Finite Group Databasebeta

← C61×#15129 #29518562

Subgroups — C61×#15129 #29518562

Order 15616.

2354 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C61C211yesmaximal, characteristic
2C2×C61×#394C212yesmaximal
3C2×C61×#389C211yesmaximal, characteristic, complemented
4C2×C61×#380C212yesmaximal, complemented
5C2×C61×#380C212yesmaximal, complemented
6C61×#1441C212yesmaximal, complemented
7C61×#1440C214yesmaximal, complemented
8C61×C4⋊#95C211yesmaximal, characteristic, complemented
9C23×D4×C61C2211yescharacteristic
10C24×C4×C61C2212yes—
11C2×C4×D4×C61—28no—
12C2×C4×D4×C61—24no—
13C22×C4⋊C4×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2212yes—
15C22×C4⋊C4×C61C2212yes—
16C22×C4⋊C4×C61C2211yescharacteristic
17C22×C4⋊C4×C61C2211yescharacteristic
18C22×C22⋊C4×C61C2214yes—
19C22×C22⋊C4×C61C2212yes—
20C22×C22⋊C4×C61C2214yes—
21C22×C42×C61C2211yescharacteristic
22C61×#394—28no—
23C61×#389—24no—
24C61×#380—28no—
25C61×#380—28no—
26C2×#95×C61C2214yes—
27C2×#95×C61C2214yes—
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C25×C61C2312yes—
32C22×D4×C61—24no—
33C22×D4×C61—24no—
34C22×D4×C61—22no—
35C22×D4×C61—22no—
36C22×D4×C61D414yes—
37C23×C4×C61—24no—
38C23×C4×C61—24no—
39C23×C4×C61—24no—
40C23×C4×C61—24no—
41C23×C4×C61—24no—
42C23×C4×C61—24no—
43C23×C4×C61D418yes—
44C23×C4×C61C2314yes—
45C23×C4×C61C2312yes—
46C23×C4×C61C2312yes—
47C23×C4×C61C2311yescharacteristic
48C23×C4×C61C2312yes—
49C23×C4×C61C2311yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C4×D4×C61—232no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61—24no—
54C2×C4⋊C4×C61—22no—
55C2×C4⋊C4×C61—22no—
56C2×C4⋊C4×C61—24no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—24no—
59C2×C4⋊C4×C61—22no—
60C2×C4⋊C4×C61—24no—
61C2×C4⋊C4×C61—24no—
62C2×C4⋊C4×C61—24no—
63C2×C4⋊C4×C61—24no—
64C2×C4⋊C4×C61D414yescomplemented
65C2×C22⋊C4×C61—24no—
66C2×C22⋊C4×C61—24no—
67C2×C22⋊C4×C61—24no—
68C2×C22⋊C4×C61—28no—
69C2×C22⋊C4×C61—28no—
70C2×C22⋊C4×C61—28no—
71C2×C22⋊C4×C61—28no—
72C2×C22⋊C4×C61—28no—
73C2×C22⋊C4×C61—44no—
74C2×C42×C61—24no—
75C2×C42×C61D414yes—
76#95×C61—28no—
77#95×C61—28no—
78#95×C61—24no—
79#95×C61—24no—
80#95×C61—42no—
81C24×C61—22no—
82C24×C61—22no—
83C24×C61—22no—
84C24×C61—22no—
85C24×C61—22no—
86C24×C61—22no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61C2×D414yes—
94C24×C61C2×D414yes—
95C24×C61#4114yes—
96C24×C61C2411yescharacteristic
97C2×D4×C61—28no—
98C2×D4×C61—28no—
99C2×D4×C61—216no—
100C2×D4×C61—216no—
101C2×D4×C61—28no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—216no—
121C22×C4×C61—216no—
122C22×C4×C61—216no—
123C22×C4×C61—24no—
124C22×C4×C61—24no—
125C22×C4×C61—24no—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—24no—
134C22×C4×C61—24no—
135C22×C4×C61—48no—
136C22×C4×C61—48no—
137C22×C4×C61—48no—
138C22×C4×C61—48no—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D414yes—
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D414yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4112yes—
148C22×C4×C61#4112yes—
149C22×C4×C61#4114yes—
150C4⋊C4×C61—28no—
151C4⋊C4×C61—28no—
152C4⋊C4×C61—44no—
153C4⋊C4×C61—48no—
154C4⋊C4×C61—48no—
155C22⋊C4×C61—216no—
156C22⋊C4×C61—416no—
157C22⋊C4×C61—416no—
158C42×C61—28no—
159C23×C61—24no—
160C23×C61—24no—
161C23×C61—24no—
162C23×C61—24no—
163C23×C61—24no—
164C23×C61—24no—
165C23×C61—24no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—24no—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—24no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61—24no—
176C23×C61—24no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61#12118yes—
194C23×C61C22×D412yes—
195C23×C61C22×D411yescharacteristic
196C23×C61C22×D411yescharacteristic
197C23×C61C22×D411yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C61C2×#4112yes—
200C23×C61C2×#4111yescharacteristic
201C23×C61C2×#4112yes—
202C23×C61H(5,𝔽2)11yescharacteristic
203C23×C612−1+411yescharacteristic
204C23×C612−1+412yes—
205D4×C61—232no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—432no—
230C2×C4×C61—42no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—42no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61—44no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61#12014yes—
243C2×C4×C61#12114yes—
244C2×C4×C61#12114yes—
245C2×C4×C61#12118yes—
246C2×C4×C61#12318yes—
247C2×C4×C61#12414yes—
248#15129C6111yesmaximal, characteristic, direct product factor, Sylow
249C22×C61—28no—
250C22×C61—28no—
251C22×C61—28no—
252C22×C61—44no—
253C22×C61—44no—
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—44no—
264C22×C61C2×#12011yescharacteristic
265C22×C61C2×#12112yes—
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12112yes—
268C22×C61C2×#12111yescharacteristic
269C22×C61C2×#12312yes—
270C22×C61#52511yescharacteristic
271C22×C61#52512yes—
272C22×C61C2×#12411yescharacteristic
273C22×C61#53612yes—
274C22×C61#53611yescharacteristic
275C22×C61#53711yescharacteristic
276C22×C61#53712yes—
277C22×C61D4211yescharacteristic
278C22×C61#54711yescharacteristic
279C22×C61#54712yes—
280C22×C61#54711yescharacteristic
281C22×C61#54812yes—
282C22×C61#55212yes—
283C22×C61#55212yes—
284C22×C61#55312yes—
285C22×C61#55511yescharacteristic
286C22×C61#55612yes—
287C4×C61—24no—
288C4×C61—42no—
289C4×C61—44no—
290C4×C61—44no—
291C4×C61—82no—
292C4×C61—84no—
293C4×C61#39418yescomplemented
294C22×C4×D4C2×C6111yescharacteristic
295C2×#394C2×C6112yes—
296C2×#389C2×C6111yescharacteristic, complemented
297C2×#380C2×C6112yescomplemented
298C2×#380C2×C6112yescomplemented
299#1441C2×C6112yescomplemented
300#1440C2×C6114yescomplemented
301C4⋊#95C2×C6111yescharacteristic, complemented
302C2×C61—48no—
303C2×C61C2×#39411yescharacteristic
304C2×C61#241512yes—
305C2×C61#241612yes—
306C2×C61#248512yes—
307C2×C61#248612yes—
308C2×C61#253611yescharacteristic
309C2×C61#253711yescharacteristic
310C2×C61#254711yescharacteristic
311C2×C61#256211yescharacteristic
312C2×C61#261611yescharacteristic
313C2×C61#284711yescharacteristic
314C23×D4C22×C6111yescharacteristic
315C24×C4C22×C6112yes—
316C2×C4×D4—28no—
317C2×C4×D4—24no—
318C22×C4⋊C4C22×C6112yes—
319C22×C4⋊C4C22×C6111yescharacteristic
320C22×C4⋊C4C22×C6112yes—
321C22×C4⋊C4C22×C6111yescharacteristic
322C22×C4⋊C4C22×C6111yescharacteristic
323C22×C22⋊C4C22×C6114yes—
324C22×C22⋊C4C22×C6112yes—
325C22×C22⋊C4C22×C6114yes—
326C22×C42C22×C6111yescharacteristic
327#394—28no—
328#389—24no—
329#380—28no—
330#380—28no—
331C2×#95C22×C6112yes—
332C2×#95C22×C6112yes—
333C2×#95C22×C6114yes—
334C2×#95C22×C6114yes—
335C2×#95C22×C6112yes—
336C61#1512911yescharacteristic, direct product factor, Sylow
337C25C23×C6112yes—
338C22×D4—24no—
339C22×D4—24no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4D4×C6114yes—
343C23×C4—24no—
344C23×C4—24no—
345C23×C4—24no—
346C23×C4—24no—
347C23×C4—24no—
348C23×C4—24no—
349C23×C4D4×C6118yes—
350C23×C4C23×C6112yes—
351C23×C4C23×C6114yes—
352C23×C4C23×C6112yes—
353C23×C4C23×C6111yescharacteristic
354C23×C4C23×C6112yes—
355C23×C4C23×C6111yescharacteristic
356C23×C4C23×C6111yescharacteristic
357C4×D4—232no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4D4×C6114yescomplemented
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—28no—
378C2×C22⋊C4—28no—
379C2×C22⋊C4—44no—
380C2×C42—24no—
381C2×C42D4×C6114yes—
382#95—28no—
383#95—28no—
384#95—24no—
385#95—24no—
386#95—42no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24C2×D4×C6114yes—
400C24C2×D4×C6114yes—
401C24#41×C6114yes—
402C24C24×C6111yescharacteristic
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—216no—
406C2×D4—216no—
407C2×D4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4C2×D4×C6112yes—
446C22×C4C2×D4×C6114yes—
447C22×C4C2×D4×C6114yes—
448C22×C4C2×D4×C6112yes—
449C22×C4C2×D4×C6112yes—
450C22×C4C2×D4×C6114yes—
451C22×C4C2×D4×C6112yes—
452C22×C4C2×D4×C6112yes—
453C22×C4#41×C6112yes—
454C22×C4#41×C6112yes—
455C22×C4#41×C6114yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C22⋊C4—216no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C42—28no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#121×C6118yes—
500C23C22×D4×C6112yes—
501C23C22×D4×C6111yescharacteristic
502C23C22×D4×C6111yescharacteristic
503C23C22×D4×C6111yescharacteristic
504C23C2×#41×C6111yescharacteristic
505C23C2×#41×C6112yes—
506C23C2×#41×C6111yescharacteristic
507C23C2×#41×C6112yes—
508C23H(5,𝔽2)×C6111yescharacteristic
509C232−1+4×C6111yescharacteristic
510C232−1+4×C6112yes—
511D4—232no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—432no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4#120×C6114yes—
549C2×C4#121×C6114yes—
550C2×C4#121×C6114yes—
551C2×C4#121×C6118yes—
552C2×C4#123×C6118yes—
553C2×C4#124×C6114yes—
554C22—28no—
555C22—28no—
556C22—28no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22C2×#120×C6111yescharacteristic
570C22C2×#121×C6111yescharacteristic
571C22C2×#121×C6111yescharacteristic
572C22C2×#121×C6112yes—
573C22C2×#121×C6112yes—
574C22C2×#123×C6112yes—
575C22C61×#52511yescharacteristic
576C22C61×#52512yes—
577C22C2×#124×C6111yescharacteristic
578C22C61×#53611yescharacteristic
579C22C61×#53612yes—
580C22C61×#53711yescharacteristic
581C22C61×#53712yes—
582C22D42×C6111yescharacteristic
583C22C61×#54711yescharacteristic
584C22C61×#54711yescharacteristic
585C22C61×#54712yes—
586C22C61×#54812yes—
587C22C61×#55212yes—
588C22C61×#55212yes—
589C22C61×#55312yes—
590C22C61×#55511yescharacteristic
591C22C61×#55612yes—
592C4—24no—
593C4—44no—
594C4—42no—
595C4—44no—
596C4—82no—
597C4—84no—
598C4C61×#39418yescomplemented
599C2—48no—
600C2C2×C61×#39411yescharacteristic
601C2C61×#241512yes—
602C2C61×#241612yes—
603C2C61×#248512yes—
604C2C61×#248612yes—
605C2C61×#253611yescharacteristic
606C2C61×#253711yescharacteristic
607C2C61×#254711yescharacteristic
608C2C61×#256211yescharacteristic
609C2C61×#261611yescharacteristic
610C2C61×#284711yescharacteristic
subgroupG/H
center, socleC24×C61C24
commutator, Frattini, cosocleC24C24×C61
prime radicalC23×D4×C61C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C61×#389C2semidirect
4. C2×C61×#380C2semidirect
5. C2×C61×#380C2semidirect
6. C61×#1441C2semidirect
7. C61×#1440C2semidirect
8. C61×C4⋊#95C2semidirect
64. C2×C4⋊C4×C61D4semidirect
248. #15129C61direct (N × H)
293. C4×C61#394semidirect
296. C2×#389C2×C61semidirect
297. C2×#380C2×C61semidirect
298. C2×#380C2×C61semidirect
299. #1441C2×C61semidirect
300. #1440C2×C61semidirect
301. C4⋊#95C2×C61semidirect
336. C61#15129direct (N × H)
370. C2×C4⋊C4D4×C61semidirect
598. C4C61×#394semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D4×C612
C2×C61×#3942
C2×C61×#3892
C2×C61×#3802
C61×#14412
C61×#14402
C61×C4⋊#952
#1512961