Finite Group Databasebeta

← C61×#24557 #29527990

Subgroups — C61×#24557 #29527990

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#481C211yesmaximal, characteristic, complemented
2C2×C61×#407C211yesmaximal, characteristic, complemented
3C61×#2847C211yesmaximal, characteristic
4C2×C61×#380C211yesmaximal, characteristic, complemented
5C61×#2077C212yesmaximal, complemented
6C61×#2033C212yesmaximal, complemented
7C61×#1895C212yesmaximal, complemented
8C61×#1874C211yesmaximal, characteristic, complemented
9C61×#1852C211yesmaximal, characteristic, complemented
10C61×#1824C211yesmaximal, characteristic, complemented
11C61×#1809C212yesmaximal, complemented
12C24×C4×C61C2211yescharacteristic
13C22×M16×C61C2212yes—
14C2×#121×C61C2211yescharacteristic
15C2×#121×C61C2211yescharacteristic
16C2×#121×C61C2211yescharacteristic
17C2×#120×C61—21no—
18C22×C4⋊C4×C61C2211yescharacteristic
19C22×C4⋊C4×C61C2211yescharacteristic
20C22×C22⋊C4×C61—21no—
21C61×#481—24no—
22C2×#107×C61C2212yescomplemented
23C2×#106×C61C2212yescomplemented
24C2×#102×C61C2212yescomplemented
25C2×#102×C61C2212yescomplemented
26C61×#407—24no—
27C2×C22⋊C8×C61C2212yes—
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#386—21no—
31C61×#386—21no—
32C61×#386—21no—
33C61×#380—21no—
34C61×#380—21no—
35C61×#380C2212yes—
36C61×#380C2212yes—
37C61×#380C2211yescharacteristic
38C61×#380C2211yescharacteristic
39C61×#380C2211yescharacteristic
40C2×#95×C61—21no—
41C61×#340C2212yescomplemented
42C61×#340C2212yescomplemented
43C61×#340C2212yescomplemented
44C61×#340C2212yescomplemented
45C25×C61D411yescharacteristic
46C22×D4×C61—21no—
47C22×D4×C61—21no—
48C22×D4×C61—21no—
49C22×D4×C61C2311yescharacteristic
50C23×C4×C61—22no—
51C23×C4×C61—22no—
52C23×C4×C61—24no—
53C23×C4×C61—21no—
54C23×C4×C61—21no—
55C23×C4×C61—42no—
56C23×C4×C61D411yescharacteristic
57C23×C4×C61D411yescharacteristic
58C23×C4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C23×C4×C61C2311yescharacteristic
61C23×C4×C61C2311yescharacteristic
62C2×M16×C61—24no—
63C2×M16×C61—24no—
64C2×M16×C61—42no—
65C22×C8×C61C2312yes—
66C22×C8×C61C2312yes—
67#121×C61—22no—
68#121×C61—22no—
69#121×C61—24no—
70#121×C61—42no—
71#120×C61—44no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—42no—
77C2×C4⋊C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61C2312yes—
79C2×C4⋊C4×C61C2312yes—
80C2×C4⋊C4×C61C2311yescharacteristic
81C2×C4⋊C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61—22no—
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—21no—
85C2×C22⋊C4×C61—21no—
86C2×C22⋊C4×C61—21no—
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—44no—
91#107×C61—42no—
92#106×C61—24no—
93#102×C61—24no—
94#102×C61—42no—
95C22⋊C8×C61—22no—
96C22⋊C8×C61—24no—
97C22⋊C8×C61D414yescomplemented
98#95×C61—22no—
99#95×C61—22no—
100#95×C61—21no—
101#95×C61—21no—
102#95×C61—21no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61—22no—
107C24×C61—22no—
108C24×C61—21no—
109C24×C61—41no—
110C24×C61—41no—
111C24×C61C2×D411yescharacteristic
112C24×C61C2×D411yescharacteristic
113C24×C61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—41no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61—48no—
151C22×C4×C61—48no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—42no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61#4112yes—
163C22×C4×C61#4114yes—
164C22×C4×C61C2411yescharacteristic
165M16×C61—48no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61C2×D414yes—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—44no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—44no—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—42no—
189C22⋊C4×C61—42no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—28no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—44no—
208C23×C61—44no—
209C23×C61—44no—
210C23×C61—44no—
211C23×C61—44no—
212C23×C61—44no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61#12112yes—
221C23×C61#12312yes—
222C23×C61AGL(1,ℤ/8)12yes—
223C23×C61AGL(1,ℤ/8)12yes—
224C23×C61#13712yes—
225C23×C61C22×D411yescharacteristic
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—84no—
229D4×C61—84no—
230D4×C61—84no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—44no—
243C2×C4×C61—44no—
244C2×C4×C61—44no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—42no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—42no—
261C2×C4×C61—41no—
262C2×C4×C61—84no—
263C2×C4×C61—84no—
264C2×C4×C61#12112yes—
265C2×C4×C61#12312yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4112yes—
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)12yes—
270C8×C61—24no—
271C8×C61—42no—
272#24557C6111yesmaximal, characteristic, direct product factor, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—416no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61#46814yes—
302C22×C61C2×#12111yescharacteristic
303C22×C61C2×#12311yescharacteristic
304C22×C61#53411yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C61C2×#13711yescharacteristic
308C22×C61#57511yescharacteristic
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—41no—
312C4×C61—81no—
313C4×C61—82no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61—82no—
317C4×C61#54612yes—
318C4×C61#55112yes—
319C2×#481C2×C6111yescharacteristic, complemented
320C2×#407C2×C6111yescharacteristic, complemented
321#2847C2×C6111yescharacteristic
322C2×#380C2×C6111yescharacteristic, complemented
323#2077C2×C6112yescomplemented
324#2033C2×C6112yescomplemented
325#1895C2×C6112yescomplemented
326#1874C2×C6111yescharacteristic, complemented
327#1852C2×C6111yescharacteristic, complemented
328#1824C2×C6111yescharacteristic, complemented
329#1809C2×C6112yescomplemented
330C2×C61—24no—
331C2×C61—42no—
332C2×C61—42no—
333C2×C61—161no—
334C2×C61#241211yescharacteristic
335C2×C61C2×#46811yescharacteristic
336C2×C61#305511yescharacteristic
337C2×C61#309211yescharacteristic
338C2×C61#309411yescharacteristic
339C2×C61#319211yescharacteristic
340C2×C61#319611yescharacteristic
341C24×C4C22×C6111yescharacteristic
342C22×M16C22×C6112yes—
343C2×#121C22×C6111yescharacteristic
344C2×#121C22×C6111yescharacteristic
345C2×#121C22×C6111yescharacteristic
346C2×#120—21no—
347C22×C4⋊C4C22×C6111yescharacteristic
348C22×C4⋊C4C22×C6111yescharacteristic
349C22×C22⋊C4—21no—
350#481—24no—
351C2×#107C22×C6112yescomplemented
352C2×#106C22×C6112yescomplemented
353C2×#102C22×C6112yescomplemented
354C2×#102C22×C6112yescomplemented
355#407—24no—
356C2×C22⋊C8C22×C6112yes—
357C2×C22⋊C8C22×C6111yescharacteristic
358C2×C22⋊C8C22×C6111yescharacteristic
359#386—21no—
360#386—21no—
361#386—21no—
362#380—21no—
363#380—21no—
364#380C22×C6111yescharacteristic
365#380C22×C6112yes—
366#380C22×C6111yescharacteristic
367#380C22×C6112yes—
368#380C22×C6111yescharacteristic
369C2×#95—21no—
370#340C22×C6112yescomplemented
371#340C22×C6112yescomplemented
372#340C22×C6112yescomplemented
373#340C22×C6112yescomplemented
374C61#2455711yescharacteristic, direct product factor, Sylow
375C25D4×C6111yescharacteristic
376C22×D4—21no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4C23×C6111yescharacteristic
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—24no—
383C23×C4—21no—
384C23×C4—21no—
385C23×C4—42no—
386C23×C4D4×C6111yescharacteristic
387C23×C4D4×C6111yescharacteristic
388C23×C4D4×C6111yescharacteristic
389C23×C4C23×C6111yescharacteristic
390C23×C4C23×C6111yescharacteristic
391C23×C4C23×C6111yescharacteristic
392C2×M16—24no—
393C2×M16—24no—
394C2×M16—42no—
395C22×C8C23×C6112yes—
396C22×C8C23×C6112yes—
397#121—22no—
398#121—22no—
399#121—24no—
400#121—42no—
401#120—44no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4C23×C6112yes—
408C2×C4⋊C4C23×C6112yes—
409C2×C4⋊C4C23×C6111yescharacteristic
410C2×C4⋊C4C23×C6111yescharacteristic
411C2×C4⋊C4C23×C6111yescharacteristic
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—44no—
421#107—42no—
422#106—24no—
423#102—24no—
424#102—42no—
425C22⋊C8—22no—
426C22⋊C8—24no—
427C22⋊C8D4×C6114yescomplemented
428#95—22no—
429#95—22no—
430#95—21no—
431#95—21no—
432#95—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—21no—
439C24—41no—
440C24—41no—
441C24C2×D4×C6111yescharacteristic
442C24C2×D4×C6111yescharacteristic
443C24C2×D4×C6111yescharacteristic
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4C2×D4×C6111yescharacteristic
489C22×C4C2×D4×C6111yescharacteristic
490C22×C4C2×D4×C6112yes—
491C22×C4C2×D4×C6111yescharacteristic
492C22×C4#41×C6112yes—
493C22×C4#41×C6114yes—
494C22×C4C24×C6111yescharacteristic
495M16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8C2×D4×C6114yes—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—28no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23#121×C6112yes—
551C23#123×C6112yes—
552C23AGL(1,ℤ/8)×C6112yes—
553C23AGL(1,ℤ/8)×C6112yes—
554C23#137×C6112yes—
555C23C22×D4×C6111yescharacteristic
556D4—42no—
557D4—42no—
558D4—84no—
559D4—84no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4#121×C6112yes—
595C2×C4#123×C6112yes—
596C2×C4C22×D4×C6111yescharacteristic
597C2×C4C2×#41×C6112yes—
598C2×C4C2×#41×C6111yescharacteristic
599C2×C4H(5,𝔽2)×C6112yes—
600C8—24no—
601C8—42no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—416no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C61×#46814yes—
631C22C2×#121×C6111yescharacteristic
632C22C2×#123×C6111yescharacteristic
633C22C61×#53411yescharacteristic
634C22C2×AGL(1,ℤ/8)×C6111yescharacteristic
635C22C2×AGL(1,ℤ/8)×C6111yescharacteristic
636C22C2×#137×C6111yescharacteristic
637C22C61×#57511yescharacteristic
638C4—22no—
639C4—22no—
640C4—41no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4C61×#54612yes—
647C4C61×#55112yes—
648C2—24no—
649C2—42no—
650C2—42no—
651C2—161no—
652C2C61×#241211yescharacteristic
653C2C2×C61×#46811yescharacteristic
654C2C61×#305511yescharacteristic
655C2C61×#309211yescharacteristic
656C2C61×#309411yescharacteristic
657C2C61×#319211yescharacteristic
658C2C61×#319611yescharacteristic
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
1. C2×C61×#481C2semidirect
2. C2×C61×#407C2semidirect
4. C2×C61×#380C2semidirect
5. C61×#2077C2semidirect
6. C61×#2033C2semidirect
7. C61×#1895C2semidirect
8. C61×#1874C2semidirect
9. C61×#1852C2semidirect
10. C61×#1824C2semidirect
11. C61×#1809C2semidirect
22. C2×#107×C61C22semidirect
23. C2×#106×C61C22semidirect
24. C2×#102×C61C22semidirect
25. C2×#102×C61C22semidirect
41. C61×#340C22semidirect
42. C61×#340C22semidirect
43. C61×#340C22semidirect
44. C61×#340C22semidirect
97. C22⋊C8×C61D4semidirect
272. #24557C61direct (N × H)
319. C2×#481C2×C61semidirect
320. C2×#407C2×C61semidirect
322. C2×#380C2×C61semidirect
323. #2077C2×C61semidirect
324. #2033C2×C61semidirect
325. #1895C2×C61semidirect
326. #1874C2×C61semidirect
327. #1852C2×C61semidirect
328. #1824C2×C61semidirect
329. #1809C2×C61semidirect
351. C2×#107C22×C61semidirect
352. C2×#106C22×C61semidirect
353. C2×#102C22×C61semidirect
354. C2×#102C22×C61semidirect
370. #340C22×C61semidirect
371. #340C22×C61semidirect
372. #340C22×C61semidirect
373. #340C22×C61semidirect
374. C61#24557direct (N × H)
427. C22⋊C8D4×C61semidirect

Maximal subgroups

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

subgroupindex
C2×C61×#4812
C2×C61×#4072
C61×#28472
C2×C61×#3802
C61×#20772
C61×#20332
C61×#18952
C61×#18742
C61×#18522
C61×#18242
C61×#18092
#2455761