Finite Group Databasebeta

← C61×#18107 #29521540

Subgroups — C61×#18107 #29521540

Order 15616.

1000 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3459C211yesmaximal, characteristic
2C61×#3091C212yesmaximal, complemented
3C61×#2395C211yesmaximal, characteristic, complemented
4C61×#1615C212yesmaximal, complemented
5C61×#1614C211yesmaximal, characteristic, complemented
6C61×#1613C211yesmaximal, characteristic, complemented
7C61×#1609C212yesmaximal, complemented
8C61×#1608C211yesmaximal, characteristic, complemented
9C61×#1607C211yesmaximal, characteristic, complemented
10C61×#1501C212yesmaximal, complemented
11C61×#1459C211yesmaximal, characteristic, complemented
12C61×#543C2211yescharacteristic
13C61×#539—22no—
14C61×#539C2212yes—
15C61×#538—22no—
16C61×#538—21no—
17C61×#534—21no—
18C61×#534C2211yescharacteristic
19C61×#529—22no—
20C61×#529—21no—
21C61×#525—22no—
22C61×#525C2212yes—
23C2×C4×D4×C61—21no—
24C2×C4×D4×C61C2211yescharacteristic
25C2×#117×C61C2211yescharacteristic, complemented
26C61×#485—22no—
27C61×#484C2212yescomplemented
28C61×#483C2212yescomplemented
29C61×#482—22no—
30C61×#481C2212yes—
31C61×#480C2212yes—
32C61×#427—22no—
33C61×#416—22no—
34C2×C22⋊C8×C61C2212yescomplemented
35C61×#400C2211yescharacteristic, complemented
36C61×#399C2211yescharacteristic
37C61×#398C2211yescharacteristic, complemented
38C61×#397—21no—
39C61×#397C2211yescharacteristic
40C61×#394—21no—
41C4×C4⋊C4×C61—21no—
42C4×C22⋊C4×C61—21no—
43C4×C22⋊C4×C61C2211yescharacteristic, complemented
44C61×#328C2212yescomplemented
45C61×#328C2212yescomplemented
46C61×#327C2212yes—
47C61×#327C2212yes—
48C61×#324C2212yescomplemented
49C61×#323C2211yescharacteristic
50C61×#323C2211yescharacteristic
51C2×#41×C61D412yes—
52C22×D4×C61—21no—
53C22×D4×C61D411yescharacteristic
54C23×C4×C61—21no—
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C22×C8×C61—22no—
58#126×C61—22no—
59#126×C61—22no—
60#126×C61—22no—
61#125×C61—21no—
62#125×C61—22no—
63#124×C61—22no—
64#124×C61—21no—
65#123×C61—22no—
66#123×C61—22no—
67#123×C61—22no—
68#123×C61—22no—
69#123×C61—22no—
70#123×C61—22no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61D412yes—
76C22⋊Q8×C61C2312yes—
77#121×C61—22no—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61—21no—
81#121×C61D412yes—
82#121×C61C2312yes—
83#120×C61—22no—
84#120×C61—22no—
85#120×C61—22no—
86C4×Q8×C61—21no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61—21no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95#117×C61—22no—
96#117×C61—22no—
97#117×C61—22no—
98#117×C61D412yes—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61C2311yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61D411yescharacteristic
117#107×C61—22no—
118#107×C61—22no—
119#106×C61—22no—
120#106×C61—22no—
121#103×C61—22no—
122#103×C61—22no—
123#102×C61—22no—
124#102×C61—22no—
125C22⋊C8×C61—22no—
126C22⋊C8×C61C2312yes—
127C22⋊C8×C61C2312yes—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61C2311yescharacteristic
133#95×C61C2311yescharacteristic
134C24×C61—21no—
135C24×C61C2×D411yescharacteristic
136C24×C61C2×D411yescharacteristic
137#41×C61—44no—
138C2×Q8×C61—21no—
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—41no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61C2×D412yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—22no—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—42no—
163C22×C4×C61—42no—
164C22×C4×C61—82no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2411yescharacteristic
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—44no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61C2×D412yes—
189C4⋊C4×C61C2×D412yes—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—42no—
203C22⋊C4×C61—42no—
204C22⋊C4×C61—42no—
205C22⋊C4×C61—42no—
206C42×C61—22no—
207C42×C61—21no—
208C42×C61—22no—
209C42×C61—21no—
210C42×C61—42no—
211C42×C61—41no—
212C42×C61—41no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—22no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—42no—
223C23×C61—81no—
224C23×C61—81no—
225C23×C61#12012yes—
226C23×C61#13512yes—
227C23×C61C22×D411yescharacteristic
228Q8×C61—42no—
229Q8×C61—81no—
230D4×C61—42no—
231D4×C61—44no—
232D4×C61—44no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61—81no—
238C2×C4×C61—22no—
239C2×C4×C61—24no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61—44no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—81no—
268C2×C4×C61—81no—
269C2×C4×C61—82no—
270C2×C4×C61—81no—
271C2×C4×C61#12011yescharacteristic
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61C22×D412yes—
274C8×C61—42no—
275C8×C61—42no—
276#18107C6111yesmaximal, characteristic, direct product factor, Sylow
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—84no—
293C22×C61C2×#12011yescharacteristic
294C22×C61C2×#13511yescharacteristic
295C22×C61#57511yescharacteristic
296C4×C61—22no—
297C4×C61—21no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—42no—
301C4×C61—41no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61#45812yes—
309#3459C2×C6111yescharacteristic
310#3091C2×C6112yescomplemented
311#2395C2×C6111yescharacteristic, complemented
312#1615C2×C6112yescomplemented
313#1614C2×C6111yescharacteristic, complemented
314#1613C2×C6111yescharacteristic, complemented
315#1609C2×C6112yescomplemented
316#1608C2×C6111yescharacteristic, complemented
317#1607C2×C6111yescharacteristic, complemented
318#1501C2×C6112yescomplemented
319#1459C2×C6111yescharacteristic, complemented
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—42no—
323C2×C61—42no—
324C2×C61—82no—
325C2×C61#300211yescharacteristic
326C2×C61#301811yescharacteristic
327C2×C61C2×#45811yescharacteristic
328#543C22×C6111yescharacteristic
329#539—22no—
330#539C22×C6112yes—
331#538—22no—
332#538—21no—
333#534—21no—
334#534C22×C6111yescharacteristic
335#529—22no—
336#529—21no—
337#525—22no—
338#525C22×C6112yes—
339C2×C4×D4—21no—
340C2×C4×D4C22×C6111yescharacteristic
341C2×#117C22×C6111yescharacteristic, complemented
342#485—22no—
343#484C22×C6112yescomplemented
344#483C22×C6112yescomplemented
345#482—22no—
346#481C22×C6112yes—
347#480C22×C6112yes—
348#427—22no—
349#416—22no—
350C2×C22⋊C8C22×C6112yescomplemented
351#400C22×C6111yescharacteristic, complemented
352#399C22×C6111yescharacteristic
353#398C22×C6111yescharacteristic, complemented
354#397—21no—
355#397C22×C6111yescharacteristic
356#394—21no—
357C4×C4⋊C4—21no—
358C4×C22⋊C4—21no—
359C4×C22⋊C4C22×C6111yescharacteristic, complemented
360#328C22×C6112yescomplemented
361#328C22×C6112yescomplemented
362#327C22×C6112yes—
363#327C22×C6112yes—
364#324C22×C6112yescomplemented
365#323C22×C6111yescharacteristic
366#323C22×C6111yescharacteristic
367C61#1810711yescharacteristic, direct product factor, Sylow
368C2×#41D4×C6112yes—
369C22×D4—21no—
370C22×D4D4×C6111yescharacteristic
371C23×C4—21no—
372C23×C4D4×C6111yescharacteristic
373C23×C4C23×C6111yescharacteristic
374C22×C8—22no—
375#126—22no—
376#126—22no—
377#126—22no—
378#125—21no—
379#125—22no—
380#124—22no—
381#124—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387#123—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8D4×C6112yes—
393C22⋊Q8C23×C6112yes—
394#121—22no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121D4×C6112yes—
399#121C23×C6112yes—
400#120—22no—
401#120—22no—
402#120—22no—
403C4×Q8—21no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412#117—22no—
413#117—22no—
414#117—22no—
415#117D4×C6112yes—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4C23×C6111yescharacteristic
420C2×C4⋊C4C23×C6111yescharacteristic
421C2×C4⋊C4C23×C6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4D4×C6111yescharacteristic
429C2×C22⋊C4C23×C6111yescharacteristic
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42D4×C6111yescharacteristic
434#107—22no—
435#107—22no—
436#106—22no—
437#106—22no—
438#103—22no—
439#103—22no—
440#102—22no—
441#102—22no—
442C22⋊C8—22no—
443C22⋊C8C23×C6112yes—
444C22⋊C8C23×C6112yes—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95C23×C6111yescharacteristic
450#95C23×C6111yescharacteristic
451C24—21no—
452C24C2×D4×C6111yescharacteristic
453C24C2×D4×C6111yescharacteristic
454#41—44no—
455C2×Q8—21no—
456C2×Q8C2×D4×C6112yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×D4×C6112yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4C2×D4×C6111yescharacteristic
483C22×C4C2×D4×C6111yescharacteristic
484C22×C4C2×D4×C6112yes—
485C22×C4C2×D4×C6111yescharacteristic
486C22×C4C2×D4×C6111yescharacteristic
487C22×C4C2×D4×C6112yes—
488C22×C4C24×C6111yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4C2×D4×C6112yes—
506C4⋊C4C2×D4×C6112yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—41no—
529C42—41no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—81no—
541C23—81no—
542C23#120×C6112yes—
543C23#135×C6112yes—
544C23C22×D4×C6111yescharacteristic
545Q8—42no—
546Q8—81no—
547D4—42no—
548D4—44no—
549D4—44no—
550D4—42no—
551D4—42no—
552D4—82no—
553D4—82no—
554D4—81no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4#120×C6111yescharacteristic
589C2×C4#120×C6111yescharacteristic
590C2×C4C22×D4×C6112yes—
591C8—42no—
592C8—42no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—84no—
609C22C2×#120×C6111yescharacteristic
610C22C2×#135×C6111yescharacteristic
611C22C61×#57511yescharacteristic
612C4—22no—
613C4—21no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4C61×#45812yes—
625C2—21no—
626C2—21no—
627C2—42no—
628C2—42no—
629C2—82no—
630C2C61×#300211yescharacteristic
631C2C61×#301811yescharacteristic
632C2C2×C61×#45811yescharacteristic
subgroupG/H
centerC2×C4×C61#120
commutator, Frattini, cosocleC22×C4C24×C61
socleC22×C61C2×#120
prime radicalC61×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3091C2semidirect
3. C61×#2395C2semidirect
4. C61×#1615C2semidirect
5. C61×#1614C2semidirect
6. C61×#1613C2semidirect
7. C61×#1609C2semidirect
8. C61×#1608C2semidirect
9. C61×#1607C2semidirect
10. C61×#1501C2semidirect
11. C61×#1459C2semidirect
25. C2×#117×C61C22semidirect
27. C61×#484C22semidirect
28. C61×#483C22semidirect
34. C2×C22⋊C8×C61C22semidirect
35. C61×#400C22semidirect
37. C61×#398C22semidirect
43. C4×C22⋊C4×C61C22semidirect
44. C61×#328C22semidirect
45. C61×#328C22semidirect
48. C61×#324C22semidirect
276. #18107C61direct (N × H)
310. #3091C2×C61semidirect
311. #2395C2×C61semidirect
312. #1615C2×C61semidirect
313. #1614C2×C61semidirect
314. #1613C2×C61semidirect
315. #1609C2×C61semidirect
316. #1608C2×C61semidirect
317. #1607C2×C61semidirect
318. #1501C2×C61semidirect
319. #1459C2×C61semidirect
341. C2×#117C22×C61semidirect
343. #484C22×C61semidirect
344. #483C22×C61semidirect
350. C2×C22⋊C8C22×C61semidirect
351. #400C22×C61semidirect
353. #398C22×C61semidirect
359. C4×C22⋊C4C22×C61semidirect
360. #328C22×C61semidirect
361. #328C22×C61semidirect
364. #324C22×C61semidirect
367. C61#18107direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#34592
C61×#30912
C61×#23952
C61×#16152
C61×#16142
C61×#16132
C61×#16092
C61×#16082
C61×#16072
C61×#15012
C61×#14592
#1810761