Finite Group Databasebeta

← C61×#15114 #29518547

Subgroups — C61×#15114 #29518547

Order 15616.

1802 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C61C211yesmaximal, characteristic
2C2×C61×#387C212yesmaximal, complemented
3C2×C61×#385C211yesmaximal, characteristic
4C61×#1444C211yesmaximal, characteristic, complemented
5C61×#1442C211yesmaximal, characteristic, complemented
6C61×#1441C211yesmaximal, characteristic, complemented
7C61×#1441C211yesmaximal, characteristic, complemented
8C61×#1440C212yesmaximal
9C61×#1439C212yesmaximal
10C61×#1434C212yesmaximal, complemented
11C61×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C61C2211yescharacteristic
13C2×#117×C61—24no—
14C2×#117×C61—28no—
15C22×C4⋊C4×C61C2211yescharacteristic
16C22×C4⋊C4×C61C2211yescharacteristic
17C22×C4⋊C4×C61C2212yes—
18C22×C4⋊C4×C61C2211yescharacteristic
19C22×C4⋊C4×C61C2211yescharacteristic
20C22×C22⋊C4×C61C2211yescharacteristic
21C22×C22⋊C4×C61C2211yescharacteristic
22C22×C22⋊C4×C61C2212yes—
23C22×C22⋊C4×C61C2212yes—
24C22×C42×C61C2212yes—
25C61×#387—28no—
26C61×#385—24no—
27C2×#95×C61C2211yescharacteristic
28C2×#95×C61C2211yescharacteristic
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2212yes—
33C2×#95×C61C2211yescharacteristic
34C2×#95×C61C2211yescharacteristic
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C2×#95×C61C2212yes—
38C2×#95×C61C2212yes—
39C25×C61C2311yescharacteristic
40C23×C4×C61—24no—
41C23×C4×C61—24no—
42C23×C4×C61—22no—
43C23×C4×C61—22no—
44C23×C4×C61D414yes—
45C23×C4×C61C2312yes—
46C23×C4×C61C2311yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C23×C4×C61C2312yes—
49C23×C4×C61C2312yes—
50C23×C4×C61C2312yes—
51C23×C4×C61C2311yescharacteristic
52C23×C4×C61C2311yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C23×C4×C61C2311yescharacteristic
55#117×C61—232no—
56C2×C4⋊C4×C61—24no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—24no—
59C2×C4⋊C4×C61—22no—
60C2×C4⋊C4×C61—22no—
61C2×C4⋊C4×C61—24no—
62C2×C4⋊C4×C61—24no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61—24no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61—24no—
70C2×C22⋊C4×C61—24no—
71C2×C22⋊C4×C61—24no—
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61—42no—
79C2×C42×C61—24no—
80C2×C42×C61—24no—
81C2×C42×C61D418yes—
82#95×C61—22no—
83#95×C61—22no—
84#95×C61—24no—
85#95×C61—24no—
86#95×C61—24no—
87#95×C61—24no—
88#95×C61—22no—
89#95×C61—24no—
90#95×C61—24no—
91#95×C61—22no—
92#95×C61—42no—
93#95×C61—42no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—21no—
100C24×C61—21no—
101C24×C61—22no—
102C24×C61—21no—
103C24×C61C2×D412yes—
104C24×C61#4114yes—
105C24×C61C2411yescharacteristic
106C22×C4×C61—216no—
107C22×C4×C61—216no—
108C22×C4×C61—28no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—28no—
115C22×C4×C61—28no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
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—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—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
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—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61C2×D414yes—
162C22×C4×C61C2×D414yes—
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D414yes—
165C22×C4×C61C2×D412yes—
166C22×C4×C61#4112yes—
167C22×C4×C61#4114yes—
168C22×C4×C61#4112yes—
169C22×C4×C61#4112yes—
170C22×C4×C61#4112yes—
171C4⋊C4×C61—216no—
172C4⋊C4×C61—28no—
173C4⋊C4×C61—28no—
174C4⋊C4×C61—44no—
175C4⋊C4×C61—44no—
176C22⋊C4×C61—28no—
177C22⋊C4×C61—216no—
178C22⋊C4×C61—48no—
179C22⋊C4×C61—44no—
180C42×C61—216no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—42no—
203C23×C61#12314yes—
204C23×C61C22×D411yescharacteristic
205C23×C61C22×D412yes—
206C23×C61C2×#4112yes—
207C23×C61C2×#4111yescharacteristic
208C23×C61C2×#4111yescharacteristic
209C23×C61C2×#4111yescharacteristic
210C23×C61C2×#4112yes—
211C23×C61C2×#4111yescharacteristic
212C23×C61H(5,𝔽2)11yescharacteristic
213C23×C612−1+411yescharacteristic
214C23×C612−1+412yes—
215C2×C4×C61—232no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
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—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—41no—
266C2×C4×C61—42no—
267C2×C4×C61—41no—
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#12014yes—
275C2×C4×C61#12314yes—
276#15114C6111yesmaximal, characteristic, direct product factor, Sylow
277C22×C61—24no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—44no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61C2×#12011yescharacteristic
289C22×C61C2×#12311yescharacteristic
290C22×C61C2×#12311yescharacteristic
291C22×C61#52512yes—
292C22×C61#52512yes—
293C22×C61#52512yes—
294C22×C61#52511yescharacteristic
295C22×C61#53212yes—
296C22×C61#53211yescharacteristic
297C22×C61#53612yes—
298C22×C61#53611yescharacteristic
299C22×C61#53712yes—
300C22×C61#53711yescharacteristic
301C22×C61#55112yes—
302C22×C61#55212yes—
303C22×C61#55212yes—
304C22×C61#55212yes—
305C22×C61#55212yes—
306C22×C61#55211yescharacteristic
307C22×C61#55311yescharacteristic
308C22×C61#55511yescharacteristic
309C22×C61#55611yescharacteristic
310C22×C61#55612yes—
311C4×C61—24no—
312C4×C61—24no—
313C4×C61—44no—
314C4×C61—44no—
315C4×C61—44no—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61—82no—
320C4×C61—81no—
321C22×#117C2×C6111yescharacteristic
322C2×#387C2×C6112yescomplemented
323C2×#385C2×C6111yescharacteristic
324#1444C2×C6111yescharacteristic, complemented
325#1442C2×C6111yescharacteristic, complemented
326#1441C2×C6111yescharacteristic, complemented
327#1441C2×C6111yescharacteristic, complemented
328#1440C2×C6112yes—
329#1439C2×C6112yes—
330#1434C2×C6112yescomplemented
331C22⋊#95C2×C6111yescharacteristic
332C2×C61—44no—
333C2×C61#238811yescharacteristic
334C2×C61#248411yescharacteristic
335C2×C61#248511yescharacteristic
336C2×C61#249811yescharacteristic
337C2×C61#249911yescharacteristic
338C2×C61#259912yes—
339C2×C61#260112yes—
340C2×C61#260912yes—
341C2×C61#261112yes—
342C2×C61#261611yescharacteristic
343C2×C61#284911yescharacteristic
344C24×C4C22×C6111yescharacteristic
345C2×#117—24no—
346C2×#117—28no—
347C22×C4⋊C4C22×C6111yescharacteristic
348C22×C4⋊C4C22×C6111yescharacteristic
349C22×C4⋊C4C22×C6112yes—
350C22×C4⋊C4C22×C6111yescharacteristic
351C22×C4⋊C4C22×C6111yescharacteristic
352C22×C22⋊C4C22×C6112yes—
353C22×C22⋊C4C22×C6111yescharacteristic
354C22×C22⋊C4C22×C6111yescharacteristic
355C22×C22⋊C4C22×C6112yes—
356C22×C42C22×C6112yes—
357#387—28no—
358#385—24no—
359C2×#95C22×C6112yes—
360C2×#95C22×C6111yescharacteristic
361C2×#95C22×C6111yescharacteristic
362C2×#95C22×C6111yescharacteristic
363C2×#95C22×C6112yes—
364C2×#95C22×C6112yes—
365C2×#95C22×C6112yes—
366C2×#95C22×C6112yes—
367C2×#95C22×C6112yes—
368C2×#95C22×C6111yescharacteristic
369C2×#95C22×C6112yes—
370C2×#95C22×C6112yes—
371C61#1511411yescharacteristic, direct product factor, Sylow
372C25C23×C6111yescharacteristic
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4D4×C6114yes—
378C23×C4C23×C6111yescharacteristic
379C23×C4C23×C6112yes—
380C23×C4C23×C6111yescharacteristic
381C23×C4C23×C6112yes—
382C23×C4C23×C6112yes—
383C23×C4C23×C6111yescharacteristic
384C23×C4C23×C6112yes—
385C23×C4C23×C6111yescharacteristic
386C23×C4C23×C6111yescharacteristic
387C23×C4C23×C6111yescharacteristic
388#117—232no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—42no—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42D4×C6118yes—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—24no—
423#95—24no—
424#95—22no—
425#95—42no—
426#95—42no—
427C24—22no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—22no—
435C24—21no—
436C24C2×D4×C6112yes—
437C24#41×C6114yes—
438C24C24×C6111yescharacteristic
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
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—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4C2×D4×C6114yes—
495C22×C4C2×D4×C6114yes—
496C22×C4C2×D4×C6112yes—
497C22×C4C2×D4×C6114yes—
498C22×C4C2×D4×C6112yes—
499C22×C4#41×C6112yes—
500C22×C4#41×C6114yes—
501C22×C4#41×C6112yes—
502C22×C4#41×C6112yes—
503C22×C4#41×C6112yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C22⋊C4—216no—
510C22⋊C4—28no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C42—216no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23#123×C6114yes—
537C23C22×D4×C6111yescharacteristic
538C23C22×D4×C6112yes—
539C23C2×#41×C6112yes—
540C23C2×#41×C6111yescharacteristic
541C23C2×#41×C6112yes—
542C23C2×#41×C6111yescharacteristic
543C23C2×#41×C6111yescharacteristic
544C23C2×#41×C6111yescharacteristic
545C23H(5,𝔽2)×C6111yescharacteristic
546C232−1+4×C6111yescharacteristic
547C232−1+4×C6112yes—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4#120×C6114yes—
608C2×C4#123×C6114yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22C2×#120×C6111yescharacteristic
621C22C2×#123×C6111yescharacteristic
622C22C2×#123×C6111yescharacteristic
623C22C61×#52512yes—
624C22C61×#52512yes—
625C22C61×#52512yes—
626C22C61×#52511yescharacteristic
627C22C61×#53212yes—
628C22C61×#53211yescharacteristic
629C22C61×#53612yes—
630C22C61×#53611yescharacteristic
631C22C61×#53712yes—
632C22C61×#53711yescharacteristic
633C22C61×#55112yes—
634C22C61×#55212yes—
635C22C61×#55212yes—
636C22C61×#55212yes—
637C22C61×#55212yes—
638C22C61×#55211yescharacteristic
639C22C61×#55311yescharacteristic
640C22C61×#55511yescharacteristic
641C22C61×#55611yescharacteristic
642C22C61×#55612yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—82no—
652C4—81no—
653C2—44no—
654C2C61×#238811yescharacteristic
655C2C61×#248411yescharacteristic
656C2C61×#248511yescharacteristic
657C2C61×#249811yescharacteristic
658C2C61×#249911yescharacteristic
659C2C61×#259912yes—
660C2C61×#260112yes—
661C2C61×#260912yes—
662C2C61×#261112yes—
663C2C61×#261611yescharacteristic
664C2C61×#284911yescharacteristic
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×#387C2semidirect
4. C61×#1444C2semidirect
5. C61×#1442C2semidirect
6. C61×#1441C2semidirect
7. C61×#1441C2semidirect
10. C61×#1434C2semidirect
276. #15114C61direct (N × H)
322. C2×#387C2×C61semidirect
324. #1444C2×C61semidirect
325. #1442C2×C61semidirect
326. #1441C2×C61semidirect
327. #1441C2×C61semidirect
330. #1434C2×C61semidirect
371. C61#15114direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#117×C612
C2×C61×#3872
C2×C61×#3852
C61×#14442
C61×#14422
C61×#14412
C61×#14402
C61×#14392
C61×#14342
C61×C22⋊#952
#1511461