Finite Group Databasebeta

← C61×#18246 #29521679

Subgroups — C61×#18246 #29521679

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3563C211yesmaximal, characteristic, complemented
2C61×#3114C211yesmaximal, characteristic
3C61×#3101C211yesmaximal, characteristic, complemented
4C61×#2633C211yesmaximal, characteristic, complemented
5C61×#1611C212yesmaximal, complemented
6C61×#1609C212yesmaximal, complemented
7C61×#1599C212yesmaximal, complemented
8C61×#1597C212yesmaximal
9C61×#1521C211yesmaximal, characteristic, complemented
10C61×#1515C211yesmaximal, characteristic, complemented
11C61×#1470C211yesmaximal, characteristic, complemented
12C61×#564—21no—
13C61×#563—21no—
14C61×#557—21no—
15C61×#554—21no—
16C61×#552—21no—
17C61×#552—22no—
18C61×#552—21no—
19C61×#544—21no—
20C61×#544C2212yes—
21C61×#541—21no—
22C61×#541—22no—
23C61×#539C2212yes—
24C61×#533C2211yescharacteristic
25C61×#529C2211yescharacteristic
26C2×#125×C61C2211yescharacteristic
27C2×C4×D4×C61C2211yescharacteristic
28C61×#483C2212yes—
29C61×#483C2211yescharacteristic
30C61×#480C2211yescharacteristic
31C61×#478—21no—
32C61×#476—21no—
33C61×#463—21no—
34C61×#461—21no—
35C61×#450C2212yes—
36C61×#450C2211yescharacteristic
37C61×#447C2211yescharacteristic
38C61×#442—21no—
39C61×#440—21no—
40C4×SD16×C61—22no—
41C61×#432C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#399—21no—
44C61×#398C2212yes—
45C61×#397—21no—
46C61×#394C2212yes—
47C61×#385—22no—
48C61×#383C2211yescharacteristic
49C61×#328C2212yes—
50C61×#328C2212yes—
51C61×#328C2212yes—
52C61×#327C2212yes—
53C61×#324C2212yes—
54C61×#324C2212yes—
55C22×D4×C61C2311yescharacteristic
56C23×C4×C61—21no—
57C2×SD16×C61—22no—
58C2×SD16×C61—21no—
59C2×D8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—22no—
62C4⋊Q8×C61—22no—
63C4⋊Q8×C61D411yescharacteristic
64C4⋊Q8×C61D411yescharacteristic
65#126×C61—22no—
66#126×C61—22no—
67#126×C61—22no—
68#126×C61—22no—
69#126×C61—21no—
70#125×C61—22no—
71#125×C61—22no—
72#125×C61—22no—
73#125×C61—22no—
74#125×C61—21no—
75#125×C61—22no—
76#125×C61—22no—
77#125×C61—22no—
78#125×C61—22no—
79#125×C61—21no—
80#125×C61—21no—
81#125×C61—21no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61D411yescharacteristic
85#124×C61D411yescharacteristic
86#123×C61—22no—
87#123×C61—22no—
88#123×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—22no—
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61C2312yes—
97C22⋊Q8×C61C2311yescharacteristic
98#121×C61C2311yescharacteristic
99C4×Q8×C61—21no—
100C4×Q8×C61—22no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61D412yes—
104C4×Q8×C61D411yescharacteristic
105C4×D4×C61—22no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61—21no—
111C4×D4×C61D411yescharacteristic
112#117×C61—22no—
113#117×C61—21no—
114#117×C61—21no—
115#117×C61—22no—
116#117×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—22no—
119C2×C4⋊C4×C61D412yes—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2312yes—
124C2×C22⋊C4×C61—22no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—22no—
127C2×C42×C61C2311yescharacteristic
128#107×C61—21no—
129#106×C61—22no—
130#106×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133#103×C61—22no—
134#103×C61—21no—
135#102×C61—21no—
136#102×C61—22no—
137#102×C61—21no—
138#102×C61—21no—
139C22⋊C8×C61C2312yes—
140C22⋊C8×C61C2311yescharacteristic
141C22⋊C8×C61C2311yescharacteristic
142#97×C61—21no—
143C4×C8×C61—21no—
144#95×C61—21no—
145#95×C61—21no—
146#95×C61—22no—
147#95×C61C2312yes—
148C24×C61—21no—
149C2×Q8×C61—22no—
150C2×Q8×C61—21no—
151C2×Q8×C61C2×D412yes—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—42no—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—22no—
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61—21no—
162C22×C4×C61—41no—
163C22×C4×C61—41no—
164C22×C4×C61—82no—
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2411yescharacteristic
170SD16×C61—42no—
171SD16×C61—44no—
172D8×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—82no—
204C4⋊C4×C61C2×D411yescharacteristic
205C4⋊C4×C61C2×D412yes—
206C4⋊C4×C61C2×D411yescharacteristic
207C22⋊C4×C61—22no—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—21no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—21no—
216C22⋊C4×C61—44no—
217C22⋊C4×C61—41no—
218C22⋊C4×C61—41no—
219C22⋊C4×C61—82no—
220C42×C61—22no—
221C42×C61—22no—
222C42×C61—22no—
223C42×C61—22no—
224C42×C61—21no—
225C42×C61C2×D411yescharacteristic
226C42×C61C2×D411yescharacteristic
227C42×C61C2×D411yescharacteristic
228C42×C61C2×D411yescharacteristic
229C23×C61—22no—
230C23×C61—21no—
231C23×C61—41no—
232C23×C61—42no—
233C23×C61—41no—
234C23×C61—42no—
235C23×C61C22×D411yescharacteristic
236Q8×C61—42no—
237Q8×C61—41no—
238Q8×C61—42no—
239Q8×C61—82no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—42no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—41no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—82no—
266C2×C4×C61—82no—
267C2×C4×C61—82no—
268C2×C4×C61#12011yescharacteristic
269C2×C4×C61#12011yescharacteristic
270C2×C4×C61#12011yescharacteristic
271C2×C4×C61#12011yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C8×C61—41no—
275C8×C61—41no—
276C8×C61—42no—
277#18246C6111yesmaximal, characteristic, direct product factor, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—44no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—42no—
285C22×C61—82no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61C2×#12011yescharacteristic
289C22×C61#57712yes—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—42no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—82no—
304C4×C61—81no—
305#3563C2×C6111yescharacteristic, complemented
306#3114C2×C6111yescharacteristic
307#3101C2×C6111yescharacteristic, complemented
308#2633C2×C6111yescharacteristic, complemented
309#1611C2×C6112yescomplemented
310#1609C2×C6112yescomplemented
311#1599C2×C6112yescomplemented
312#1597C2×C6112yes—
313#1521C2×C6111yescharacteristic, complemented
314#1515C2×C6111yescharacteristic, complemented
315#1470C2×C6111yescharacteristic, complemented
316C2×C61—22no—
317C2×C61—81no—
318C2×C61—82no—
319C2×C61#301011yescharacteristic
320C2×C61#302011yescharacteristic
321C2×C61#302511yescharacteristic
322#564—21no—
323#563—21no—
324#557—21no—
325#554—21no—
326#552—21no—
327#552—22no—
328#552—21no—
329#544—21no—
330#544C22×C6112yes—
331#541—21no—
332#541—22no—
333#539C22×C6112yes—
334#533C22×C6111yescharacteristic
335#529C22×C6111yescharacteristic
336C2×#125C22×C6111yescharacteristic
337C2×C4×D4C22×C6111yescharacteristic
338#483C22×C6112yes—
339#483C22×C6111yescharacteristic
340#480C22×C6111yescharacteristic
341#478—21no—
342#476—21no—
343#463—21no—
344#461—21no—
345#450C22×C6112yes—
346#450C22×C6111yescharacteristic
347#447C22×C6111yescharacteristic
348#442—21no—
349#440—21no—
350C4×SD16—22no—
351#432C22×C6111yescharacteristic
352#431C22×C6111yescharacteristic
353#399—21no—
354#398C22×C6112yes—
355#397—21no—
356#394C22×C6112yes—
357#385—22no—
358#383C22×C6111yescharacteristic
359#328C22×C6112yes—
360#328C22×C6112yes—
361#328C22×C6112yes—
362#327C22×C6112yes—
363#324C22×C6112yes—
364#324C22×C6112yes—
365C61#1824611yescharacteristic, direct product factor, Sylow
366C22×D4C23×C6111yescharacteristic
367C23×C4—21no—
368C2×SD16—22no—
369C2×SD16—21no—
370C2×D8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—22no—
373C4⋊Q8—22no—
374C4⋊Q8D4×C6111yescharacteristic
375C4⋊Q8D4×C6111yescharacteristic
376#126—22no—
377#126—22no—
378#126—22no—
379#126—22no—
380#126—21no—
381#125—22no—
382#125—22no—
383#125—22no—
384#125—22no—
385#125—21no—
386#125—22no—
387#125—22no—
388#125—22no—
389#125—22no—
390#125—21no—
391#125—21no—
392#125—21no—
393#125—21no—
394#125—21no—
395#125D4×C6111yescharacteristic
396#124D4×C6111yescharacteristic
397#123—22no—
398#123—21no—
399#123—22no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405C22⋊Q8—21no—
406C22⋊Q8—22no—
407C22⋊Q8C23×C6111yescharacteristic
408C22⋊Q8C23×C6112yes—
409#121C23×C6111yescharacteristic
410C4×Q8—21no—
411C4×Q8—22no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8D4×C6111yescharacteristic
415C4×Q8D4×C6112yes—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4D4×C6111yescharacteristic
423#117—22no—
424#117—21no—
425#117—21no—
426#117—22no—
427#117—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4D4×C6112yes—
431C2×C4⋊C4D4×C6111yescharacteristic
432C2×C4⋊C4D4×C6111yescharacteristic
433C2×C4⋊C4C23×C6111yescharacteristic
434C2×C4⋊C4C23×C6112yes—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C42C23×C6111yescharacteristic
439#107—21no—
440#106—22no—
441#106—21no—
442C4⋊C8—21no—
443C4⋊C8—21no—
444#103—22no—
445#103—21no—
446#102—21no—
447#102—22no—
448#102—21no—
449#102—21no—
450C22⋊C8C23×C6112yes—
451C22⋊C8C23×C6111yescharacteristic
452C22⋊C8C23×C6111yescharacteristic
453#97—21no—
454C4×C8—21no—
455#95—21no—
456#95—21no—
457#95—22no—
458#95C23×C6112yes—
459C24—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8C2×D4×C6112yes—
463C2×Q8C2×D4×C6111yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—42no—
468C2×D4C2×D4×C6111yescharacteristic
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—82no—
476C22×C4C2×D4×C6112yes—
477C22×C4C2×D4×C6111yescharacteristic
478C22×C4C2×D4×C6112yes—
479C22×C4C2×D4×C6111yescharacteristic
480C22×C4C24×C6111yescharacteristic
481SD16—42no—
482SD16—44no—
483D8—42no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—82no—
515C4⋊C4C2×D4×C6111yescharacteristic
516C4⋊C4C2×D4×C6112yes—
517C4⋊C4C2×D4×C6111yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—44no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42C2×D4×C6111yescharacteristic
537C42C2×D4×C6111yescharacteristic
538C42C2×D4×C6111yescharacteristic
539C42C2×D4×C6111yescharacteristic
540C23—22no—
541C23—21no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23C22×D4×C6111yescharacteristic
547Q8—41no—
548Q8—42no—
549Q8—42no—
550Q8—82no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—42no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4#120×C6111yescharacteristic
580C2×C4#120×C6111yescharacteristic
581C2×C4#120×C6111yescharacteristic
582C2×C4#120×C6111yescharacteristic
583C2×C4C22×D4×C6111yescharacteristic
584C2×C4C22×D4×C6111yescharacteristic
585C8—41no—
586C8—41no—
587C8—42no—
588C22—21no—
589C22—21no—
590C22—44no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22C2×#120×C6111yescharacteristic
599C22C61×#57712yes—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—82no—
614C4—81no—
615C2—22no—
616C2—82no—
617C2—81no—
618C2C61×#301011yescharacteristic
619C2C61×#302011yescharacteristic
620C2C61×#302511yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#447C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3563C2semidirect
3. C61×#3101C2semidirect
4. C61×#2633C2semidirect
5. C61×#1611C2semidirect
6. C61×#1609C2semidirect
7. C61×#1599C2semidirect
9. C61×#1521C2semidirect
10. C61×#1515C2semidirect
11. C61×#1470C2semidirect
277. #18246C61direct (N × H)
305. #3563C2×C61semidirect
307. #3101C2×C61semidirect
308. #2633C2×C61semidirect
309. #1611C2×C61semidirect
310. #1609C2×C61semidirect
311. #1599C2×C61semidirect
313. #1521C2×C61semidirect
314. #1515C2×C61semidirect
315. #1470C2×C61semidirect
365. C61#18246direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35632
C61×#31142
C61×#31012
C61×#26332
C61×#16112
C61×#16092
C61×#15992
C61×#15972
C61×#15212
C61×#15152
C61×#14702
#1824661