Finite Group Databasebeta

← C61×#23901 #29527334

Subgroups — C61×#23901 #29527334

Order 15616.

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
1C61×#3484C211yesmaximal, characteristic
2C22×#123×C61C211yesmaximal, characteristic, complemented
3C2×C61×#458C212yesmaximal, complemented
4C61×#2849C212yesmaximal, complemented
5C61×#2022C214yesmaximal, complemented
6C61×#1898C212yesmaximal, complemented
7C61×#1890C212yesmaximal, complemented
8C61×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C61C2211yescharacteristic
10C23×D4×C61C2211yescharacteristic
11C24×C4×C61C2211yescharacteristic, complemented
12C61×#546—24no—
13D42×C61—24no—
14C61×#535—21no—
15C61×#534—21no—
16C61×#534C2212yes—
17C61×#525—21no—
18C2×#123×C61—21no—
19C2×#123×C61—21no—
20C2×#123×C61—28no—
21C2×#123×C61C2212yes—
22C2×#123×C61C2212yescomplemented
23C2×#123×C61C2214yescomplemented
24C2×#120×C61—21no—
25C2×#120×C61C2212yes—
26C61×#518C2211yescharacteristic
27C22×C4⋊C4×C61—21no—
28C22×C22⋊C4×C61—21no—
29C22×C22⋊C4×C61C2211yescharacteristic, complemented
30C61×#458—28no—
31C2×#99×C61C2212yes—
32C2×#99×C61C2214yescomplemented
33C2×#99×C61C2212yescomplemented
34C61×#386—24no—
35C61×#385—24no—
36C61×#381C2214yescomplemented
37C61×#379C2212yescomplemented
38C61×#342C2214yescomplemented
39C25×C61D411yescharacteristic
40C25×C61C2311yescharacteristic
41H(5,𝔽2)×C61—24no—
42H(5,𝔽2)×C61—42no—
43C2×#41×C61—21no—
44C2×#41×C61—22no—
45C22×D4×C61—21no—
46C22×D4×C61—22no—
47C22×D4×C61—24no—
48C22×D4×C61—22no—
49C22×D4×C61—24no—
50C22×D4×C61—42no—
51C22×D4×C61D411yescharacteristic
52C22×D4×C61D411yescharacteristic
53C22×D4×C61D414yes—
54C22×D4×C61C2311yescharacteristic
55C22×D4×C61C2312yes—
56C22×D4×C61C2311yescharacteristic
57C23×C4×C61—21no—
58C23×C4×C61—21no—
59C23×C4×C61—21no—
60C23×C4×C61—24no—
61C23×C4×C61—24no—
62C23×C4×C61—42no—
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2312yes—
65#127×C61—22no—
66#124×C61—22no—
67#123×C61—22no—
68#123×C61—28no—
69#123×C61—28no—
70#123×C61—24no—
71#123×C61—22no—
72#123×C61—22no—
73#123×C61—24no—
74#123×C61—24no—
75#123×C61—28no—
76#123×C61—44no—
77#123×C61D418yescomplemented
78C22⋊Q8×C61—22no—
79#121×C61—24no—
80#121×C61—24no—
81#121×C61—22no—
82#121×C61—24no—
83#120×C61—22no—
84#120×C61—24no—
85#120×C61—22no—
86#120×C61—22no—
87#120×C61—22no—
88#120×C61—24no—
89#120×C61—22no—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92#117×C61—21no—
93C2×C4⋊C4×C61—28no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—24no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—28no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—24no—
104C2×C22⋊C4×C61—24no—
105C2×C22⋊C4×C61—24no—
106C2×C22⋊C4×C61D414yes—
107C2×C22⋊C4×C61C2314yes—
108C2×C22⋊C4×C61C2312yes—
109C2×C22⋊C4×C61C2312yes—
110#99×C61—24no—
111#99×C61—28no—
112#99×C61—42no—
113#95×C61—28no—
114C24×C61—22no—
115C24×C61—22no—
116C24×C61—21no—
117C24×C61—22no—
118C24×C61—22no—
119C24×C61—44no—
120C24×C61—41no—
121C24×C61—44no—
122C24×C61C2×D411yescharacteristic
123C24×C61C2×D414yes—
124C24×C61C2×D411yescharacteristic
125C24×C61C2×D412yes—
126C24×C61C2×D411yescharacteristic
127C24×C61C2×D411yescharacteristic
128C24×C61C2411yescharacteristic
129#41×C61—48no—
130#41×C61—42no—
131#41×C61—42no—
132C2×Q8×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—24no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—24no—
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—22no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—416no—
154C2×D4×C61—48no—
155C2×D4×C61—42no—
156C2×D4×C61—42no—
157C2×D4×C61—48no—
158C2×D4×C61—84no—
159C2×D4×C61C2×D414yes—
160C22×C4×C61—24no—
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—22no—
164C22×C4×C61—24no—
165C22×C4×C61—22no—
166C22×C4×C61—24no—
167C22×C4×C61—22no—
168C22×C4×C61—22no—
169C22×C4×C61—22no—
170C22×C4×C61—24no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—24no—
174C22×C4×C61—41no—
175C22×C4×C61—416no—
176C22×C4×C61—44no—
177C22×C4×C61—42no—
178C22×C4×C61—42no—
179C22×C4×C61—42no—
180C22×C4×C61—42no—
181C22×C4×C61C2×D412yes—
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D414yes—
185C22×C4×C61C2×D414yescomplemented
186C4⋊C4×C61—28no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—24no—
189C4⋊C4×C61—28no—
190C4⋊C4×C61—48no—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—28no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—24no—
195C22⋊C4×C61—24no—
196C22⋊C4×C61—28no—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—24no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—24no—
203C22⋊C4×C61—24no—
204C22⋊C4×C61—48no—
205C22⋊C4×C61—48no—
206C22⋊C4×C61C2×D414yescomplemented
207C42×C61—22no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—24no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—28no—
215C23×C61—28no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—22no—
220C23×C61—21no—
221C23×C61—24no—
222C23×C61—44no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—44no—
226C23×C61—48no—
227C23×C61—48no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61—48no—
231C23×C61—42no—
232C23×C61—44no—
233C23×C61—44no—
234C23×C61—82no—
235C23×C61—84no—
236C23×C61—82no—
237C23×C61#12012yes—
238C23×C61#12012yes—
239C23×C61#12012yes—
240C23×C61C22×D411yescharacteristic
241C23×C61C22×D412yes—
242C23×C61C22×D412yes—
243C23×C61H(5,𝔽2)12yes—
244Q8×C61—42no—
245D4×C61—28no—
246D4×C61—44no—
247D4×C61—42no—
248D4×C61—44no—
249D4×C61—44no—
250D4×C61—42no—
251D4×C61—42no—
252D4×C61—44no—
253D4×C61—44no—
254D4×C61—48no—
255D4×C61—84no—
256D4×C61—84no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—24no—
260C2×C4×C61—24no—
261C2×C4×C61—21no—
262C2×C4×C61—24no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—28no—
267C2×C4×C61—24no—
268C2×C4×C61—44no—
269C2×C4×C61—44no—
270C2×C4×C61—48no—
271C2×C4×C61—44no—
272C2×C4×C61—48no—
273C2×C4×C61—48no—
274C2×C4×C61—42no—
275C2×C4×C61—42no—
276C2×C4×C61—42no—
277C2×C4×C61—44no—
278C2×C4×C61—44no—
279C2×C4×C61—48no—
280C2×C4×C61—44no—
281C2×C4×C61—82no—
282C2×C4×C61#12012yes—
283#23901C6111yesmaximal, characteristic, direct product factor, Sylow
284C22×C61—24no—
285C22×C61—24no—
286C22×C61—22no—
287C22×C61—24no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—416no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—41no—
303C22×C61—44no—
304C22×C61—42no—
305C22×C61—81no—
306C22×C61—88no—
307C22×C61—84no—
308C22×C61—81no—
309C22×C61—162no—
310C22×C61#45814yescomplemented
311C22×C61C2×#12011yescharacteristic
312C22×C61C2×#12011yescharacteristic
313C22×C61#53411yescharacteristic
314C22×C61D4214yes—
315C4×C61—22no—
316C4×C61—44no—
317C4×C61—44no—
318C4×C61—41no—
319C4×C61—42no—
320C4×C61—44no—
321C4×C61—84no—
322#3484C2×C6111yescharacteristic
323C22×#123C2×C6111yescharacteristic, complemented
324C2×#458C2×C6112yescomplemented
325#2849C2×C6112yescomplemented
326#2022C2×C6114yescomplemented
327#1898C2×C6112yescomplemented
328#1890C2×C6112yescomplemented
329#1781C2×C6111yescharacteristic, complemented
330C2×C61—21no—
331C2×C61—24no—
332C2×C61—21no—
333C2×C61—24no—
334C2×C61—42no—
335C2×C61—42no—
336C2×C61—81no—
337C2×C61—82no—
338C2×C61#240311yescharacteristic
339C2×C61C2×#45812yes—
340C2×H(5,𝔽2)C22×C6111yescharacteristic
341C23×D4C22×C6111yescharacteristic
342C24×C4C22×C6111yescharacteristic, complemented
343#546—24no—
344D42—24no—
345#535—21no—
346#534—21no—
347#534C22×C6112yes—
348#525—21no—
349C2×#123—21no—
350C2×#123—21no—
351C2×#123—28no—
352C2×#123C22×C6112yes—
353C2×#123C22×C6112yescomplemented
354C2×#123C22×C6114yescomplemented
355C2×#120—21no—
356C2×#120C22×C6112yes—
357#518C22×C6111yescharacteristic
358C22×C4⋊C4—21no—
359C22×C22⋊C4—21no—
360C22×C22⋊C4C22×C6111yescharacteristic, complemented
361#458—28no—
362C2×#99C22×C6112yes—
363C2×#99C22×C6114yescomplemented
364C2×#99C22×C6112yescomplemented
365#386—24no—
366#385—24no—
367#381C22×C6114yescomplemented
368#379C22×C6112yescomplemented
369#342C22×C6114yescomplemented
370C61#2390111yescharacteristic, direct product factor, Sylow
371C25D4×C6111yescharacteristic
372C25C23×C6111yescharacteristic
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×C6111yescharacteristic
384C22×D4D4×C6111yescharacteristic
385C22×D4D4×C6114yes—
386C22×D4C23×C6112yes—
387C22×D4C23×C6111yescharacteristic
388C22×D4C23×C6111yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C6111yescharacteristic
396C23×C4C23×C6112yes—
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×C6118yescomplemented
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×C6114yes—
439C2×C22⋊C4C23×C6114yes—
440C2×C22⋊C4C23×C6112yes—
441C2×C22⋊C4C23×C6112yes—
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×C6111yescharacteristic
455C24C2×D4×C6114yes—
456C24C2×D4×C6111yescharacteristic
457C24C2×D4×C6112yes—
458C24C2×D4×C6111yescharacteristic
459C24C2×D4×C6111yescharacteristic
460C24C24×C6111yescharacteristic
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×C6114yes—
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×C6112yes—
514C22×C4C2×D4×C6111yescharacteristic
515C22×C4C2×D4×C6111yescharacteristic
516C22×C4C2×D4×C6114yes—
517C22×C4C2×D4×C6114yescomplemented
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×C6114yescomplemented
539C42—22no—
540C23—21no—
541C23—24no—
542C23—21no—
543C23—21no—
544C23—28no—
545C23—28no—
546C23—21no—
547C23—21no—
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×C6112yes—
570C23#120×C6112yes—
571C23#120×C6112yes—
572C23C22×D4×C6111yescharacteristic
573C23C22×D4×C6112yes—
574C23C22×D4×C6112yes—
575C23H(5,𝔽2)×C6112yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—42no—
583D4—44no—
584D4—44no—
585D4—48no—
586D4—42no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C6112yes—
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—84no—
639C22—81no—
640C22—162no—
641C22C61×#45814yescomplemented
642C22C2×#120×C6111yescharacteristic
643C22C2×#120×C6111yescharacteristic
644C22C61×#53411yescharacteristic
645C22D42×C6114yes—
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—
661C2C61×#240311yescharacteristic
662C2C2×C61×#45812yes—
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC24C24×C61
prime radicalC61×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C25×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C61C2semidirect
3. C2×C61×#458C2semidirect
4. C61×#2849C2semidirect
5. C61×#2022C2semidirect
6. C61×#1898C2semidirect
7. C61×#1890C2semidirect
8. C61×#1781C2semidirect
11. C24×C4×C61C22semidirect
22. C2×#123×C61C22semidirect
23. C2×#123×C61C22semidirect
29. C22×C22⋊C4×C61C22semidirect
32. C2×#99×C61C22semidirect
33. C2×#99×C61C22semidirect
36. C61×#381C22semidirect
37. C61×#379C22semidirect
38. C61×#342C22semidirect
77. #123×C61D4semidirect
185. C22×C4×C61C2×D4semidirect
206. C22⋊C4×C61C2×D4semidirect
283. #23901C61direct (N × H)
310. C22×C61#458semidirect
323. C22×#123C2×C61semidirect
324. C2×#458C2×C61semidirect
325. #2849C2×C61semidirect
326. #2022C2×C61semidirect
327. #1898C2×C61semidirect
328. #1890C2×C61semidirect
329. #1781C2×C61semidirect
342. C24×C4C22×C61semidirect
353. C2×#123C22×C61semidirect
354. C2×#123C22×C61semidirect
360. C22×C22⋊C4C22×C61semidirect
363. C2×#99C22×C61semidirect
364. C2×#99C22×C61semidirect
367. #381C22×C61semidirect
368. #379C22×C61semidirect
369. #342C22×C61semidirect
370. C61#23901direct (N × H)
409. #123D4×C61semidirect
517. C22×C4C2×D4×C61semidirect
538. C22⋊C4C2×D4×C61semidirect
641. C22C61×#458semidirect

Maximal subgroups

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

subgroupindex
C61×#34842
C22×#123×C612
C2×C61×#4582
C61×#28492
C61×#20222
C61×#18982
C61×#18902
C61×#17812
#2390161