Finite Group Databasebeta

← C41×#26854 #23803055

Subgroups — C41×#26854 #23803055

Order 10496.

724 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3137C211yesmaximal, characteristic, complemented
2C41×#3097C211yesmaximal, characteristic, complemented
3C41×#3081C211yesmaximal, characteristic, complemented
4C41×#2946C211yesmaximal, characteristic
5C41×#2064C211yesmaximal, characteristic, complemented
6C41×#2060C211yesmaximal, characteristic, complemented
7C41×#2051C211yesmaximal, characteristic
8C41×#2003C211yesmaximal, characteristic
9C41×#1959C211yesmaximal, characteristic, complemented
10C41×#1938C211yesmaximal, characteristic, complemented
11C41×#1931C211yesmaximal, characteristic, complemented
12C41×#1930C211yesmaximal, characteristic, complemented
13C41×#1885C211yesmaximal, characteristic, complemented
14C41×#1859C211yesmaximal, characteristic, complemented
15C41×#1773C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)×C41C2211yescharacteristic
17C41×#544C2211yescharacteristic
18C41×#536C2211yescharacteristic
19C2×#124×C41C2211yescharacteristic
20C41×#520C2211yescharacteristic
21C41×#518C2211yescharacteristic
22C41×#491—21no—
23C41×#489—21no—
24C41×#488—21no—
25C41×#487—21no—
26C41×#485—21no—
27C41×#485—21no—
28C41×#484—21no—
29C41×#483—21no—
30C41×#478—21no—
31C41×#478—21no—
32C41×#476—21no—
33C41×#474—21no—
34C41×#442—21no—
35C41×#440—21no—
36C4×SD16×C41—21no—
37C4×D8×C41—21no—
38C2×#108×C41C2211yescharacteristic
39C41×#428C2211yescharacteristic
40C41×#428C2211yescharacteristic
41C41×#428C2211yescharacteristic
42C41×#427C2211yescharacteristic
43C41×#424C2211yescharacteristic
44C2×#104×C41C2211yescharacteristic
45C41×#419C2211yescharacteristic
46C41×#419C2211yescharacteristic
47C41×#418C2211yescharacteristic
48C41×#417C2211yescharacteristic
49C41×#416C2211yescharacteristic
50C2×#103×C41C2211yescharacteristic
51C2×#103×C41C2211yescharacteristic
52C2×#102×C41C2211yescharacteristic
53C2×#102×C41C2211yescharacteristic
54C2×#101×C41C2211yescharacteristic
55C2×#100×C41C2211yescharacteristic
56C41×#408C2211yescharacteristic
57C41×#405C2211yescharacteristic
58C41×#405C2211yescharacteristic
59C2×#97×C41C2211yescharacteristic
60C41×#384C2211yescharacteristic
61C41×#343C2211yescharacteristic
62C41×#343C2211yescharacteristic
63C41×#341C2211yescharacteristic
64C41×#339C2211yescharacteristic
65C41×#339C2211yescharacteristic
66C41×#339C2211yescharacteristic
67C2×#41×C41C2311yescharacteristic
68C22×Q8×C41C2311yescharacteristic
69C22×D4×C41C2311yescharacteristic
70AGL(1,ℤ/8)×C41—22no—
71AGL(1,ℤ/8)×C41—22no—
72C2×SD16×C41—21no—
73C2×D8×C41—21no—
74C2×M16×C41C2311yescharacteristic
75C2×M16×C41C2311yescharacteristic
76C2×M16×C41C2311yescharacteristic
77C2×M16×C41C2311yescharacteristic
78C22×C8×C41C2311yescharacteristic
79C22×C8×C41C2311yescharacteristic
80C4⋊Q8×C41—21no—
81C4⋊Q8×C41—21no—
82C4⋊Q8×C41—21no—
83#125×C41—21no—
84#125×C41—21no—
85#124×C41—21no—
86#124×C41—22no—
87#124×C41—22no—
88#123×C41—41no—
89C22⋊Q8×C41—21no—
90C22⋊Q8×C41—21no—
91C22⋊Q8×C41—41no—
92C4×Q8×C41—21no—
93C4×D4×C41—21no—
94C4×D4×C41—21no—
95C4×D4×C41—21no—
96C4×D4×C41—21no—
97C4×D4×C41—21no—
98C4×D4×C41—41no—
99#117×C41—21no—
100#117×C41C2311yescharacteristic
101#117×C41C2311yescharacteristic
102#117×C41C2311yescharacteristic
103C2×C4⋊C4×C41—21no—
104C2×C4⋊C4×C41—41no—
105C2×C4⋊C4×C41C2311yescharacteristic
106C2×C4⋊C4×C41C2311yescharacteristic
107C2×C22⋊C4×C41—41no—
108C2×C22⋊C4×C41—41no—
109C2×C42×C41C2311yescharacteristic
110#108×C41—22no—
111#107×C41—21no—
112#107×C41—21no—
113#107×C41—21no—
114#107×C41D411yescharacteristic
115#107×C41D411yescharacteristic
116#106×C41—21no—
117#106×C41—21no—
118#106×C41—21no—
119#106×C41D411yescharacteristic
120#106×C41D411yescharacteristic
121C4⋊C8×C41—21no—
122C4⋊C8×C41—21no—
123#104×C41—22no—
124#103×C41—21no—
125#103×C41—21no—
126#103×C41—21no—
127#103×C41—21no—
128#103×C41—21no—
129#103×C41—21no—
130#103×C41—21no—
131#103×C41—21no—
132#103×C41—21no—
133#103×C41D411yescharacteristic
134#103×C41D411yescharacteristic
135#102×C41—21no—
136#102×C41—21no—
137#102×C41—21no—
138#102×C41—21no—
139#102×C41—21no—
140#102×C41—21no—
141#102×C41—21no—
142#102×C41D411yescharacteristic
143#102×C41D411yescharacteristic
144#101×C41—22no—
145#100×C41—22no—
146C22⋊C8×C41—21no—
147C22⋊C8×C41—21no—
148#97×C41—21no—
149#97×C41—21no—
150#97×C41—22no—
151C4×C8×C41—21no—
152C4×C8×C41—21no—
153C24×C41—21no—
154#41×C41—21no—
155#41×C41—21no—
156#41×C41—21no—
157#41×C41—21no—
158C2×Q8×C41—21no—
159C2×Q8×C41—21no—
160C2×Q8×C41—21no—
161C2×Q8×C41—21no—
162C2×Q8×C41—21no—
163C2×Q8×C41—21no—
164C2×Q8×C41—41no—
165C2×Q8×C41C2×D411yescharacteristic
166C2×D4×C41—21no—
167C2×D4×C41—21no—
168C2×D4×C41—21no—
169C2×D4×C41—21no—
170C2×D4×C41—21no—
171C2×D4×C41—21no—
172C2×D4×C41C2×D411yescharacteristic
173C2×D4×C41#4112yes—
174C22×C4×C41—21no—
175C22×C4×C41—21no—
176C22×C4×C41—21no—
177C22×C4×C41—21no—
178C22×C4×C41—21no—
179C22×C4×C41—41no—
180C22×C4×C41—41no—
181C22×C4×C41—41no—
182C22×C4×C41C2411yescharacteristic
183SD16×C41—24no—
184D8×C41—24no—
185M16×C41—21no—
186M16×C41—21no—
187M16×C41—21no—
188M16×C41—21no—
189M16×C41—21no—
190M16×C41—21no—
191M16×C41—21no—
192M16×C41#4112yes—
193C2×C8×C41—21no—
194C2×C8×C41—21no—
195C2×C8×C41—21no—
196C2×C8×C41—21no—
197C2×C8×C41—21no—
198C2×C8×C41—21no—
199C2×C8×C41—21no—
200C2×C8×C41—21no—
201C2×C8×C41C2×D411yescharacteristic
202C2×C8×C41C2×D411yescharacteristic
203C2×C8×C41C2×D411yescharacteristic
204C2×C8×C41C2×D411yescharacteristic
205C4⋊C4×C41—21no—
206C4⋊C4×C41—21no—
207C4⋊C4×C41—21no—
208C4⋊C4×C41—21no—
209C4⋊C4×C41—41no—
210C4⋊C4×C41—41no—
211C4⋊C4×C41—41no—
212C4⋊C4×C41—41no—
213C4⋊C4×C41—41no—
214C4⋊C4×C41—41no—
215C4⋊C4×C41—82no—
216C4⋊C4×C41C2×D411yescharacteristic
217C4⋊C4×C41C2×D411yescharacteristic
218C4⋊C4×C41C2×D411yescharacteristic
219C4⋊C4×C41C2×D411yescharacteristic
220C4⋊C4×C41C2×D411yescharacteristic
221C4⋊C4×C41C2×D411yescharacteristic
222C22⋊C4×C41—21no—
223C22⋊C4×C41—21no—
224C22⋊C4×C41—21no—
225C22⋊C4×C41—41no—
226C22⋊C4×C41—41no—
227C22⋊C4×C41—44no—
228C22⋊C4×C41—41no—
229C22⋊C4×C41—41no—
230C22⋊C4×C41—41no—
231C42×C41—21no—
232C42×C41—21no—
233C42×C41—21no—
234C42×C41—21no—
235C42×C41—41no—
236C42×C41#4112yes—
237C23×C41—21no—
238C23×C41—22no—
239C23×C41—41no—
240C23×C41—41no—
241C23×C41—81no—
242C23×C41—81no—
243C23×C41H(5,𝔽2)11yescharacteristic
244Q8×C41—21no—
245Q8×C41—21no—
246Q8×C41—41no—
247Q8×C41—41no—
248Q8×C41—41no—
249Q8×C41—41no—
250Q8×C41—161no—
251D4×C41—21no—
252D4×C41—22no—
253D4×C41—21no—
254D4×C41—22no—
255D4×C41—41no—
256D4×C41—41no—
257D4×C41—41no—
258D4×C41—41no—
259C2×C4×C41—21no—
260C2×C4×C41—21no—
261C2×C4×C41—21no—
262C2×C4×C41—21no—
263C2×C4×C41—21no—
264C2×C4×C41—21no—
265C2×C4×C41—21no—
266C2×C4×C41—22no—
267C2×C4×C41—41no—
268C2×C4×C41—41no—
269C2×C4×C41—41no—
270C2×C4×C41—41no—
271C2×C4×C41—41no—
272C2×C4×C41—41no—
273C2×C4×C41—41no—
274C2×C4×C41—41no—
275C2×C4×C41—41no—
276C2×C4×C41—81no—
277C2×C4×C41—81no—
278C2×C4×C41—161no—
279C2×C4×C41—161no—
280C2×C4×C41—161no—
281C2×C4×C41C22×D411yescharacteristic
282C2×C4×C41C22×D411yescharacteristic
283C2×C4×C41C2×#4111yescharacteristic
284C2×C4×C41C2×#4111yescharacteristic
285C2×C4×C41C2×#4111yescharacteristic
286C2×C4×C412−1+411yescharacteristic
287C8×C41—22no—
288C8×C41—21no—
289C8×C41—21no—
290C8×C41—21no—
291C8×C41—21no—
292C8×C41—41no—
293C8×C41—41no—
294C8×C41—41no—
295#26854C4111yesmaximal, characteristic, direct product factor, Sylow
296C22×C41—21no—
297C22×C41—21no—
298C22×C41—41no—
299C22×C41—42no—
300C22×C41—41no—
301C22×C41—41no—
302C22×C41—82no—
303C22×C41—81no—
304C22×C41—81no—
305C22×C41—161no—
306C22×C41#54211yescharacteristic
307C22×C41D4211yescharacteristic
308C22×C41#55111yescharacteristic
309C4×C41—42no—
310C4×C41—41no—
311C4×C41—41no—
312C4×C41—41no—
313C4×C41—41no—
314C4×C41—41no—
315C4×C41—41no—
316C4×C41—41no—
317C4×C41—81no—
318C4×C41—81no—
319C4×C41—81no—
320C4×C41#52511yescharacteristic
321C4×C41#53211yescharacteristic
322C4×C41#54711yescharacteristic
323C4×C41#54811yescharacteristic
324#3137C2×C4111yescharacteristic, complemented
325#3097C2×C4111yescharacteristic, complemented
326#3081C2×C4111yescharacteristic, complemented
327#2946C2×C4111yescharacteristic
328#2064C2×C4111yescharacteristic, complemented
329#2060C2×C4111yescharacteristic, complemented
330#2051C2×C4111yescharacteristic
331#2003C2×C4111yescharacteristic
332#1959C2×C4111yescharacteristic, complemented
333#1938C2×C4111yescharacteristic, complemented
334#1931C2×C4111yescharacteristic, complemented
335#1930C2×C4111yescharacteristic, complemented
336#1885C2×C4111yescharacteristic, complemented
337#1859C2×C4111yescharacteristic, complemented
338#1773C2×C4111yescharacteristic, complemented
339C2×C41—21no—
340C2×C41—21no—
341C2×C41—81no—
342C2×C41—82no—
343C2×C41#246111yescharacteristic
344C2×C41#328811yescharacteristic
345C2×C41#328911yescharacteristic
346C2×AGL(1,ℤ/8)C22×C4111yescharacteristic
347#544C22×C4111yescharacteristic
348#536C22×C4111yescharacteristic
349C2×#124C22×C4111yescharacteristic
350#520C22×C4111yescharacteristic
351#518C22×C4111yescharacteristic
352#491—21no—
353#489—21no—
354#488—21no—
355#487—21no—
356#485—21no—
357#485—21no—
358#484—21no—
359#483—21no—
360#478—21no—
361#478—21no—
362#476—21no—
363#474—21no—
364#442—21no—
365#440—21no—
366C4×SD16—21no—
367C4×D8—21no—
368C2×#108C22×C4111yescharacteristic
369#428C22×C4111yescharacteristic
370#428C22×C4111yescharacteristic
371#428C22×C4111yescharacteristic
372#427C22×C4111yescharacteristic
373#424C22×C4111yescharacteristic
374C2×#104C22×C4111yescharacteristic
375#419C22×C4111yescharacteristic
376#419C22×C4111yescharacteristic
377#418C22×C4111yescharacteristic
378#417C22×C4111yescharacteristic
379#416C22×C4111yescharacteristic
380C2×#103C22×C4111yescharacteristic
381C2×#103C22×C4111yescharacteristic
382C2×#102C22×C4111yescharacteristic
383C2×#102C22×C4111yescharacteristic
384C2×#101C22×C4111yescharacteristic
385C2×#100C22×C4111yescharacteristic
386#408C22×C4111yescharacteristic
387#405C22×C4111yescharacteristic
388#405C22×C4111yescharacteristic
389C2×#97C22×C4111yescharacteristic
390#384C22×C4111yescharacteristic
391#343C22×C4111yescharacteristic
392#343C22×C4111yescharacteristic
393#341C22×C4111yescharacteristic
394#339C22×C4111yescharacteristic
395#339C22×C4111yescharacteristic
396#339C22×C4111yescharacteristic
397C41#2685411yescharacteristic, direct product factor, Sylow
398C2×#41C23×C4111yescharacteristic
399C22×Q8C23×C4111yescharacteristic
400C22×D4C23×C4111yescharacteristic
401AGL(1,ℤ/8)—22no—
402AGL(1,ℤ/8)—22no—
403C2×SD16—21no—
404C2×D8—21no—
405C2×M16C23×C4111yescharacteristic
406C2×M16C23×C4111yescharacteristic
407C2×M16C23×C4111yescharacteristic
408C2×M16C23×C4111yescharacteristic
409C22×C8C23×C4111yescharacteristic
410C22×C8C23×C4111yescharacteristic
411C4⋊Q8—21no—
412C4⋊Q8—21no—
413C4⋊Q8—21no—
414#125—21no—
415#125—21no—
416#124—21no—
417#124—22no—
418#124—22no—
419#123—41no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—41no—
423C4×Q8—21no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—41no—
430#117—21no—
431#117C23×C4111yescharacteristic
432#117C23×C4111yescharacteristic
433#117C23×C4111yescharacteristic
434C2×C4⋊C4—21no—
435C2×C4⋊C4—41no—
436C2×C4⋊C4C23×C4111yescharacteristic
437C2×C4⋊C4C23×C4111yescharacteristic
438C2×C22⋊C4—41no—
439C2×C22⋊C4—41no—
440C2×C42C23×C4111yescharacteristic
441#108—22no—
442#107—21no—
443#107—21no—
444#107—21no—
445#107D4×C4111yescharacteristic
446#107D4×C4111yescharacteristic
447#106—21no—
448#106—21no—
449#106—21no—
450#106D4×C4111yescharacteristic
451#106D4×C4111yescharacteristic
452C4⋊C8—21no—
453C4⋊C8—21no—
454#104—22no—
455#103—21no—
456#103—21no—
457#103—21no—
458#103—21no—
459#103—21no—
460#103—21no—
461#103—21no—
462#103—21no—
463#103—21no—
464#103D4×C4111yescharacteristic
465#103D4×C4111yescharacteristic
466#102—21no—
467#102—21no—
468#102—21no—
469#102—21no—
470#102—21no—
471#102—21no—
472#102—21no—
473#102D4×C4111yescharacteristic
474#102D4×C4111yescharacteristic
475#101—22no—
476#100—22no—
477C22⋊C8—21no—
478C22⋊C8—21no—
479#97—21no—
480#97—21no—
481#97—22no—
482C4×C8—21no—
483C4×C8—21no—
484C24—21no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—41no—
496C2×Q8C2×D4×C4111yescharacteristic
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4C2×D4×C4111yescharacteristic
504C2×D4#41×C4112yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4—41no—
513C22×C4C24×C4111yescharacteristic
514SD16—24no—
515D8—24no—
516M16—21no—
517M16—21no—
518M16—21no—
519M16—21no—
520M16—21no—
521M16—21no—
522M16—21no—
523M16#41×C4112yes—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8C2×D4×C4111yescharacteristic
533C2×C8C2×D4×C4111yescharacteristic
534C2×C8C2×D4×C4111yescharacteristic
535C2×C8C2×D4×C4111yescharacteristic
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—82no—
547C4⋊C4C2×D4×C4111yescharacteristic
548C4⋊C4C2×D4×C4111yescharacteristic
549C4⋊C4C2×D4×C4111yescharacteristic
550C4⋊C4C2×D4×C4111yescharacteristic
551C4⋊C4C2×D4×C4111yescharacteristic
552C4⋊C4C2×D4×C4111yescharacteristic
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—44no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—41no—
567C42#41×C4112yes—
568C23—21no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—81no—
573C23—81no—
574C23H(5,𝔽2)×C4111yescharacteristic
575Q8—21no—
576Q8—21no—
577Q8—41no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—161no—
582D4—21no—
583D4—22no—
584D4—21no—
585D4—22no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—161no—
610C2×C4—161no—
611C2×C4—161no—
612C2×C4C22×D4×C4111yescharacteristic
613C2×C4C22×D4×C4111yescharacteristic
614C2×C4C2×#41×C4111yescharacteristic
615C2×C4C2×#41×C4111yescharacteristic
616C2×C4C2×#41×C4111yescharacteristic
617C2×C42−1+4×C4111yescharacteristic
618C8—22no—
619C8—21no—
620C8—21no—
621C8—21no—
622C8—21no—
623C8—41no—
624C8—41no—
625C8—41no—
626C22—21no—
627C22—21no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—82no—
633C22—81no—
634C22—81no—
635C22—161no—
636C22C41×#54211yescharacteristic
637C22D42×C4111yescharacteristic
638C22C41×#55111yescharacteristic
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4C41×#52511yescharacteristic
651C4C41×#53211yescharacteristic
652C4C41×#54711yescharacteristic
653C4C41×#54811yescharacteristic
654C2—21no—
655C2—21no—
656C2—81no—
657C2—82no—
658C2C41×#246111yescharacteristic
659C2C41×#328811yescharacteristic
660C2C41×#328911yescharacteristic
subgroupG/H
center, socleC22×C41D42
commutator, Frattini, cosocleC22×C4C24×C41
prime radicalC2×AGL(1,ℤ/8)×C41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C22×C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C41×#3137C2semidirect
2. C41×#3097C2semidirect
3. C41×#3081C2semidirect
5. C41×#2064C2semidirect
6. C41×#2060C2semidirect
9. C41×#1959C2semidirect
10. C41×#1938C2semidirect
11. C41×#1931C2semidirect
12. C41×#1930C2semidirect
13. C41×#1885C2semidirect
14. C41×#1859C2semidirect
15. C41×#1773C2semidirect
295. #26854C41direct (N × H)
324. #3137C2×C41semidirect
325. #3097C2×C41semidirect
326. #3081C2×C41semidirect
328. #2064C2×C41semidirect
329. #2060C2×C41semidirect
332. #1959C2×C41semidirect
333. #1938C2×C41semidirect
334. #1931C2×C41semidirect
335. #1930C2×C41semidirect
336. #1885C2×C41semidirect
337. #1859C2×C41semidirect
338. #1773C2×C41semidirect
397. C41#26854direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#31372
C41×#30972
C41×#30812
C41×#29462
C41×#20642
C41×#20602
C41×#20512
C41×#20032
C41×#19592
C41×#19382
C41×#19312
C41×#19302
C41×#18852
C41×#18592
C41×#17732
#2685441