no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—24no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—81no—
568C23—81no—
569C23#123×C4112yes—
570C23C22×D4×C4111yescharacteristic
571C23C22×D4×C4112yes—
572C23C2×#41×C4112yes—
573C23H(5,𝔽2)×C4112yes—
574Q8—44no—
575Q8—42no—
576D4—28no—
577D4—44no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—44no—
583D4—44no—
584D4—82no—
585D4—82no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4#120×C4111yescharacteristic
623C2×C4#120×C4111yescharacteristic
624C2×C4#120×C4112yes—
625C2×C4#123×C4112yes—
626C22—22no—
627C22—21no—
628C22—22no—
629C22—21no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—84no—
646C22—162no—
647C22C2×#120×C4111yescharacteristic
648C22C2×#123×C4111yescharacteristic
649C22C41×#53411yescharacteristic
650C22C41×#54614yes—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—82no—
662C2—21no—
663C2—24no—
664C2—21no—
665C2—42no—
666C2—81no—
667C2—82no—
668C2C41×#240511yescharacteristic
669C2C41×#302711yescharacteristic
670C2C41×#302811yescharacteristic
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC24C24×C41
prime radicalC41×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C23×C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2623C2semidirect
3. C41×#2311C2semidirect
4. C41×#2044C2semidirect
5. C41×#1899C2semidirect
6. C41×#1867C2semidirect
7. C41×#1790C2semidirect
8. C41×#1782C2semidirect
22. C2×#117×C41C22semidirect
23. C22×C4⋊C4×C41C22semidirect
26. C2×#99×C41C22semidirect
35. C41×#386C22semidirect
36. C41×#385C22semidirect
37. C41×#381C22semidirect
40. C41×#342C22semidirect
290. #24059C41direct (N × H)
328. #2623C2×C41semidirect
329. #2311C2×C41semidirect
330. #2044C2×C41semidirect
331. #1899C2×C41semidirect
332. #1867C2×C41semidirect
333. #1790C2×C41semidirect
334. #1782C2×C41semidirect
357. C2×#117C22×C41semidirect
358. C22×C4⋊C4C22×C41semidirect
361. C2×#99C22×C41semidirect
370. #386C22×C41semidirect
371. #385C22×C41semidirect
372. #381C22×C41semidirect
375. #342C22×C41semidirect
376. C41#24059direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#34852
C41×#26232
C41×#23112
C41×#20442
C41×#18992
C41×#18672
C41×#17902
C41×#17822
#2405941