Finite Group Databasebeta

← C61×#24172 #29527605

Subgroups — C61×#24172 #29527605

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3511C211yesmaximal, characteristic
2C61×#3037C211yesmaximal, characteristic, complemented
3C4×#137×C61C211yesmaximal, characteristic, complemented
4C61×#2568C211yesmaximal, characteristic, complemented
5C61×#2040C212yesmaximal, complemented
6C61×#2007C212yesmaximal, complemented
7C61×#1997C211yesmaximal, characteristic, complemented
8C61×#1980C211yesmaximal, characteristic, complemented
9C61×#1882C212yesmaximal, complemented
10C61×#1860C212yesmaximal, complemented
11C61×#1806C211yesmaximal, characteristic, complemented
12C2×#137×C61C2211yescharacteristic
13C2×#137×C61C2212yes—
14C61×#556—21no—
15C61×#555—21no—
16C61×#554—21no—
17C61×#553—22no—
18C61×#548—22no—
19C61×#547—21no—
20C61×#542—22no—
21C61×#537C2212yes—
22C61×#533—21no—
23C61×#532—21no—
24C61×#532C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#520C2212yes—
27C4×#41×C61C2211yescharacteristic
28C2×C4×Q8×C61C2211yescharacteristic
29C61×#464—22no—
30C61×#462—21no—
31C61×#460—21no—
32C61×#441—21no—
33C61×#440—21no—
34C4×Q16×C61—21no—
35C4×SD16×C61—21no—
36C61×#428C2211yescharacteristic
37C61×#423C2211yescharacteristic
38C2×#104×C61C2212yes—
39C2×#104×C61C2212yes—
40C2×#104×C61C2212yes—
41C61×#419C2211yescharacteristic
42C61×#417C2212yes—
43C61×#417C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#400—22no—
46C61×#391—22no—
47C61×#391C2212yes—
48C61×#382C2212yes—
49C4×C4⋊C4×C61C2211yescharacteristic
50C61×#343C2212yes—
51C61×#343C2212yes—
52C61×#339C2212yes—
53C2×#41×C61—22no—
54C2×#41×C61C2312yes—
55C2×#41×C61C2311yescharacteristic
56C22×Q8×C61D412yes—
57C22×Q8×C61C2311yescharacteristic
58#137×C61—22no—
59#137×C61—24no—
60#137×C61—22no—
61#137×C61—42no—
62C2×Q16×C61—21no—
63C2×Q16×C61—22no—
64C2×SD16×C61—21no—
65C2×SD16×C61—22no—
66C2×M16×C61C2312yes—
67C2×M16×C61C2311yescharacteristic
68C2×M16×C61C2311yescharacteristic
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61D411yescharacteristic
74#127×C61D411yescharacteristic
75#126×C61—24no—
76#125×C61—22no—
77#124×C61—22no—
78#124×C61—24no—
79#124×C61D412yes—
80#123×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61—22no—
88#121×C61—21no—
89#121×C61—21no—
90#121×C61—24no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61D411yescharacteristic
98C4×Q8×C61D411yescharacteristic
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—21no—
102C4×D4×C61—24no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61D411yescharacteristic
107C4×D4×C61D411yescharacteristic
108#117×C61—22no—
109#117×C61—22no—
110#117×C61—21no—
111#117×C61C2311yescharacteristic
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61D412yes—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2312yes—
118C2×C42×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61C2312yes—
121C2×C42×C61C2311yescharacteristic
122#107×C61—21no—
123#106×C61—21no—
124C4⋊C8×C61—21no—
125C4⋊C8×C61—21no—
126#104×C61—24no—
127#104×C61—42no—
128#104×C61—42no—
129#103×C61—21no—
130#103×C61—22no—
131#103×C61—21no—
132#103×C61—21no—
133#102×C61—21no—
134#102×C61—22no—
135#97×C61—21no—
136C4×C8×C61—21no—
137#95×C61—21no—
138#95×C61—24no—
139#95×C61—24no—
140#41×C61—22no—
141#41×C61—22no—
142#41×C61—21no—
143#41×C61—21no—
144#41×C61—21no—
145#41×C61—22no—
146#41×C61—22no—
147#41×C61—21no—
148#41×C61—84no—
149C2×Q8×C61—22no—
150C2×Q8×C61—22no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—44no—
157C2×Q8×C61C2×D412yes—
158C2×Q8×C61C2×D411yescharacteristic
159C2×Q8×C61#4112yescomplemented
160C2×D4×C61—21no—
161C2×D4×C61—22no—
162C2×D4×C61—24no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61C2×D412yes—
166C2×D4×C61C2×D411yescharacteristic
167C22×C4×C61—21no—
168C22×C4×C61—24no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—21no—
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D414yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2411yescharacteristic
179Q16×C61—24no—
180Q16×C61—44no—
181SD16×C61—24no—
182SD16×C61—44no—
183M16×C61—21no—
184M16×C61—22no—
185M16×C61—21no—
186M16×C61—22no—
187M16×C61—21no—
188M16×C61#4112yescomplemented
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—22no—
192C4⋊C4×C61—24no—
193C4⋊C4×C61—24no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—42no—
205C4⋊C4×C61—41no—
206C4⋊C4×C61—41no—
207C4⋊C4×C61C2×D411yescharacteristic
208C4⋊C4×C61C2×D411yescharacteristic
209C22⋊C4×C61—24no—
210C22⋊C4×C61—24no—
211C22⋊C4×C61—24no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C42×C61—24no—
218C42×C61—21no—
219C42×C61—21no—
220C42×C61—21no—
221C42×C61—22no—
222C42×C61—22no—
223C42×C61—41no—
224C42×C61—41no—
225C42×C61C2×D411yescharacteristic
226C42×C61C2×D411yescharacteristic
227C42×C61C2×D411yescharacteristic
228C42×C61C2×D411yescharacteristic
229C23×C61—21no—
230C23×C61—22no—
231C23×C61C22×D411yescharacteristic
232Q8×C61—22no—
233Q8×C61—21no—
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—44no—
241Q8×C61—82no—
242D4×C61—21no—
243D4×C61—21no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—42no—
247D4×C61—42no—
248D4×C61—41no—
249D4×C61—42no—
250D4×C61—41no—
251D4×C61—42no—
252D4×C61—84no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—24no—
259C2×C4×C61—22no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—42no—
269C2×C4×C61—44no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—41no—
274C2×C4×C61—42no—
275C2×C4×C61—84no—
276C2×C4×C61#12011yescharacteristic
277C2×C4×C61#12011yescharacteristic
278C2×C4×C61#12011yescharacteristic
279C2×C4×C61#12011yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C2×#4111yescharacteristic
283C2×C4×C61C2×#4111yescharacteristic
284C2×C4×C61H(5,𝔽2)11yescharacteristic
285C2×C4×C612−1+411yescharacteristic
286C8×C61—22no—
287C8×C61—41no—
288C8×C61—42no—
289#24172C6111yesmaximal, characteristic, direct product factor, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—42no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—42no—
296C22×C61—41no—
297C22×C61—42no—
298C22×C61C2×#12011yescharacteristic
299C22×C61#54811yescharacteristic
300C22×C61#55011yescharacteristic
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—42no—
307C4×C61—41no—
308C4×C61—44no—
309C4×C61—41no—
310C4×C61—82no—
311C4×C61—81no—
312C4×C61#45512yes—
313C4×C61#52511yescharacteristic
314C4×C61#53711yescharacteristic
315C4×C61#54611yescharacteristic
316C4×C61#54711yescharacteristic
317#3511C2×C6111yescharacteristic
318#3037C2×C6111yescharacteristic, complemented
319C4×#137C2×C6111yescharacteristic, complemented
320#2568C2×C6111yescharacteristic, complemented
321#2040C2×C6112yescomplemented
322#2007C2×C6112yescomplemented
323#1997C2×C6111yescharacteristic, complemented
324#1980C2×C6111yescharacteristic, complemented
325#1882C2×C6112yescomplemented
326#1860C2×C6112yescomplemented
327#1806C2×C6111yescharacteristic, complemented
328C2×C61—21no—
329C2×C61—21no—
330C2×C61—81no—
331C2×C61—82no—
332C2×C61#240711yescharacteristic
333C2×C61C2×#45511yescharacteristic
334C2×C61#302011yescharacteristic
335C2×#137C22×C6111yescharacteristic
336C2×#137C22×C6112yes—
337#556—21no—
338#555—21no—
339#554—21no—
340#553—22no—
341#548—22no—
342#547—21no—
343#542—22no—
344#537C22×C6112yes—
345#533—21no—
346#532—21no—
347#532C22×C6111yescharacteristic
348C2×C4⋊Q8C22×C6111yescharacteristic
349#520C22×C6112yes—
350C4×#41C22×C6111yescharacteristic
351C2×C4×Q8C22×C6111yescharacteristic
352#464—22no—
353#462—21no—
354#460—21no—
355#441—21no—
356#440—21no—
357C4×Q16—21no—
358C4×SD16—21no—
359#428C22×C6111yescharacteristic
360#423C22×C6111yescharacteristic
361C2×#104C22×C6112yes—
362C2×#104C22×C6112yes—
363C2×#104C22×C6112yes—
364#419C22×C6111yescharacteristic
365#417C22×C6112yes—
366#417C22×C6111yescharacteristic
367C4×M16C22×C6111yescharacteristic
368#400—22no—
369#391—22no—
370#391C22×C6112yes—
371#382C22×C6112yes—
372C4×C4⋊C4C22×C6111yescharacteristic
373#343C22×C6112yes—
374#343C22×C6112yes—
375#339C22×C6112yes—
376C61#2417211yescharacteristic, direct product factor, Sylow
377C2×#41—22no—
378C2×#41C23×C6111yescharacteristic
379C2×#41C23×C6112yes—
380C22×Q8D4×C6112yes—
381C22×Q8C23×C6111yescharacteristic
382#137—22no—
383#137—24no—
384#137—22no—
385#137—42no—
386C2×Q16—21no—
387C2×Q16—22no—
388C2×SD16—21no—
389C2×SD16—22no—
390C2×M16C23×C6112yes—
391C2×M16C23×C6111yescharacteristic
392C2×M16C23×C6111yescharacteristic
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8D4×C6111yescharacteristic
398#127D4×C6111yescharacteristic
399#126—24no—
400#125—22no—
401#124—22no—
402#124—24no—
403#124D4×C6112yes—
404#123—24no—
405C22⋊Q8—22no—
406C22⋊Q8—24no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409#121—22no—
410#121—22no—
411#121—22no—
412#121—21no—
413#121—21no—
414#121—24no—
415C4×Q8—22no—
416C4×Q8—22no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8D4×C6111yescharacteristic
422C4×Q8D4×C6111yescharacteristic
423C4×D4—22no—
424C4×D4—22no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—24no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4D4×C6111yescharacteristic
431C4×D4D4×C6111yescharacteristic
432#117—22no—
433#117—22no—
434#117—21no—
435#117C23×C6111yescharacteristic
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4D4×C6112yes—
440C2×C4⋊C4C23×C6111yescharacteristic
441C2×C4⋊C4C23×C6112yes—
442C2×C42—21no—
443C2×C42—21no—
444C2×C42C23×C6112yes—
445C2×C42C23×C6111yescharacteristic
446#107—21no—
447#106—21no—
448C4⋊C8—21no—
449C4⋊C8—21no—
450#104—24no—
451#104—42no—
452#104—42no—
453#103—21no—
454#103—22no—
455#103—21no—
456#103—21no—
457#102—21no—
458#102—22no—
459#97—21no—
460C4×C8—21no—
461#95—21no—
462#95—24no—
463#95—24no—
464#41—22no—
465#41—21no—
466#41—21no—
467#41—22no—
468#41—21no—
469#41—22no—
470#41—22no—
471#41—21no—
472#41—84no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—44no—
481C2×Q8C2×D4×C6112yes—
482C2×Q8C2×D4×C6111yescharacteristic
483C2×Q8#41×C6112yescomplemented
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—24no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4C2×D4×C6112yes—
490C2×D4C2×D4×C6111yescharacteristic
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4C2×D4×C6111yescharacteristic
500C22×C4C2×D4×C6114yes—
501C22×C4C2×D4×C6111yescharacteristic
502C22×C4C24×C6111yescharacteristic
503Q16—24no—
504Q16—44no—
505SD16—24no—
506SD16—44no—
507M16—21no—
508M16—22no—
509M16—21no—
510M16—22no—
511M16—21no—
512M16#41×C6112yescomplemented
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×D4×C6111yescharacteristic
532C4⋊C4C2×D4×C6111yescharacteristic
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C42—24no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42C2×D4×C6111yescharacteristic
550C42C2×D4×C6111yescharacteristic
551C42C2×D4×C6111yescharacteristic
552C42C2×D4×C6111yescharacteristic
553C23—21no—
554C23—22no—
555C23C22×D4×C6111yescharacteristic
556Q8—21no—
557Q8—22no—
558Q8—22no—
559Q8—22no—
560Q8—22no—
561Q8—21no—
562Q8—41no—
563Q8—41no—
564Q8—44no—
565Q8—82no—
566D4—21no—
567D4—22no—
568D4—22no—
569D4—21no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—41no—
575D4—42no—
576D4—84no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4#120×C6111yescharacteristic
601C2×C4#120×C6111yescharacteristic
602C2×C4#120×C6111yescharacteristic
603C2×C4#120×C6111yescharacteristic
604C2×C4C22×D4×C6111yescharacteristic
605C2×C4C22×D4×C6111yescharacteristic
606C2×C4C2×#41×C6111yescharacteristic
607C2×C4C2×#41×C6111yescharacteristic
608C2×C4H(5,𝔽2)×C6111yescharacteristic
609C2×C42−1+4×C6111yescharacteristic
610C8—22no—
611C8—41no—
612C8—42no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22C2×#120×C6111yescharacteristic
622C22C61×#54811yescharacteristic
623C22C61×#55011yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—44no—
632C4—41no—
633C4—82no—
634C4—81no—
635C4C61×#45512yes—
636C4C61×#52511yescharacteristic
637C4C61×#53711yescharacteristic
638C4C61×#54611yescharacteristic
639C4C61×#54711yescharacteristic
640C2—21no—
641C2—21no—
642C2—81no—
643C2—82no—
644C2C61×#240711yescharacteristic
645C2C2×C61×#45511yescharacteristic
646C2C61×#302011yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#3511C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3037C2semidirect
3. C4×#137×C61C2semidirect
4. C61×#2568C2semidirect
5. C61×#2040C2semidirect
6. C61×#2007C2semidirect
7. C61×#1997C2semidirect
8. C61×#1980C2semidirect
9. C61×#1882C2semidirect
10. C61×#1860C2semidirect
11. C61×#1806C2semidirect
159. C2×Q8×C61#41semidirect
188. M16×C61#41semidirect
289. #24172C61direct (N × H)
318. #3037C2×C61semidirect
319. C4×#137C2×C61semidirect
320. #2568C2×C61semidirect
321. #2040C2×C61semidirect
322. #2007C2×C61semidirect
323. #1997C2×C61semidirect
324. #1980C2×C61semidirect
325. #1882C2×C61semidirect
326. #1860C2×C61semidirect
327. #1806C2×C61semidirect
376. C61#24172direct (N × H)
483. C2×Q8#41×C61semidirect
512. M16#41×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#35112
C61×#30372
C4×#137×C612
C61×#25682
C61×#20402
C61×#20072
C61×#19972
C61×#19802
C61×#18822
C61×#18602
C61×#18062
#2417261