Finite Group Databasebeta

← C61×#19310 #29522743

Subgroups — C61×#19310 #29522743

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3184C211yesmaximal, characteristic
2C61×#3112C211yesmaximal, characteristic, complemented
3C61×#3037C211yesmaximal, characteristic, complemented
4Q8×M16×C61C211yesmaximal, characteristic, complemented
5C61×#1702C211yesmaximal, characteristic, complemented
6C61×#1698C211yesmaximal, characteristic, complemented
7C61×#1692C211yesmaximal, characteristic, complemented
8C61×#1688C211yesmaximal, characteristic, complemented
9C61×#1682C211yesmaximal, characteristic, complemented
10C61×#1678C211yesmaximal, characteristic, complemented
11C61×#1672C211yesmaximal, characteristic, complemented
12C61×#1668C211yesmaximal, characteristic, complemented
13C61×#1569C211yesmaximal, characteristic, complemented
14C61×#1560C211yesmaximal, characteristic, complemented
15C61×#1557C211yesmaximal, characteristic, complemented
16C2×#137×C61—21no—
17C61×#532C2211yescharacteristic
18C61×#532C2211yescharacteristic
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#498C2211yescharacteristic
21C61×#493C2211yescharacteristic
22C61×#486C2211yescharacteristic
23C61×#486C2211yescharacteristic
24C61×#480—21no—
25C61×#477C2211yescharacteristic
26C61×#477C2211yescharacteristic
27C61×#469—21no—
28C61×#469—21no—
29C61×#464—21no—
30C61×#464—21no—
31C61×#462C2211yescharacteristic
32C61×#462C2211yescharacteristic
33C61×#460C2211yescharacteristic
34C61×#460C2211yescharacteristic
35C61×#460C2211yescharacteristic
36C61×#460C2211yescharacteristic
37C61×#448—21no—
38C61×#446—21no—
39C61×#446—21no—
40C61×#446C2211yescharacteristic
41C61×#446C2211yescharacteristic
42C8×Q8×C61C2211yescharacteristic
43C8×Q8×C61C2211yescharacteristic
44C61×#442—21no—
45C61×#442—21no—
46C61×#440—21no—
47C61×#440—21no—
48C61×#432C2211yescharacteristic
49C61×#423—21no—
50C61×#423C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C61×#417—21no—
53C4×M16×C61—21no—
54C4×M16×C61C2211yescharacteristic
55C61×#335C2211yescharacteristic
56C61×#335C2211yescharacteristic
57C61×#334C2211yescharacteristic
58C61×#334C2211yescharacteristic
59C61×#333C2211yescharacteristic
60C61×#333C2211yescharacteristic
61C61×#332C2211yescharacteristic
62C61×#332C2211yescharacteristic
63C61×#331C2211yescharacteristic
64C61×#331C2211yescharacteristic
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#41×C61—21no—
68C2×#41×C61—21no—
69C22×Q8×C61D411yescharacteristic
70#137×C61—44no—
71C2×Q16×C61—21no—
72C2×Q16×C61—21no—
73C2×SD16×C61—21no—
74C2×SD16×C61—21no—
75C2×SD16×C61—21no—
76C2×SD16×C61—21no—
77C2×D8×C61—21no—
78C2×D8×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61D411yescharacteristic
82C2×M16×C61D411yescharacteristic
83C4⋊Q8×C61C2311yescharacteristic
84C4⋊Q8×C61C2311yescharacteristic
85#127×C61C2311yescharacteristic
86#127×C61C2311yescharacteristic
87#124×C61—21no—
88#124×C61—21no—
89#121×C61—21no—
90#121×C61—21no—
91#121×C61—21no—
92#121×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D411yescharacteristic
96C4×Q8×C61D411yescharacteristic
97C4×Q8×C61C2311yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#107×C61—21no—
108#107×C61—21no—
109#106×C61—21no—
110#106×C61—21no—
111C4⋊C8×C61—21no—
112C4⋊C8×C61—21no—
113C4⋊C8×C61—21no—
114C4⋊C8×C61—21no—
115C4⋊C8×C61—21no—
116C4⋊C8×C61—21no—
117C4⋊C8×C61C2311yescharacteristic
118C4⋊C8×C61C2311yescharacteristic
119C4⋊C8×C61C2311yescharacteristic
120C4⋊C8×C61C2311yescharacteristic
121C4⋊C8×C61C2311yescharacteristic
122C4⋊C8×C61C2311yescharacteristic
123#103×C61—21no—
124#103×C61—21no—
125#103×C61—21no—
126#103×C61—21no—
127#102×C61—21no—
128#102×C61—21no—
129#102×C61—21no—
130#102×C61—21no—
131#102×C61—21no—
132#102×C61—21no—
133#102×C61—21no—
134#102×C61—21no—
135C22⋊C8×C61—21no—
136#97×C61—21no—
137#97×C61—21no—
138#97×C61—21no—
139#97×C61D411yescharacteristic
140#97×C61D411yescharacteristic
141C4×C8×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—82no—
148C2×Q8×C61—21no—
149C2×Q8×C61—22no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—41no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—41no—
162C2×D4×C61—41no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C22×C4×C61—41no—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61#4111yescharacteristic
170Q16×C61—42no—
171Q16×C61—42no—
172SD16×C61—42no—
173SD16×C61—42no—
174SD16×C61—42no—
175SD16×C61—42no—
176D8×C61—42no—
177D8×C61—42no—
178M16×C61—22no—
179M16×C61—41no—
180M16×C61—42no—
181M16×C61—42no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D411yescharacteristic
190C2×C8×C61C2×D411yescharacteristic
191C2×C8×C61C2×D411yescharacteristic
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—41no—
204C22⋊C4×C61—41no—
205C22⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C42×C61—21no—
208C42×C61—21no—
209C42×C61—21no—
210C42×C61—21no—
211C42×C61C2×D411yescharacteristic
212C42×C61C2×D411yescharacteristic
213C42×C61#4111yescharacteristic
214C42×C61C2411yescharacteristic
215C23×C61—41no—
216C23×C61—41no—
217C23×C61#12111yescharacteristic
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—42no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—81no—
225Q8×C61AGL(1,ℤ/8)12yes—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—81no—
235D4×C61—81no—
236D4×C61—82no—
237D4×C61—82no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—82no—
251C2×C4×C61—82no—
252C2×C4×C61—81no—
253C2×C4×C61#12111yescharacteristic
254C2×C4×C61#12111yescharacteristic
255C2×C4×C61#12111yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61—41no—
268C8×C61#13712yescomplemented
269#19310C6111yesmaximal, characteristic, direct product factor, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×#12111yescharacteristic
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C61C2×#13711yescharacteristic
292C4×C61#57511yescharacteristic
293C4×C61#57511yescharacteristic
294#3184C2×C6111yescharacteristic
295#3112C2×C6111yescharacteristic, complemented
296#3037C2×C6111yescharacteristic, complemented
297Q8×M16C2×C6111yescharacteristic, complemented
298#1702C2×C6111yescharacteristic, complemented
299#1698C2×C6111yescharacteristic, complemented
300#1692C2×C6111yescharacteristic, complemented
301#1688C2×C6111yescharacteristic, complemented
302#1682C2×C6111yescharacteristic, complemented
303#1678C2×C6111yescharacteristic, complemented
304#1672C2×C6111yescharacteristic, complemented
305#1668C2×C6111yescharacteristic, complemented
306#1569C2×C6111yescharacteristic, complemented
307#1560C2×C6111yescharacteristic, complemented
308#1557C2×C6111yescharacteristic, complemented
309C2×C61—41no—
310C2×C61—161no—
311C2×C61—161no—
312C2×C61#303711yescharacteristic
313C2×C61#305511yescharacteristic
314C2×C61#306811yescharacteristic
315C2×#137—21no—
316#532C22×C6111yescharacteristic
317#532C22×C6111yescharacteristic
318C2×C4×Q8C22×C6111yescharacteristic
319#498C22×C6111yescharacteristic
320#493C22×C6111yescharacteristic
321#486C22×C6111yescharacteristic
322#486C22×C6111yescharacteristic
323#480—21no—
324#477C22×C6111yescharacteristic
325#477C22×C6111yescharacteristic
326#469—21no—
327#469—21no—
328#464—21no—
329#464—21no—
330#462C22×C6111yescharacteristic
331#462C22×C6111yescharacteristic
332#460C22×C6111yescharacteristic
333#460C22×C6111yescharacteristic
334#460C22×C6111yescharacteristic
335#460C22×C6111yescharacteristic
336#448—21no—
337#446—21no—
338#446—21no—
339#446C22×C6111yescharacteristic
340#446C22×C6111yescharacteristic
341C8×Q8C22×C6111yescharacteristic
342C8×Q8C22×C6111yescharacteristic
343#442—21no—
344#442—21no—
345#440—21no—
346#440—21no—
347#432C22×C6111yescharacteristic
348#423—21no—
349#423C22×C6111yescharacteristic
350#423C22×C6111yescharacteristic
351#417—21no—
352C4×M16—21no—
353C4×M16C22×C6111yescharacteristic
354#335C22×C6111yescharacteristic
355#335C22×C6111yescharacteristic
356#334C22×C6111yescharacteristic
357#334C22×C6111yescharacteristic
358#333C22×C6111yescharacteristic
359#333C22×C6111yescharacteristic
360#332C22×C6111yescharacteristic
361#332C22×C6111yescharacteristic
362#331C22×C6111yescharacteristic
363#331C22×C6111yescharacteristic
364#331C22×C6111yescharacteristic
365#331C22×C6111yescharacteristic
366C61#1931011yescharacteristic, direct product factor, Sylow
367C2×#41—21no—
368C2×#41—21no—
369C22×Q8D4×C6111yescharacteristic
370#137—44no—
371C2×Q16—21no—
372C2×Q16—21no—
373C2×SD16—21no—
374C2×SD16—21no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×D8—21no—
378C2×D8—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16D4×C6111yescharacteristic
382C2×M16D4×C6111yescharacteristic
383C4⋊Q8C23×C6111yescharacteristic
384C4⋊Q8C23×C6111yescharacteristic
385#127C23×C6111yescharacteristic
386#127C23×C6111yescharacteristic
387#124—21no—
388#124—21no—
389#121—21no—
390#121—21no—
391#121—21no—
392#121—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8D4×C6111yescharacteristic
396C4×Q8D4×C6111yescharacteristic
397C4×Q8C23×C6111yescharacteristic
398C4×Q8C23×C6111yescharacteristic
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4D4×C6111yescharacteristic
405C2×C42—21no—
406C2×C42C23×C6111yescharacteristic
407#107—21no—
408#107—21no—
409#106—21no—
410#106—21no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—21no—
415C4⋊C8—21no—
416C4⋊C8—21no—
417C4⋊C8C23×C6111yescharacteristic
418C4⋊C8C23×C6111yescharacteristic
419C4⋊C8C23×C6111yescharacteristic
420C4⋊C8C23×C6111yescharacteristic
421C4⋊C8C23×C6111yescharacteristic
422C4⋊C8C23×C6111yescharacteristic
423#103—21no—
424#103—21no—
425#103—21no—
426#103—21no—
427#102—21no—
428#102—21no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435C22⋊C8—21no—
436#97—21no—
437#97—21no—
438#97—21no—
439#97D4×C6111yescharacteristic
440#97D4×C6111yescharacteristic
441C4×C8—21no—
442C4×C8—21no—
443C4×C8C23×C6111yescharacteristic
444C4×C8C23×C6111yescharacteristic
445#41—42no—
446#41—42no—
447#41—82no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—41no—
453C2×Q8C2×D4×C6111yescharacteristic
454C2×Q8C2×D4×C6111yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—41no—
462C2×D4—41no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4C2×D4×C6111yescharacteristic
468C22×C4C2×D4×C6111yescharacteristic
469C22×C4#41×C6111yescharacteristic
470Q16—42no—
471Q16—42no—
472SD16—42no—
473SD16—42no—
474SD16—42no—
475SD16—42no—
476D8—42no—
477D8—42no—
478M16—22no—
479M16—41no—
480M16—42no—
481M16—42no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8C2×D4×C6111yescharacteristic
489C2×C8C2×D4×C6111yescharacteristic
490C2×C8C2×D4×C6111yescharacteristic
491C2×C8C2×D4×C6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4C2×D4×C6111yescharacteristic
502C4⋊C4C2×D4×C6111yescharacteristic
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42C2×D4×C6111yescharacteristic
512C42C2×D4×C6111yescharacteristic
513C42#41×C6111yescharacteristic
514C42C24×C6111yescharacteristic
515C23—41no—
516C23—41no—
517C23#121×C6111yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—21no—
521Q8—42no—
522Q8—41no—
523Q8—41no—
524Q8—81no—
525Q8AGL(1,ℤ/8)×C6112yes—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—81no—
535D4—81no—
536D4—82no—
537D4—82no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4#121×C6111yescharacteristic
554C2×C4#121×C6111yescharacteristic
555C2×C4#121×C6111yescharacteristic
556C2×C4C22×D4×C6111yescharacteristic
557C2×C4C22×D4×C6111yescharacteristic
558C2×C4C2×#41×C6111yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8#137×C6112yescomplemented
569C22—21no—
570C22—21no—
571C22—21no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C2×#121×C6111yescharacteristic
579C4—21no—
580C4—21no—
581C4—22no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4C2×AGL(1,ℤ/8)×C6111yescharacteristic
590C4C2×#137×C6111yescharacteristic
591C4C61×#57511yescharacteristic
592C4C61×#57511yescharacteristic
593C2—41no—
594C2—161no—
595C2—161no—
596C2C61×#303711yescharacteristic
597C2C61×#305511yescharacteristic
598C2C61×#306811yescharacteristic
subgroupG/H
center, socleC22×C61C2×#121
commutator, Frattini, cosocleC42C24×C61
prime radicalC61×#3184C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3112C2semidirect
3. C61×#3037C2semidirect
4. Q8×M16×C61C2semidirect
5. C61×#1702C2semidirect
6. C61×#1698C2semidirect
7. C61×#1692C2semidirect
8. C61×#1688C2semidirect
9. C61×#1682C2semidirect
10. C61×#1678C2semidirect
11. C61×#1672C2semidirect
12. C61×#1668C2semidirect
13. C61×#1569C2semidirect
14. C61×#1560C2semidirect
15. C61×#1557C2semidirect
268. C8×C61#137semidirect
269. #19310C61direct (N × H)
295. #3112C2×C61semidirect
296. #3037C2×C61semidirect
297. Q8×M16C2×C61semidirect
298. #1702C2×C61semidirect
299. #1698C2×C61semidirect
300. #1692C2×C61semidirect
301. #1688C2×C61semidirect
302. #1682C2×C61semidirect
303. #1678C2×C61semidirect
304. #1672C2×C61semidirect
305. #1668C2×C61semidirect
306. #1569C2×C61semidirect
307. #1560C2×C61semidirect
308. #1557C2×C61semidirect
366. C61#19310direct (N × H)
568. C8#137×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#31842
C61×#31122
C61×#30372
Q8×M16×C612
C61×#17022
C61×#16982
C61×#16922
C61×#16882
C61×#16822
C61×#16782
C61×#16722
C61×#16682
C61×#15692
C61×#15602
C61×#15572
#1931061