Finite Group Databasebeta

← C61×#29891 #29533324

Subgroups — C61×#29891 #29533324

Order 15616.

1128 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#2847C211yesmaximal, characteristic
2C2×C61×#380C211yesmaximal, characteristic, complemented
3C61×#2077C212yesmaximal, complemented
4C61×#2076C212yesmaximal, complemented
5C61×#2034C212yesmaximal, complemented
6C61×#2033C212yesmaximal, complemented
7C61×#1874C211yesmaximal, characteristic, complemented
8C61×#1874C211yesmaximal, characteristic, complemented
9C61×#1853C211yesmaximal, characteristic, complemented
10C61×#1852C211yesmaximal, characteristic, complemented
11C61×#1779C211yesmaximal, characteristic, complemented
12C24×C4×C61C2211yescharacteristic
13C2×#121×C61C2211yescharacteristic
14C2×#121×C61C2211yescharacteristic
15C2×#121×C61C2211yescharacteristic
16C2×#120×C61—21no—
17C22×C4⋊C4×C61C2211yescharacteristic
18C22×C4⋊C4×C61C2211yescharacteristic
19C22×C22⋊C4×C61—21no—
20C2×#107×C61C2212yescomplemented
21C2×#106×C61C2212yescomplemented
22C2×#102×C61C2212yescomplemented
23C2×#102×C61C2212yescomplemented
24C2×C22⋊C8×C61C2211yescharacteristic
25C2×C22⋊C8×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2212yes—
27C2×C22⋊C8×C61C2212yes—
28C61×#386—21no—
29C61×#386—21no—
30C61×#386—21no—
31C61×#380—21no—
32C61×#380—21no—
33C61×#380C2212yes—
34C61×#380C2212yes—
35C61×#380C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C61×#380C2211yescharacteristic
38C2×#95×C61—21no—
39C61×#340C2212yescomplemented
40C61×#340C2212yescomplemented
41C61×#340C2212yescomplemented
42C61×#340C2212yescomplemented
43C25×C61D411yescharacteristic
44C22×D4×C61—21no—
45C22×D4×C61—21no—
46C22×D4×C61—21no—
47C22×D4×C61C2311yescharacteristic
48C23×C4×C61—22no—
49C23×C4×C61—22no—
50C23×C4×C61—21no—
51C23×C4×C61—21no—
52C23×C4×C61—42no—
53C23×C4×C61—42no—
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61D411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C22×C8×C61C2312yes—
61C22×C8×C61C2312yes—
62#121×C61—22no—
63#121×C61—22no—
64#121×C61—42no—
65#121×C61—42no—
66#120×C61—44no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—42no—
70C2×C4⋊C4×C61—42no—
71C2×C4⋊C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61C2312yes—
73C2×C4⋊C4×C61C2312yes—
74C2×C4⋊C4×C61C2311yescharacteristic
75C2×C4⋊C4×C61C2311yescharacteristic
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—21no—
82C2×C22⋊C4×C61—21no—
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—44no—
85#107×C61—42no—
86#106×C61—42no—
87#102×C61—42no—
88#102×C61—42no—
89C22⋊C8×C61—22no—
90C22⋊C8×C61—22no—
91C22⋊C8×C61—42no—
92C22⋊C8×C61—42no—
93#95×C61—22no—
94#95×C61—22no—
95#95×C61—21no—
96#95×C61—21no—
97#95×C61—21no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—21no—
104C24×C61—41no—
105C24×C61—41no—
106C24×C61C2×D411yescharacteristic
107C24×C61C2×D411yescharacteristic
108C24×C61C2×D411yescharacteristic
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61—41no—
146C22×C4×C61—48no—
147C22×C4×C61—48no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—44no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—44no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—28no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—44no—
202C23×C61—44no—
203C23×C61—44no—
204C23×C61—44no—
205C23×C61—44no—
206C23×C61—44no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61#12312yes—
215C23×C61#13512yes—
216C23×C61AGL(1,ℤ/8)12yes—
217C23×C61AGL(1,ℤ/8)12yes—
218C23×C61#13712yes—
219C23×C61C22×D411yescharacteristic
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—84no—
223D4×C61—84no—
224D4×C61—84no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—44no—
236C2×C4×C61—44no—
237C2×C4×C61—44no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—84no—
258C2×C4×C61—84no—
259C2×C4×C61#12312yes—
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)12yes—
263C2×C4×C61H(5,𝔽2)12yes—
264C8×C61—42no—
265C8×C61—42no—
266#29891C6111yesmaximal, characteristic, direct product factor, Sylow
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—416no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61#48214yes—
296C22×C61C2×#12311yescharacteristic
297C22×C61#53411yescharacteristic
298C22×C61#53411yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—82no—
308C4×C61—82no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61#53812yes—
312C4×C61#53812yes—
313#2847C2×C6111yescharacteristic
314C2×#380C2×C6111yescharacteristic, complemented
315#2077C2×C6112yescomplemented
316#2076C2×C6112yescomplemented
317#2034C2×C6112yescomplemented
318#2033C2×C6112yescomplemented
319#1874C2×C6111yescharacteristic, complemented
320#1874C2×C6111yescharacteristic, complemented
321#1853C2×C6111yescharacteristic, complemented
322#1852C2×C6111yescharacteristic, complemented
323#1779C2×C6111yescharacteristic, complemented
324C2×C61—24no—
325C2×C61—42no—
326C2×C61—42no—
327C2×C61—161no—
328C2×C61#262211yescharacteristic
329C2×C61C2×#48211yescharacteristic
330C2×C61#309211yescharacteristic
331C2×C61#318811yescharacteristic
332C2×C61#319011yescharacteristic
333C2×C61#319211yescharacteristic
334C2×C61#319311yescharacteristic
335C24×C4C22×C6111yescharacteristic
336C2×#121C22×C6111yescharacteristic
337C2×#121C22×C6111yescharacteristic
338C2×#121C22×C6111yescharacteristic
339C2×#120—21no—
340C22×C4⋊C4C22×C6111yescharacteristic
341C22×C4⋊C4C22×C6111yescharacteristic
342C22×C22⋊C4—21no—
343C2×#107C22×C6112yescomplemented
344C2×#106C22×C6112yescomplemented
345C2×#102C22×C6112yescomplemented
346C2×#102C22×C6112yescomplemented
347C2×C22⋊C8C22×C6111yescharacteristic
348C2×C22⋊C8C22×C6111yescharacteristic
349C2×C22⋊C8C22×C6112yes—
350C2×C22⋊C8C22×C6112yes—
351#386—21no—
352#386—21no—
353#386—21no—
354#380—21no—
355#380—21no—
356#380C22×C6111yescharacteristic
357#380C22×C6112yes—
358#380C22×C6111yescharacteristic
359#380C22×C6112yes—
360#380C22×C6111yescharacteristic
361C2×#95—21no—
362#340C22×C6112yescomplemented
363#340C22×C6112yescomplemented
364#340C22×C6112yescomplemented
365#340C22×C6112yescomplemented
366C61#2989111yescharacteristic, direct product factor, Sylow
367C25D4×C6111yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4C23×C6111yescharacteristic
372C23×C4—22no—
373C23×C4—22no—
374C23×C4—21no—
375C23×C4—21no—
376C23×C4—42no—
377C23×C4—42no—
378C23×C4D4×C6111yescharacteristic
379C23×C4D4×C6111yescharacteristic
380C23×C4D4×C6111yescharacteristic
381C23×C4C23×C6111yescharacteristic
382C23×C4C23×C6111yescharacteristic
383C23×C4C23×C6111yescharacteristic
384C22×C8C23×C6112yes—
385C22×C8C23×C6112yes—
386#121—22no—
387#121—22no—
388#121—42no—
389#121—42no—
390#120—44no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—42no—
394C2×C4⋊C4—42no—
395C2×C4⋊C4C23×C6112yes—
396C2×C4⋊C4C23×C6112yes—
397C2×C4⋊C4C23×C6111yescharacteristic
398C2×C4⋊C4C23×C6111yescharacteristic
399C2×C4⋊C4C23×C6111yescharacteristic
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—44no—
409#107—42no—
410#106—42no—
411#102—42no—
412#102—42no—
413C22⋊C8—22no—
414C22⋊C8—22no—
415C22⋊C8—42no—
416C22⋊C8—42no—
417#95—22no—
418#95—22no—
419#95—21no—
420#95—21no—
421#95—21no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—41no—
429C24—41no—
430C24C2×D4×C6111yescharacteristic
431C24C2×D4×C6111yescharacteristic
432C24C2×D4×C6111yescharacteristic
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4C2×D4×C6111yescharacteristic
479C22×C4C2×D4×C6111yescharacteristic
480C22×C4C2×D4×C6111yescharacteristic
481C22×C4#41×C6112yes—
482C22×C4#41×C6112yes—
483C22×C4C24×C6111yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—28no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#123×C6112yes—
539C23#135×C6112yes—
540C23AGL(1,ℤ/8)×C6112yes—
541C23AGL(1,ℤ/8)×C6112yes—
542C23#137×C6112yes—
543C23C22×D4×C6111yescharacteristic
544D4—42no—
545D4—42no—
546D4—84no—
547D4—84no—
548D4—84no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4#123×C6112yes—
584C2×C4C2×#41×C6111yescharacteristic
585C2×C4C2×#41×C6111yescharacteristic
586C2×C4H(5,𝔽2)×C6112yes—
587C2×C4H(5,𝔽2)×C6112yes—
588C8—42no—
589C8—42no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—416no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C61×#48214yes—
619C22C2×#123×C6111yescharacteristic
620C22C61×#53411yescharacteristic
621C22C61×#53411yescharacteristic
622C22C2×#135×C6111yescharacteristic
623C22C2×AGL(1,ℤ/8)×C6111yescharacteristic
624C22C2×AGL(1,ℤ/8)×C6111yescharacteristic
625C22C2×#137×C6111yescharacteristic
626C4—22no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—82no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4C61×#53812yes—
635C4C61×#53812yes—
636C2—24no—
637C2—42no—
638C2—42no—
639C2—161no—
640C2C61×#262211yescharacteristic
641C2C2×C61×#48211yescharacteristic
642C2C61×#309211yescharacteristic
643C2C61×#318811yescharacteristic
644C2C61×#319011yescharacteristic
645C2C61×#319211yescharacteristic
646C2C61×#319311yescharacteristic
subgroupG/H
center, socleC23×C61C22×D4
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C61 ◁ C24×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#380C2semidirect
3. C61×#2077C2semidirect
4. C61×#2076C2semidirect
5. C61×#2034C2semidirect
6. C61×#2033C2semidirect
7. C61×#1874C2semidirect
8. C61×#1874C2semidirect
9. C61×#1853C2semidirect
10. C61×#1852C2semidirect
11. C61×#1779C2semidirect
20. C2×#107×C61C22semidirect
21. C2×#106×C61C22semidirect
22. C2×#102×C61C22semidirect
23. C2×#102×C61C22semidirect
39. C61×#340C22semidirect
40. C61×#340C22semidirect
41. C61×#340C22semidirect
42. C61×#340C22semidirect
266. #29891C61direct (N × H)
314. C2×#380C2×C61semidirect
315. #2077C2×C61semidirect
316. #2076C2×C61semidirect
317. #2034C2×C61semidirect
318. #2033C2×C61semidirect
319. #1874C2×C61semidirect
320. #1874C2×C61semidirect
321. #1853C2×C61semidirect
322. #1852C2×C61semidirect
323. #1779C2×C61semidirect
343. C2×#107C22×C61semidirect
344. C2×#106C22×C61semidirect
345. C2×#102C22×C61semidirect
346. C2×#102C22×C61semidirect
362. #340C22×C61semidirect
363. #340C22×C61semidirect
364. #340C22×C61semidirect
365. #340C22×C61semidirect
366. C61#29891direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#28472
C2×C61×#3802
C61×#20772
C61×#20762
C61×#20342
C61×#20332
C61×#18742
C61×#18532
C61×#18522
C61×#17792
#2989161