Finite Group Databasebeta

← C61×#15050 #29518483

Subgroups — C61×#15050 #29518483

Order 15616.

1590 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#397C211yesmaximal, characteristic
2C2×C61×#393C211yesmaximal, characteristic
3C2×C61×#391C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
5C61×#1446C211yesmaximal, characteristic, complemented
6C61×#1445C211yesmaximal, characteristic, complemented
7C61×#1443C212yesmaximal, complemented
8C61×#1440C212yesmaximal
9C61×#1434C212yesmaximal, complemented
10C61×#1434C211yesmaximal, characteristic, complemented
11C61×C22⋊#95C212yesmaximal
12C23×Q8×C61C2211yescharacteristic
13C24×C4×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2211yescharacteristic
15C22×C22⋊C4×C61C2211yescharacteristic
16C22×C22⋊C4×C61C2212yes—
17C22×C22⋊C4×C61C2212yes—
18C22×C22⋊C4×C61C2211yescharacteristic
19C22×C42×C61C2211yescharacteristic
20C22×C42×C61C2211yescharacteristic
21C61×#397—24no—
22C61×#393—24no—
23C61×#391—24no—
24C4×C22⋊C4×C61—24no—
25C2×#95×C61C2211yescharacteristic
26C2×#95×C61C2211yescharacteristic
27C2×#95×C61C2212yes—
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2211yescharacteristic
31C2×#95×C61C2212yes—
32C2×#95×C61C2211yescharacteristic
33C2×#95×C61C2212yes—
34C2×#95×C61C2212yes—
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C2×#95×C61C2212yes—
38C2×#95×C61C2212yes—
39C25×C61C2311yescharacteristic
40C22×Q8×C61—22no—
41C22×Q8×C61—22no—
42C22×Q8×C61—24no—
43C22×Q8×C61—22no—
44C22×Q8×C61—42no—
45C23×C4×C61—22no—
46C23×C4×C61—22no—
47C23×C4×C61—22no—
48C23×C4×C61—24no—
49C23×C4×C61—42no—
50C23×C4×C61C2312yes—
51C23×C4×C61C2312yes—
52C23×C4×C61C2312yes—
53C23×C4×C61C2311yescharacteristic
54C23×C4×C61C2311yescharacteristic
55C23×C4×C61C2311yescharacteristic
56C23×C4×C61C2312yes—
57C23×C4×C61C2311yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C2×C4⋊C4×C61—22no—
61C2×C4⋊C4×C61—22no—
62C2×C4⋊C4×C61—22no—
63C2×C22⋊C4×C61—22no—
64C2×C22⋊C4×C61—22no—
65C2×C22⋊C4×C61—22no—
66C2×C22⋊C4×C61—22no—
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61—24no—
69C2×C22⋊C4×C61—24no—
70C2×C22⋊C4×C61—24no—
71C2×C22⋊C4×C61—24no—
72C2×C22⋊C4×C61—24no—
73C2×C22⋊C4×C61—42no—
74C2×C22⋊C4×C61D414yes—
75C2×C42×C61—22no—
76C2×C42×C61—24no—
77C2×C42×C61—42no—
78C2×C42×C61D414yes—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—24no—
82#95×C61—24no—
83#95×C61—24no—
84#95×C61—22no—
85#95×C61—22no—
86#95×C61—24no—
87#95×C61—24no—
88#95×C61—24no—
89#95×C61—24no—
90#95×C61—24no—
91#95×C61—42no—
92#95×C61—42no—
93C24×C61—21no—
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—22no—
97C24×C61—21no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61C2×D412yes—
102C24×C61#4112yes—
103C24×C61#4112yes—
104C24×C61C2411yescharacteristic
105C2×Q8×C61—28no—
106C2×Q8×C61—48no—
107C2×Q8×C61—48no—
108C2×Q8×C61—44no—
109C2×Q8×C61—44no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—28no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
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—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
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—22no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—48no—
160C22×C4×C61—48no—
161C22×C4×C61—44no—
162C22×C4×C61—44no—
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D412yes—
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61#4112yes—
169C22×C4×C61#4112yes—
170C22×C4×C61#4114yes—
171C22×C4×C61#4112yes—
172C22×C4×C61#4114yes—
173C4⋊C4×C61—44no—
174C22⋊C4×C61—28no—
175C22⋊C4×C61—28no—
176C22⋊C4×C61—48no—
177C22⋊C4×C61—48no—
178C42×C61—44no—
179C42×C61—44no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—22no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61#12414yes—
203C23×C61C22×D411yescharacteristic
204C23×C61C22×D411yescharacteristic
205C23×C61C2×#4111yescharacteristic
206C23×C61C2×#4111yescharacteristic
207C23×C61C2×#4111yescharacteristic
208C23×C61C2×#4111yescharacteristic
209C23×C61C2×#4112yes—
210C23×C61C2×#4111yescharacteristic
211C23×C61C2×#4112yes—
212C23×C61H(5,𝔽2)12yes—
213C23×C612−1+412yes—
214Q8×C61—416no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—416no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
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—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#12114yes—
273C2×C4×C61#12414yes—
274#15050C6111yesmaximal, characteristic, direct product factor, Sylow
275C22×C61—24no—
276C22×C61—28no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61C2×#12111yescharacteristic
287C22×C61#52511yescharacteristic
288C22×C61#52511yescharacteristic
289C22×C61#52511yescharacteristic
290C22×C61C2×#12411yescharacteristic
291C22×C61C2×#12411yescharacteristic
292C22×C61#52911yescharacteristic
293C22×C61#52911yescharacteristic
294C22×C61#52912yes—
295C22×C61#52912yes—
296C22×C61#53211yescharacteristic
297C22×C61#53411yescharacteristic
298C22×C61#53611yescharacteristic
299C22×C61#53912yes—
300C22×C61#54212yes—
301C22×C61#54612yes—
302C22×C61#54712yes—
303C22×C61#55112yes—
304C22×C61#55112yes—
305C22×C61#55112yes—
306C22×C61#55212yes—
307C22×C61#55612yes—
308C22×C61#55612yes—
309C4×C61—24no—
310C4×C61—24no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—44no—
315C4×C61—42no—
316C4×C61—82no—
317C4×C61—82no—
318C4×C61—82no—
319C2×#397C2×C6111yescharacteristic
320C2×#393C2×C6111yescharacteristic
321C2×#391C2×C6111yescharacteristic, complemented
322C2×C4×C22⋊C4C2×C6111yescharacteristic
323#1446C2×C6111yescharacteristic, complemented
324#1445C2×C6111yescharacteristic, complemented
325#1443C2×C6112yescomplemented
326#1440C2×C6112yes—
327#1434C2×C6112yescomplemented
328#1434C2×C6111yescharacteristic, complemented
329C22⋊#95C2×C6112yes—
330C2×C61—44no—
331C2×C61#239811yescharacteristic
332C2×C61#248812yes—
333C2×C61#252112yes—
334C2×C61#252312yes—
335C2×C61#252712yes—
336C2×C61#253411yescharacteristic
337C2×C61#254711yescharacteristic
338C2×C61#256111yescharacteristic
339C2×C61#259911yescharacteristic
340C2×C61#263811yescharacteristic
341C2×C61#285011yescharacteristic
342C23×Q8C22×C6111yescharacteristic
343C24×C4C22×C6111yescharacteristic
344C22×C4⋊C4C22×C6111yescharacteristic
345C22×C22⋊C4C22×C6112yes—
346C22×C22⋊C4C22×C6111yescharacteristic
347C22×C22⋊C4C22×C6112yes—
348C22×C22⋊C4C22×C6111yescharacteristic
349C22×C42C22×C6111yescharacteristic
350C22×C42C22×C6111yescharacteristic
351#397—24no—
352#393—24no—
353#391—24no—
354C4×C22⋊C4—24no—
355C2×#95C22×C6112yes—
356C2×#95C22×C6111yescharacteristic
357C2×#95C22×C6112yes—
358C2×#95C22×C6111yescharacteristic
359C2×#95C22×C6112yes—
360C2×#95C22×C6112yes—
361C2×#95C22×C6112yes—
362C2×#95C22×C6112yes—
363C2×#95C22×C6112yes—
364C2×#95C22×C6112yes—
365C2×#95C22×C6111yescharacteristic
366C2×#95C22×C6111yescharacteristic
367C2×#95C22×C6112yes—
368C2×#95C22×C6112yes—
369C61#1505011yescharacteristic, direct product factor, Sylow
370C25C23×C6111yescharacteristic
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—24no—
374C22×Q8—22no—
375C22×Q8—42no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—42no—
381C23×C4C23×C6112yes—
382C23×C4C23×C6112yes—
383C23×C4C23×C6112yes—
384C23×C4C23×C6111yescharacteristic
385C23×C4C23×C6111yescharacteristic
386C23×C4C23×C6112yes—
387C23×C4C23×C6111yescharacteristic
388C23×C4C23×C6111yescharacteristic
389C23×C4C23×C6111yescharacteristic
390C23×C4C23×C6111yescharacteristic
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4D4×C6114yes—
406C2×C42—22no—
407C2×C42—24no—
408C2×C42—42no—
409C2×C42D4×C6114yes—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—42no—
423#95—42no—
424C24—21no—
425C24—21no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C2×D4×C6112yes—
433C24#41×C6112yes—
434C24#41×C6112yes—
435C24C24×C6111yescharacteristic
436C2×Q8—28no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—44no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
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—22no—
470C22×C4—22no—
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—22no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4C2×D4×C6112yes—
495C22×C4C2×D4×C6112yes—
496C22×C4C2×D4×C6112yes—
497C22×C4C2×D4×C6112yes—
498C22×C4C2×D4×C6112yes—
499C22×C4#41×C6112yes—
500C22×C4#41×C6112yes—
501C22×C4#41×C6114yes—
502C22×C4#41×C6112yes—
503C22×C4#41×C6114yes—
504C4⋊C4—44no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C42—44no—
510C42—44no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23#124×C6114yes—
534C23C22×D4×C6111yescharacteristic
535C23C22×D4×C6111yescharacteristic
536C23C2×#41×C6111yescharacteristic
537C23C2×#41×C6111yescharacteristic
538C23C2×#41×C6111yescharacteristic
539C23C2×#41×C6112yes—
540C23C2×#41×C6112yes—
541C23C2×#41×C6111yescharacteristic
542C23C2×#41×C6111yescharacteristic
543C23H(5,𝔽2)×C6112yes—
544C232−1+4×C6112yes—
545Q8—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—416no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
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#121×C6114yes—
604C2×C4#124×C6114yes—
605C22—24no—
606C22—28no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22C2×#121×C6111yescharacteristic
617C22C61×#52511yescharacteristic
618C22C61×#52511yescharacteristic
619C22C61×#52511yescharacteristic
620C22C2×#124×C6111yescharacteristic
621C22C2×#124×C6111yescharacteristic
622C22C61×#52912yes—
623C22C61×#52911yescharacteristic
624C22C61×#52912yes—
625C22C61×#52911yescharacteristic
626C22C61×#53211yescharacteristic
627C22C61×#53411yescharacteristic
628C22C61×#53611yescharacteristic
629C22C61×#53912yes—
630C22C61×#54212yes—
631C22C61×#54612yes—
632C22C61×#54712yes—
633C22C61×#55112yes—
634C22C61×#55112yes—
635C22C61×#55112yes—
636C22C61×#55212yes—
637C22C61×#55612yes—
638C22C61×#55612yes—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4—82no—
649C2—44no—
650C2C61×#239811yescharacteristic
651C2C61×#248812yes—
652C2C61×#252112yes—
653C2C61×#252312yes—
654C2C61×#252712yes—
655C2C61×#253411yescharacteristic
656C2C61×#254711yescharacteristic
657C2C61×#256111yescharacteristic
658C2C61×#259911yescharacteristic
659C2C61×#263811yescharacteristic
660C2C61×#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
3. C2×C61×#391C2semidirect
5. C61×#1446C2semidirect
6. C61×#1445C2semidirect
7. C61×#1443C2semidirect
9. C61×#1434C2semidirect
10. C61×#1434C2semidirect
274. #15050C61direct (N × H)
321. C2×#391C2×C61semidirect
323. #1446C2×C61semidirect
324. #1445C2×C61semidirect
325. #1443C2×C61semidirect
327. #1434C2×C61semidirect
328. #1434C2×C61semidirect
369. C61#15050direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#3972
C2×C61×#3932
C2×C61×#3912
C2×C4×C22⋊C4×C612
C61×#14462
C61×#14452
C61×#14432
C61×#14402
C61×#14342
C61×C22⋊#952
#1505061