Finite Group Databasebeta

← C61×#15236 #29518669

Subgroups — C61×#15236 #29518669

Order 15616.

1902 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C61C211yesmaximal, characteristic
2C2×C61×#391C211yesmaximal, characteristic, complemented
3C2×C61×#391C211yesmaximal, characteristic, complemented
4C2×C61×#382C211yesmaximal, characteristic, complemented
5C2×C61×#380C211yesmaximal, characteristic
6C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
7C61×#1442C214yesmaximal, complemented
8C61×#1440C214yesmaximal
9C61×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×Q8×C61C2211yescharacteristic
11C24×C4×C61C2211yescharacteristic
12C2×C22⋊Q8×C61—24no—
13C2×C22⋊Q8×C61—24no—
14C2×C22⋊Q8×C61—24no—
15C22×C4⋊C4×C61C2211yescharacteristic
16C22×C4⋊C4×C61C2211yescharacteristic
17C22×C4⋊C4×C61C2212yes—
18C22×C4⋊C4×C61C2211yescharacteristic
19C22×C22⋊C4×C61C2212yes—
20C22×C22⋊C4×C61C2212yes—
21C22×C22⋊C4×C61C2212yes—
22C22×C42×C61C2211yescharacteristic
23C22×C42×C61C2211yescharacteristic
24C61×#391—24no—
25C61×#391—24no—
26C61×#382—24no—
27C61×#380—24no—
28C4×C22⋊C4×C61—24no—
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2214yes—
32C2×#95×C61C2214yes—
33C2×#95×C61C2214yes—
34C2×#95×C61C2214yes—
35C25×C61C2311yescharacteristic
36C22×Q8×C61—22no—
37C22×Q8×C61—22no—
38C22×Q8×C61—22no—
39C22×Q8×C61—22no—
40C22×Q8×C61—22no—
41C22×Q8×C61—22no—
42C22×Q8×C61D414yes—
43C23×C4×C61—22no—
44C23×C4×C61—22no—
45C23×C4×C61—22no—
46C23×C4×C61—22no—
47C23×C4×C61—22no—
48C23×C4×C61—22no—
49C23×C4×C61D414yes—
50C23×C4×C61C2312yes—
51C23×C4×C61C2312yes—
52C23×C4×C61C2311yescharacteristic
53C23×C4×C61C2312yes—
54C23×C4×C61C2312yes—
55C23×C4×C61C2312yes—
56C23×C4×C61C2312yes—
57C23×C4×C61C2311yescharacteristic
58C22⋊Q8×C61—232no—
59C2×C4⋊C4×C61—24no—
60C2×C4⋊C4×C61—24no—
61C2×C4⋊C4×C61—24no—
62C2×C4⋊C4×C61—22no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61D414yes—
71C2×C22⋊C4×C61—24no—
72C2×C22⋊C4×C61—24no—
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61—28no—
79C2×C42×C61—22no—
80C2×C42×C61—22no—
81C2×C42×C61—22no—
82C2×C42×C61—22no—
83C2×C42×C61—22no—
84C2×C42×C61D414yes—
85#95×C61—24no—
86#95×C61—24no—
87#95×C61—28no—
88#95×C61—28no—
89#95×C61—28no—
90#95×C61—28no—
91C24×C61—21no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—21no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61C2×D412yes—
101C24×C61#4114yes—
102C24×C61C2411yescharacteristic
103C2×Q8×C61—28no—
104C2×Q8×C61—28no—
105C2×Q8×C61—28no—
106C2×Q8×C61—44no—
107C2×Q8×C61—44no—
108C2×Q8×C61—44no—
109C2×Q8×C61—44no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—24no—
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—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—28no—
139C22×C4×C61—28no—
140C22×C4×C61—28no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—24no—
148C22×C4×C61—24no—
149C22×C4×C61—22no—
150C22×C4×C61—24no—
151C22×C4×C61—44no—
152C22×C4×C61—44no—
153C22×C4×C61—44no—
154C22×C4×C61—44no—
155C22×C4×C61C2×D414yes—
156C22×C4×C61C2×D414yes—
157C22×C4×C61C2×D414yes—
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D412yes—
163C22×C4×C61#4114yes—
164C22×C4×C61#4112yes—
165C22×C4×C61#4112yes—
166C22×C4×C61#4112yes—
167C4⋊C4×C61—216no—
168C4⋊C4×C61—28no—
169C4⋊C4×C61—44no—
170C4⋊C4×C61—44no—
171C22⋊C4×C61—216no—
172C22⋊C4×C61—216no—
173C22⋊C4×C61—48no—
174C42×C61—28no—
175C42×C61—44no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61#12414yes—
199C23×C61C22×D411yescharacteristic
200C23×C61C22×D411yescharacteristic
201C23×C61C22×D411yescharacteristic
202C23×C61C22×D411yescharacteristic
203C23×C61C2×#4111yescharacteristic
204C23×C61C2×#4111yescharacteristic
205C23×C61C2×#4111yescharacteristic
206C23×C61C2×#4112yes—
207C23×C61C2×#4112yes—
208C23×C612−1+412yes—
209C23×C612−1+412yes—
210Q8×C61—416no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—28no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—44no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—416no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—44no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—44no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—44no—
254C2×C4×C61—44no—
255C2×C4×C61#12014yes—
256C2×C4×C61#12114yes—
257C2×C4×C61#12114yes—
258C2×C4×C61#12114yes—
259C2×C4×C61#12314yes—
260C2×C4×C61#12314yes—
261C2×C4×C61#12414yes—
262#15236C6111yesmaximal, characteristic, direct product factor, Sylow
263C22×C61—24no—
264C22×C61—24no—
265C22×C61—24no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—42no—
271C22×C61—44no—
272C22×C61—42no—
273C22×C61—44no—
274C22×C61C2×#12011yescharacteristic
275C22×C61C2×#12111yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12111yescharacteristic
278C22×C61C2×#12311yescharacteristic
279C22×C61C2×#12311yescharacteristic
280C22×C61#52512yes—
281C22×C61C2×#12411yescharacteristic
282C22×C61C2×#12411yescharacteristic
283C22×C61#52912yes—
284C22×C61#53211yescharacteristic
285C22×C61#53612yes—
286C22×C61#53611yescharacteristic
287C22×C61#53611yescharacteristic
288C22×C61#54112yes—
289C22×C61#54712yes—
290C22×C61#54812yes—
291C22×C61#54812yes—
292C22×C61#54812yes—
293C22×C61#55212yes—
294C22×C61#55212yes—
295C22×C61#55212yes—
296C22×C61#55612yes—
297C4×C61—42no—
298C4×C61—44no—
299C4×C61—44no—
300C4×C61—44no—
301C4×C61—44no—
302C4×C61—82no—
303C4×C61—82no—
304C4×C61#39418yes—
305C22×C22⋊Q8C2×C6111yescharacteristic
306C2×#391C2×C6111yescharacteristic, complemented
307C2×#391C2×C6111yescharacteristic, complemented
308C2×#382C2×C6111yescharacteristic, complemented
309C2×#380C2×C6111yescharacteristic
310C2×C4×C22⋊C4C2×C6111yescharacteristic
311#1442C2×C6114yescomplemented
312#1440C2×C6114yes—
313C4⋊#95C2×C6111yescharacteristic, complemented
314C2×C61—44no—
315C2×C61C2×#39411yescharacteristic
316C2×C61#241512yes—
317C2×C61#242112yes—
318C2×C61#243211yescharacteristic
319C2×C61#243611yescharacteristic
320C2×C61#248912yes—
321C2×C61#249911yescharacteristic
322C2×C61#250311yescharacteristic
323C2×C61#253612yes—
324C2×C61#262111yescharacteristic
325C2×C61#285011yescharacteristic
326C23×Q8C22×C6111yescharacteristic
327C24×C4C22×C6111yescharacteristic
328C2×C22⋊Q8—24no—
329C2×C22⋊Q8—24no—
330C2×C22⋊Q8—24no—
331C22×C4⋊C4C22×C6112yes—
332C22×C4⋊C4C22×C6111yescharacteristic
333C22×C4⋊C4C22×C6111yescharacteristic
334C22×C4⋊C4C22×C6111yescharacteristic
335C22×C22⋊C4C22×C6112yes—
336C22×C22⋊C4C22×C6112yes—
337C22×C22⋊C4C22×C6112yes—
338C22×C42C22×C6111yescharacteristic
339C22×C42C22×C6111yescharacteristic
340#391—24no—
341#391—24no—
342#382—24no—
343#380—24no—
344C4×C22⋊C4—24no—
345C2×#95C22×C6114yes—
346C2×#95C22×C6114yes—
347C2×#95C22×C6114yes—
348C2×#95C22×C6114yes—
349C2×#95C22×C6112yes—
350C2×#95C22×C6112yes—
351C61#1523611yescharacteristic, direct product factor, Sylow
352C25C23×C6111yescharacteristic
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8D4×C6114yes—
360C23×C4—22no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4D4×C6114yes—
367C23×C4C23×C6112yes—
368C23×C4C23×C6112yes—
369C23×C4C23×C6112yes—
370C23×C4C23×C6112yes—
371C23×C4C23×C6112yes—
372C23×C4C23×C6111yescharacteristic
373C23×C4C23×C6112yes—
374C23×C4C23×C6111yescharacteristic
375C22⋊Q8—232no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4D4×C6114yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—28no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42D4×C6114yes—
402#95—24no—
403#95—24no—
404#95—28no—
405#95—28no—
406#95—28no—
407#95—28no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—22no—
417C24C2×D4×C6112yes—
418C24#41×C6114yes—
419C24C24×C6111yescharacteristic
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4C2×D4×C6114yes—
473C22×C4C2×D4×C6114yes—
474C22×C4C2×D4×C6114yes—
475C22×C4C2×D4×C6112yes—
476C22×C4C2×D4×C6112yes—
477C22×C4C2×D4×C6112yes—
478C22×C4C2×D4×C6112yes—
479C22×C4C2×D4×C6112yes—
480C22×C4#41×C6114yes—
481C22×C4#41×C6112yes—
482C22×C4#41×C6112yes—
483C22×C4#41×C6112yes—
484C4⋊C4—216no—
485C4⋊C4—28no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—48no—
491C42—28no—
492C42—44no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23#124×C6114yes—
516C23C22×D4×C6111yescharacteristic
517C23C22×D4×C6111yescharacteristic
518C23C22×D4×C6111yescharacteristic
519C23C22×D4×C6111yescharacteristic
520C23C2×#41×C6112yes—
521C23C2×#41×C6111yescharacteristic
522C23C2×#41×C6112yes—
523C23C2×#41×C6111yescharacteristic
524C23C2×#41×C6111yescharacteristic
525C232−1+4×C6112yes—
526C232−1+4×C6112yes—
527Q8—416no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4#120×C6114yes—
573C2×C4#121×C6114yes—
574C2×C4#121×C6114yes—
575C2×C4#121×C6114yes—
576C2×C4#123×C6114yes—
577C2×C4#123×C6114yes—
578C2×C4#124×C6114yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22C2×#120×C6111yescharacteristic
591C22C2×#121×C6111yescharacteristic
592C22C2×#121×C6111yescharacteristic
593C22C2×#121×C6111yescharacteristic
594C22C2×#123×C6111yescharacteristic
595C22C2×#123×C6111yescharacteristic
596C22C61×#52512yes—
597C22C2×#124×C6111yescharacteristic
598C22C2×#124×C6111yescharacteristic
599C22C61×#52912yes—
600C22C61×#53211yescharacteristic
601C22C61×#53612yes—
602C22C61×#53611yescharacteristic
603C22C61×#53611yescharacteristic
604C22C61×#54112yes—
605C22C61×#54712yes—
606C22C61×#54812yes—
607C22C61×#54812yes—
608C22C61×#54812yes—
609C22C61×#55212yes—
610C22C61×#55212yes—
611C22C61×#55212yes—
612C22C61×#55612yes—
613C4—44no—
614C4—42no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4C61×#39418yes—
621C2—44no—
622C2C2×C61×#39411yescharacteristic
623C2C61×#241512yes—
624C2C61×#242112yes—
625C2C61×#243211yescharacteristic
626C2C61×#243611yescharacteristic
627C2C61×#248912yes—
628C2C61×#249911yescharacteristic
629C2C61×#250311yescharacteristic
630C2C61×#253612yes—
631C2C61×#262111yescharacteristic
632C2C61×#285011yescharacteristic
subgroupG/H
center, socleC24×C61C24
commutator, Frattini, cosocleC24C24×C61
prime radicalC25×C61C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#391C2semidirect
3. C2×C61×#391C2semidirect
4. C2×C61×#382C2semidirect
7. C61×#1442C2semidirect
9. C61×C4⋊#95C2semidirect
262. #15236C61direct (N × H)
306. C2×#391C2×C61semidirect
307. C2×#391C2×C61semidirect
308. C2×#382C2×C61semidirect
311. #1442C2×C61semidirect
313. C4⋊#95C2×C61semidirect
351. C61#15236direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C612
C2×C61×#3912
C2×C61×#3822
C2×C61×#3802
C2×C4×C22⋊C4×C612
C61×#14422
C61×#14402
C61×C4⋊#952
#1523661