Finite Group Databasebeta

← C61×#14506 #29517939

Subgroups — C61×#14506 #29517939

Order 15616.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C61C211yesmaximal, characteristic
2C2×C61×#388C212yesmaximal, complemented
3C61×#1445C212yesmaximal, complemented
4C61×#1439C212yesmaximal, complemented
5C61×#1438C211yesmaximal, characteristic
6C61×#1438C211yesmaximal, characteristic
7C61×#1437C212yesmaximal, complemented
8C61×C4⋊#95C212yesmaximal, complemented
9C61×C22⋊#95C212yesmaximal, complemented
10C23×D4×C61C2211yescharacteristic
11C24×C4×C61C2211yescharacteristic
12C2×#123×C61—28no—
13C2×#123×C61C418yescomplemented
14C22×C4⋊C4×C61C2212yes—
15C22×C4⋊C4×C61C2212yes—
16C22×C22⋊C4×C61C2212yes—
17C22×C22⋊C4×C61C2212yes—
18C22×C22⋊C4×C61C2211yescharacteristic
19C22×C22⋊C4×C61C2212yes—
20C22×C22⋊C4×C61C2212yes—
21C22×C22⋊C4×C61C2211yescharacteristic
22C22×C22⋊C4×C61C2211yescharacteristic
23C22×C42×C61C2212yes—
24C61×#388—28no—
25C2×#95×C61C2212yes—
26C2×#95×C61C2212yes—
27C2×#95×C61C2212yes—
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2211yescharacteristic
32C2×#95×C61C2212yes—
33C2×#95×C61C2211yescharacteristic
34C25×C61C2311yescharacteristic
35C25×C61C2311yescharacteristic
36C22×D4×C61—24no—
37C22×D4×C61—42no—
38C22×D4×C61—41no—
39C22×D4×C61—41no—
40C22×D4×C61C2×C414yes—
41C23×C4×C61—24no—
42C23×C4×C61—42no—
43C23×C4×C61—41no—
44C23×C4×C61—41no—
45C23×C4×C61C2×C414yes—
46C23×C4×C61C2312yes—
47C23×C4×C61C2312yes—
48C23×C4×C61C2312yes—
49C23×C4×C61C2311yescharacteristic
50C23×C4×C61C2312yes—
51C23×C4×C61C2312yes—
52C23×C4×C61C2312yes—
53#123×C61—232no—
54C2×C4⋊C4×C61—24no—
55C2×C4⋊C4×C61—24no—
56C2×C4⋊C4×C61—24no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—42no—
59C2×C4⋊C4×C61C2×C418yes—
60C2×C22⋊C4×C61—24no—
61C2×C22⋊C4×C61—24no—
62C2×C22⋊C4×C61—24no—
63C2×C22⋊C4×C61—24no—
64C2×C22⋊C4×C61—24no—
65C2×C22⋊C4×C61—24no—
66C2×C22⋊C4×C61—24no—
67C2×C22⋊C4×C61—24no—
68C2×C22⋊C4×C61—41no—
69C2×C22⋊C4×C61—42no—
70C2×C22⋊C4×C61—42no—
71C2×C22⋊C4×C61—42no—
72C2×C22⋊C4×C61—42no—
73C2×C22⋊C4×C61—41no—
74C2×C22⋊C4×C61—42no—
75C2×C22⋊C4×C61C2×C418yes—
76C2×C22⋊C4×C61C2×C414yes—
77C2×C22⋊C4×C61D418yes—
78C2×C42×C61—24no—
79C2×C42×C61—24no—
80C2×C42×C61—42no—
81#95×C61—24no—
82#95×C61—24no—
83#95×C61—41no—
84#95×C61—42no—
85#95×C61—42no—
86#95×C61—42no—
87#95×C61—42no—
88#95×C61—42no—
89#95×C61—41no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—21no—
103C24×C61—21no—
104C24×C61—22no—
105C24×C61C22×C412yes—
106C24×C61C22×C412yes—
107C24×C61C2×D414yes—
108C24×C61C2411yescharacteristic
109C2×D4×C61—28no—
110C2×D4×C61—44no—
111C2×D4×C61—44no—
112C2×D4×C61—48no—
113C2×D4×C61—48no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—28no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
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—21no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61—22no—
155C22×C4×C61—44no—
156C22×C4×C61—44no—
157C22×C4×C61—48no—
158C22×C4×C61—48no—
159C22×C4×C61C22×C414yes—
160C22×C4×C61C22×C412yes—
161C22×C4×C61C22×C414yes—
162C22×C4×C61C2×D414yes—
163C22×C4×C61C2×D414yes—
164C22×C4×C61#4114yes—
165C22×C4×C61#4114yes—
166C4⋊C4×C61—216no—
167C4⋊C4×C61—48no—
168C22⋊C4×C61—216no—
169C22⋊C4×C61—44no—
170C22⋊C4×C61—48no—
171C22⋊C4×C61—48no—
172C22⋊C4×C61—48no—
173C22⋊C4×C61—44no—
174C22⋊C4×C61C22⋊C4116yescomplemented
175C42×C61—48no—
176C23×C61—24no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—22no—
187C23×C61—24no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—24no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—42no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—41no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—42no—
216C23×C61—41no—
217C23×C61C2×C22⋊C414yes—
218C23×C61C23×C411yescharacteristic
219C23×C61C22×D412yes—
220C23×C61C2×#4112yes—
221C23×C61C2×#4112yes—
222C23×C61H(5,𝔽2)12yes—
223C23×C61H(5,𝔽2)11yescharacteristic
224C23×C61H(5,𝔽2)11yescharacteristic
225C23×C61H(5,𝔽2)12yes—
226C23×C612−1+412yes—
227D4×C61—416no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—416no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—42no—
274C2×C4×C61—42no—
275C2×C4×C61C2×C22⋊C418yes—
276#14506C6111yesmaximal, characteristic, direct product factor, Sylow
277C22×C61—24no—
278C22×C61—28no—
279C22×C61—44no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61—44no—
286C22×C61—42no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—82no—
292C22×C61—82no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61C4×#4112yes—
297C22×C61#51811yescharacteristic
298C22×C61#51811yescharacteristic
299C22×C61#52012yes—
300C22×C61#53512yes—
301C22×C61#53512yes—
302C22×C61#53812yes—
303C22×C61#53912yes—
304C22×C61#54012yes—
305C22×C61#54212yes—
306C22×C61#54612yes—
307C22×C61#54612yes—
308C22×C61#54712yes—
309C22×C61#54712yes—
310C22×C61#55111yescharacteristic
311C22×C61#55312yes—
312C22×C61#55911yescharacteristic
313C22×C61#56011yescharacteristic
314C22×C61#56212yes—
315C22×C61#56311yescharacteristic
316C4×C61—28no—
317C4×C61—44no—
318C4×C61—42no—
319C4×C61—44no—
320C4×C61—44no—
321C4×C61—82no—
322C4×C61—82no—
323C22×#123C2×C6111yescharacteristic
324C2×#388C2×C6112yescomplemented
325#1445C2×C6112yescomplemented
326#1439C2×C6112yescomplemented
327#1438C2×C6111yescharacteristic
328#1438C2×C6111yescharacteristic
329#1437C2×C6112yescomplemented
330C4⋊#95C2×C6112yescomplemented
331C22⋊#95C2×C6112yescomplemented
332C2×C61—44no—
333C2×C61—82no—
334C2×C61#230911yescharacteristic
335C2×C61#233512yes—
336C2×C61#233912yes—
337C2×C61#237411yescharacteristic
338C2×C61#237611yescharacteristic
339C2×C61#270312yes—
340C2×C61#270612yes—
341C2×C61#271012yes—
342C2×C61#271812yes—
343C23×D4C22×C6111yescharacteristic
344C24×C4C22×C6111yescharacteristic
345C2×#123—28no—
346C2×#123C4×C6118yescomplemented
347C22×C4⋊C4C22×C6112yes—
348C22×C4⋊C4C22×C6112yes—
349C22×C22⋊C4C22×C6112yes—
350C22×C22⋊C4C22×C6112yes—
351C22×C22⋊C4C22×C6111yescharacteristic
352C22×C22⋊C4C22×C6112yes—
353C22×C22⋊C4C22×C6112yes—
354C22×C22⋊C4C22×C6111yescharacteristic
355C22×C22⋊C4C22×C6111yescharacteristic
356C22×C42C22×C6112yes—
357#388—28no—
358C2×#95C22×C6112yes—
359C2×#95C22×C6112yes—
360C2×#95C22×C6112yes—
361C2×#95C22×C6112yes—
362C2×#95C22×C6112yes—
363C2×#95C22×C6112yes—
364C2×#95C22×C6111yescharacteristic
365C2×#95C22×C6112yes—
366C2×#95C22×C6111yescharacteristic
367C61#1450611yescharacteristic, direct product factor, Sylow
368C25C23×C6111yescharacteristic
369C25C23×C6111yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C6114yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C6114yes—
380C23×C4C23×C6112yes—
381C23×C4C23×C6112yes—
382C23×C4C23×C6112yes—
383C23×C4C23×C6111yescharacteristic
384C23×C4C23×C6112yes—
385C23×C4C23×C6112yes—
386C23×C4C23×C6112yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C6118yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C6118yes—
410C2×C22⋊C4C2×C4×C6114yes—
411C2×C22⋊C4D4×C6118yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C6112yes—
440C24C22×C4×C6112yes—
441C24C2×D4×C6114yes—
442C24C24×C6111yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
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—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C6114yes—
494C22×C4C22×C4×C6112yes—
495C22×C4C22×C4×C6114yes—
496C22×C4C2×D4×C6114yes—
497C22×C4C2×D4×C6114yes—
498C22×C4#41×C6114yes—
499C22×C4#41×C6114yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C61116yescomplemented
509C42—48no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C6114yes—
552C23C23×C4×C6111yescharacteristic
553C23C22×D4×C6112yes—
554C23C2×#41×C6112yes—
555C23C2×#41×C6112yes—
556C23H(5,𝔽2)×C6112yes—
557C23H(5,𝔽2)×C6111yescharacteristic
558C23H(5,𝔽2)×C6111yescharacteristic
559C23H(5,𝔽2)×C6112yes—
560C232−1+4×C6112yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C6118yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C6111yescharacteristic
629C22C4×#41×C6112yes—
630C22C61×#51811yescharacteristic
631C22C61×#51811yescharacteristic
632C22C61×#52012yes—
633C22C61×#53512yes—
634C22C61×#53512yes—
635C22C61×#53812yes—
636C22C61×#53912yes—
637C22C61×#54012yes—
638C22C61×#54212yes—
639C22C61×#54612yes—
640C22C61×#54612yes—
641C22C61×#54712yes—
642C22C61×#54712yes—
643C22C61×#55111yescharacteristic
644C22C61×#55312yes—
645C22C61×#55911yescharacteristic
646C22C61×#56011yescharacteristic
647C22C61×#56212yes—
648C22C61×#56311yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C61×#230911yescharacteristic
659C2C61×#233512yes—
660C2C61×#233912yes—
661C2C61×#237411yescharacteristic
662C2C61×#237611yescharacteristic
663C2C61×#270312yes—
664C2C61×#270612yes—
665C2C61×#271012yes—
666C2C61×#271812yes—
subgroupG/H
center, socleC24×C61C24
commutatorC23C23×C4×C61
Frattini, cosocleC24C24×C61
prime radicalC23×D4×C61C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#388C2semidirect
3. C61×#1445C2semidirect
4. C61×#1439C2semidirect
7. C61×#1437C2semidirect
8. C61×C4⋊#95C2semidirect
9. C61×C22⋊#95C2semidirect
13. C2×#123×C61C4semidirect
174. C22⋊C4×C61C22⋊C4semidirect
276. #14506C61direct (N × H)
324. C2×#388C2×C61semidirect
325. #1445C2×C61semidirect
326. #1439C2×C61semidirect
329. #1437C2×C61semidirect
330. C4⋊#95C2×C61semidirect
331. C22⋊#95C2×C61semidirect
346. C2×#123C4×C61semidirect
367. C61#14506direct (N × H)
508. C22⋊C4C22⋊C4×C61semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C612
C2×C61×#3882
C61×#14452
C61×#14392
C61×#14382
C61×#14372
C61×C4⋊#952
C61×C22⋊#952
#1450661