Finite Group Databasebeta

← C61×#9113 #29512546

Subgroups — C61×#9113 #29512546

Order 15616.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C8×C61C211yesmaximal, characteristic, complemented
2C61×#2847C211yesmaximal, characteristic
3C2×C61×#327C212yesmaximal, complemented
4C61×C22⋊#95C211yesmaximal, characteristic, complemented
5C61×#1294C212yesmaximal, complemented
6C24×C4×C61C2211yescharacteristic
7C23×C8×C61—21no—
8C2×#121×C61C411yescharacteristic, complemented
9C2×#121×C61C411yescharacteristic, complemented
10C2×#121×C61C2211yescharacteristic
11C2×#120×C61—21no—
12C22×C22⋊C4×C61—21no—
13C2×C22⋊C8×C61—21no—
14C2×C22⋊C8×C61—24no—
15C2×C22⋊C8×C61C2212yes—
16C61×#386—21no—
17C61×#386—21no—
18C61×#386—21no—
19C61×#380C411yescharacteristic, complemented
20C61×#380C411yescharacteristic, complemented
21C61×#380C2211yescharacteristic
22C2×#95×C61—21no—
23C2×#95×C61C2212yes—
24C61×#327—24no—
25C25×C61D411yescharacteristic
26C22×D4×C61—21no—
27C22×D4×C61—21no—
28C22×D4×C61—21no—
29C22×D4×C61C2×C411yescharacteristic
30C23×C4×C61—22no—
31C23×C4×C61—22no—
32C23×C4×C61—22no—
33C23×C4×C61—42no—
34C23×C4×C61—42no—
35C23×C4×C61C2×C411yescharacteristic
36C23×C4×C61C2×C411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C22×C8×C61—24no—
42C22×C8×C61—22no—
43C22×C8×C61—42no—
44C22×C8×C61—42no—
45#121×C61—22no—
46#121×C61—22no—
47#121×C61—42no—
48#121×C61—42no—
49#120×C61—42no—
50#120×C61—42no—
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61—24no—
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—42no—
62C2×C22⋊C4×C61—42no—
63C22⋊C8×C61—24no—
64C22⋊C8×C61—22no—
65C22⋊C8×C61D414yescomplemented
66#95×C61—21no—
67#95×C61—21no—
68#95×C61—21no—
69#95×C61—22no—
70#95×C61—22no—
71#95×C61—42no—
72C24×C61—22no—
73C24×C61—22no—
74C24×C61—22no—
75C24×C61—22no—
76C24×C61—21no—
77C24×C61—41no—
78C24×C61—41no—
79C24×C61C22⋊C411yescharacteristic
80C24×C61C22⋊C411yescharacteristic
81C24×C61D812yes—
82C24×C61SD1612yes—
83C24×C61C2×D411yescharacteristic
84C2×D4×C61—22no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—42no—
88C2×D4×C61—42no—
89C2×D4×C61—42no—
90C2×D4×C61—42no—
91C2×D4×C61—42no—
92C2×D4×C61—42no—
93C2×D4×C61—42no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61—42no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—24no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61—48no—
121C22×C4×C61—48no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C22⋊C411yescharacteristic
129C22×C4×C61C22⋊C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C2×C8×C61—24no—
135C2×C8×C61—48no—
136C2×C8×C61—42no—
137C2×C8×C61—42no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—42no—
142C22⋊C4×C61—48no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—42no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—44no—
171C23×C61—44no—
172C23×C61—44no—
173C23×C61—44no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61#9912yes—
182C23×C61#10212yes—
183C23×C61#10212yes—
184C23×C61#10412yes—
185C23×C61C2×C22⋊C411yescharacteristic
186C23×C61#12112yes—
187C23×C61C2×D811yescharacteristic
188C23×C61C2×SD1611yescharacteristic
189C23×C61#13512yes—
190D4×C61—82no—
191D4×C61—84no—
192D4×C61—84no—
193D4×C61—82no—
194D4×C61—82no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—44no—
209C2×C4×C61—44no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—44no—
214C2×C4×C61—44no—
215C2×C4×C61—44no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—82no—
226C2×C4×C61—82no—
227C2×C4×C61—82no—
228C2×C4×C61—82no—
229C2×C4×C61#9912yes—
230C2×C4×C61C4×D412yes—
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12311yescharacteristic
233C8×C61—44no—
234#9113C6111yesmaximal, characteristic, direct product factor, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—48no—
253C22×C61—48no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61#32714yescomplemented
266C22×C61#38611yescharacteristic
267C22×C61C2×#9911yescharacteristic
268C22×C61C2×#9911yescharacteristic
269C22×C61C2×#10211yescharacteristic
270C22×C61#41611yescharacteristic
271C22×C61C2×#10411yescharacteristic
272C22×C61#42111yescharacteristic
273C22×C61#46512yes—
274C22×C61#46612yes—
275C4×C61—21no—
276C4×C61—22no—
277C4×C61—21no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—84no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61—81no—
284C22×C22⋊C8C2×C6111yescharacteristic, complemented
285#2847C2×C6111yescharacteristic
286C2×#327C2×C6112yescomplemented
287C22⋊#95C2×C6111yescharacteristic, complemented
288#1294C2×C6112yescomplemented
289C2×C61—22no—
290C2×C61—24no—
291C2×C61—22no—
292C2×C61—42no—
293C2×C61—161no—
294C2×C61C2×#32711yescharacteristic
295C2×C61#150111yescharacteristic
296C2×C61#151611yescharacteristic
297C2×C61#151911yescharacteristic
298C2×C61#187111yescharacteristic
299C2×C61#189311yescharacteristic
300C2×C61#189811yescharacteristic
301C24×C4C22×C6111yescharacteristic
302C23×C8—21no—
303C2×#121C4×C6111yescharacteristic, complemented
304C2×#121C4×C6111yescharacteristic, complemented
305C2×#121C22×C6111yescharacteristic
306C2×#120—21no—
307C22×C22⋊C4—21no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8—24no—
310C2×C22⋊C8C22×C6112yes—
311#386—21no—
312#386—21no—
313#386—21no—
314#380C4×C6111yescharacteristic, complemented
315#380C4×C6111yescharacteristic, complemented
316#380C22×C6111yescharacteristic
317C2×#95—21no—
318C2×#95C22×C6112yes—
319#327—24no—
320C61#911311yescharacteristic, direct product factor, Sylow
321C25D4×C6111yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×C4×C6111yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C2×C4×C6111yescharacteristic
332C23×C4C2×C4×C6111yescharacteristic
333C23×C4D4×C6111yescharacteristic
334C23×C4D4×C6111yescharacteristic
335C23×C4D4×C6111yescharacteristic
336C23×C4C23×C6111yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×C4×C6111yescharacteristic
348C2×C4⋊C4C2×C4×C6111yescharacteristic
349C2×C4⋊C4C2×C4×C6111yescharacteristic
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8D4×C6114yescomplemented
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—42no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24C22⋊C4×C6111yescharacteristic
376C24C22⋊C4×C6111yescharacteristic
377C24D8×C6112yes—
378C24SD16×C6112yes—
379C24C2×D4×C6111yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22⋊C4×C6111yescharacteristic
425C22×C4C22⋊C4×C6111yescharacteristic
426C22×C4C22×C4×C6111yescharacteristic
427C22×C4C2×D4×C6111yescharacteristic
428C22×C4C2×D4×C6112yes—
429C22×C4#41×C6112yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#99×C6112yes—
478C23#102×C6112yes—
479C23#102×C6112yes—
480C23#104×C6112yes—
481C23C2×C22⋊C4×C6111yescharacteristic
482C23#121×C6112yes—
483C23C2×D8×C6111yescharacteristic
484C23C2×SD16×C6111yescharacteristic
485C23#135×C6112yes—
486D4—82no—
487D4—84no—
488D4—84no—
489D4—82no—
490D4—82no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#99×C6112yes—
526C2×C4C4×D4×C6112yes—
527C2×C4#120×C6111yescharacteristic
528C2×C4#123×C6111yescharacteristic
529C8—44no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C61×#32714yescomplemented
561C22C61×#38611yescharacteristic
562C22C2×#99×C6111yescharacteristic
563C22C2×#99×C6111yescharacteristic
564C22C2×#102×C6111yescharacteristic
565C22C61×#41611yescharacteristic
566C22C2×#104×C6111yescharacteristic
567C22C61×#42111yescharacteristic
568C22C61×#46512yes—
569C22C61×#46612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—81no—
577C4—81no—
578C4—81no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—161no—
584C2C2×C61×#32711yescharacteristic
585C2C61×#150111yescharacteristic
586C2C61×#151611yescharacteristic
587C2C61×#151911yescharacteristic
588C2C61×#187111yescharacteristic
589C2C61×#189311yescharacteristic
590C2C61×#189811yescharacteristic
subgroupG/H
center, socleC23×C61C2×C22⋊C4
commutatorC22×C4C22×C4×C61
Frattini, cosocleC23×C4C23×C61
prime radicalC61×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C61 ◁ C24×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8×C61C2semidirect
3. C2×C61×#327C2semidirect
4. C61×C22⋊#95C2semidirect
5. C61×#1294C2semidirect
8. C2×#121×C61C4semidirect
9. C2×#121×C61C4semidirect
19. C61×#380C4semidirect
20. C61×#380C4semidirect
65. C22⋊C8×C61D4semidirect
234. #9113C61direct (N × H)
265. C22×C61#327semidirect
284. C22×C22⋊C8C2×C61semidirect
286. C2×#327C2×C61semidirect
287. C22⋊#95C2×C61semidirect
288. #1294C2×C61semidirect
303. C2×#121C4×C61semidirect
304. C2×#121C4×C61semidirect
314. #380C4×C61semidirect
315. #380C4×C61semidirect
320. C61#9113direct (N × H)
361. C22⋊C8D4×C61semidirect
560. C22C61×#327semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C8×C612
C61×#28472
C2×C61×#3272
C61×C22⋊#952
C61×#12942
#911361