Finite Group Databasebeta

← C61×#18257 #29521690

Subgroups — C61×#18257 #29521690

Order 15616.

1192 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3536C211yesmaximal, characteristic
2C61×#3004C211yesmaximal, characteristic, complemented
3C2×C61×#451C211yesmaximal, characteristic, complemented
4C61×#2436C211yesmaximal, characteristic, complemented
5C61×#1606C214yesmaximal, complemented
6C61×#1602C214yesmaximal, complemented
7C61×#1508C211yesmaximal, characteristic, complemented
8C61×#1501C211yesmaximal, characteristic, complemented
9C61×#1471C211yesmaximal, characteristic, complemented
10C23×Q8×C61C2211yescharacteristic
11C2×#137×C61—21no—
12C22×Q16×C61—21no—
13C61×#562—22no—
14C61×#554—21no—
15C61×#548—21no—
16C61×#547—21no—
17C61×#546—21no—
18C61×#546—22no—
19C61×#543C2212yes—
20C61×#542—21no—
21C61×#539—21no—
22C61×#538C2212yes—
23C61×#537—21no—
24C61×#537—21no—
25C2×#124×C61—21no—
26C61×#525C2211yescharacteristic
27C2×C22⋊Q8×C61C2211yescharacteristic
28C2×#121×C61C2211yescharacteristic
29C61×#451—22no—
30C61×#451—21no—
31C61×#451C2212yes—
32C61×#451C2214yes—
33C61×#448—21no—
34C61×#448C2212yes—
35C61×#419—21no—
36C2×#103×C61—21no—
37C61×#407C2211yescharacteristic
38C2×C22⋊C8×C61C2211yescharacteristic
39C61×#393C2214yes—
40C61×#391—21no—
41C61×#386—22no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C61×#328C2212yes—
44C61×#328C2214yes—
45C61×#327C2212yes—
46C61×#323C2214yes—
47C2×#41×C61—22no—
48C2×#41×C61—21no—
49C2×#41×C61D411yescharacteristic
50C22×Q8×C61—21no—
51C22×Q8×C61—21no—
52C22×Q8×C61—21no—
53C22×Q8×C61—21no—
54C22×Q8×C61—21no—
55C22×Q8×C61—44no—
56C22×Q8×C61D411yescharacteristic
57C22×Q8×C61C2312yes—
58C22×D4×C61D412yes—
59C22×D4×C61D411yescharacteristic
60C23×C4×C61—21no—
61C23×C4×C61C2311yescharacteristic
62#137×C61—44no—
63C2×Q16×C61—24no—
64C2×Q16×C61—22no—
65C2×Q16×C61—44no—
66C2×SD16×C61—22no—
67C2×M16×C61—21no—
68C22×C8×C61—21no—
69C4⋊Q8×C61—21no—
70#126×C61—22no—
71#125×C61—22no—
72#124×C61—21no—
73#124×C61—22no—
74#124×C61—22no—
75#124×C61—44no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—21no—
80#123×C61—22no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61D411yescharacteristic
91C22⋊Q8×C61D412yes—
92C22⋊Q8×C61C2312yes—
93C22⋊Q8×C61C2311yescharacteristic
94#121×C61—21no—
95#121×C61—22no—
96#121×C61—22no—
97#121×C61—22no—
98#121×C61—22no—
99#121×C61—22no—
100#121×C61—21no—
101#121×C61—21no—
102#121×C61—21no—
103#121×C61—21no—
104#121×C61D411yescharacteristic
105#121×C61C2311yescharacteristic
106#120×C61—22no—
107#120×C61—21no—
108C4×Q8×C61—21no—
109C4×D4×C61—22no—
110C4×D4×C61—21no—
111C4×D4×C61—22no—
112C4×D4×C61—21no—
113C4×D4×C61—21no—
114#117×C61D411yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C22⋊C4×C61—22no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—24no—
123C2×C22⋊C4×C61C2312yes—
124C2×C42×C61—21no—
125#103×C61—24no—
126#103×C61—22no—
127#103×C61—22no—
128C22⋊C8×C61—21no—
129C22⋊C8×C61—21no—
130C22⋊C8×C61C2312yes—
131C22⋊C8×C61C2312yes—
132#95×C61—24no—
133#95×C61C2312yes—
134C24×C61C2×D411yescharacteristic
135C24×C61C2×D412yes—
136#41×C61—42no—
137#41×C61—84no—
138#41×C61—82no—
139C2×Q8×C61—24no—
140C2×Q8×C61—22no—
141C2×Q8×C61—21no—
142C2×Q8×C61—24no—
143C2×Q8×C61—24no—
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—44no—
147C2×Q8×C61—42no—
148C2×Q8×C61—42no—
149C2×Q8×C61—42no—
150C2×Q8×C61—42no—
151C2×Q8×C61—44no—
152C2×Q8×C61—44no—
153C2×Q8×C61—41no—
154C2×Q8×C61—41no—
155C2×Q8×C61C2×D411yescharacteristic
156C2×Q8×C61C2×D412yes—
157C2×D4×C61—21no—
158C2×D4×C61—22no—
159C2×D4×C61—21no—
160C2×D4×C61—22no—
161C2×D4×C61—22no—
162C2×D4×C61—21no—
163C2×D4×C61—22no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—42no—
167C2×D4×C61—44no—
168C2×D4×C61C2×D411yescharacteristic
169C22×C4×C61—21no—
170C22×C4×C61—22no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—22no—
177C22×C4×C61—42no—
178C22×C4×C61—44no—
179C22×C4×C61—41no—
180C22×C4×C61—41no—
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D412yes—
185C22×C4×C61C2×D411yescharacteristic
186C22×C4×C61C2×D411yescharacteristic
187C22×C4×C61C2411yescharacteristic
188Q16×C61—48no—
189Q16×C61—44no—
190SD16×C61—44no—
191M16×C61—42no—
192C2×C8×C61—22no—
193C2×C8×C61—22no—
194C2×C8×C61—42no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61C2×D412yes—
207C4⋊C4×C61C2×D411yescharacteristic
208C4⋊C4×C61C2×D411yescharacteristic
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—21no—
216C22⋊C4×C61—21no—
217C22⋊C4×C61—22no—
218C22⋊C4×C61—21no—
219C22⋊C4×C61—21no—
220C22⋊C4×C61—22no—
221C22⋊C4×C61—22no—
222C22⋊C4×C61—21no—
223C22⋊C4×C61—48no—
224C22⋊C4×C61—42no—
225C42×C61—22no—
226C42×C61—21no—
227C42×C61—42no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—21no—
232C23×C61—24no—
233C23×C61—41no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61#12011yescharacteristic
237C23×C61#12011yescharacteristic
238C23×C61C22×D411yescharacteristic
239Q8×C61—24no—
240Q8×C61—24no—
241Q8×C61—42no—
242Q8×C61—42no—
243Q8×C61—44no—
244Q8×C61—42no—
245Q8×C61—41no—
246Q8×C61—44no—
247Q8×C61—42no—
248Q8×C61—82no—
249D4×C61—41no—
250D4×C61—42no—
251D4×C61—44no—
252D4×C61—42no—
253D4×C61—42no—
254D4×C61—81no—
255D4×C61—82no—
256D4×C61—81no—
257D4×C61—82no—
258D4×C61—82no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—24no—
263C2×C4×C61—21no—
264C2×C4×C61—22no—
265C2×C4×C61—21no—
266C2×C4×C61—22no—
267C2×C4×C61—21no—
268C2×C4×C61—22no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—42no—
273C2×C4×C61—44no—
274C2×C4×C61—44no—
275C2×C4×C61—42no—
276C2×C4×C61—42no—
277C2×C4×C61—42no—
278C2×C4×C61—42no—
279C2×C4×C61—42no—
280C2×C4×C61—42no—
281C2×C4×C61—42no—
282C2×C4×C61—81no—
283C2×C4×C61—82no—
284C2×C4×C61—82no—
285C2×C4×C61—82no—
286C2×C4×C61#12011yescharacteristic
287C2×C4×C61#12011yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C8×C61—42no—
291C8×C61—42no—
292#18257C6111yesmaximal, characteristic, direct product factor, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—44no—
301C22×C61—41no—
302C22×C61—42no—
303C22×C61—41no—
304C22×C61—42no—
305C22×C61—41no—
306C22×C61—42no—
307C22×C61—84no—
308C22×C61#45512yes—
309C22×C61C2×#12011yescharacteristic
310C22×C61#57812yes—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61—41no—
318C4×C61—44no—
319C4×C61—41no—
320C4×C61—42no—
321C4×C61—82no—
322C4×C61B(4,𝔽2)12yes—
323#3536C2×C6111yescharacteristic
324#3004C2×C6111yescharacteristic, complemented
325C2×#451C2×C6111yescharacteristic, complemented
326#2436C2×C6111yescharacteristic, complemented
327#1606C2×C6114yescomplemented
328#1602C2×C6114yescomplemented
329#1508C2×C6111yescharacteristic, complemented
330#1501C2×C6111yescharacteristic, complemented
331#1471C2×C6111yescharacteristic, complemented
332C2×C61—22no—
333C2×C61—22no—
334C2×C61—41no—
335C2×C61—81no—
336C2×C61—82no—
337C2×C61#301111yescharacteristic
338C2×C61C2×#45511yescharacteristic
339C2×C61C2×B(4,𝔽2)11yescharacteristic
340C23×Q8C22×C6111yescharacteristic
341C2×#137—21no—
342C22×Q16—21no—
343#562—22no—
344#554—21no—
345#548—21no—
346#547—21no—
347#546—22no—
348#546—21no—
349#543C22×C6112yes—
350#542—21no—
351#539—21no—
352#538C22×C6112yes—
353#537—21no—
354#537—21no—
355C2×#124—21no—
356#525C22×C6111yescharacteristic
357C2×C22⋊Q8C22×C6111yescharacteristic
358C2×#121C22×C6111yescharacteristic
359#451—22no—
360#451—21no—
361#451C22×C6112yes—
362#451C22×C6114yes—
363#448—21no—
364#448C22×C6112yes—
365#419—21no—
366C2×#103—21no—
367#407C22×C6111yescharacteristic
368C2×C22⋊C8C22×C6111yescharacteristic
369#393C22×C6114yes—
370#391—21no—
371#386—22no—
372C4×C22⋊C4C22×C6111yescharacteristic
373#328C22×C6112yes—
374#328C22×C6114yes—
375#327C22×C6112yes—
376#323C22×C6114yes—
377C61#1825711yescharacteristic, direct product factor, Sylow
378C2×#41—22no—
379C2×#41—21no—
380C2×#41D4×C6111yescharacteristic
381C22×Q8—21no—
382C22×Q8—21no—
383C22×Q8—21no—
384C22×Q8—21no—
385C22×Q8—21no—
386C22×Q8—44no—
387C22×Q8D4×C6111yescharacteristic
388C22×Q8C23×C6112yes—
389C22×D4D4×C6111yescharacteristic
390C22×D4D4×C6112yes—
391C23×C4—21no—
392C23×C4C23×C6111yescharacteristic
393#137—44no—
394C2×Q16—24no—
395C2×Q16—22no—
396C2×Q16—44no—
397C2×SD16—22no—
398C2×M16—21no—
399C22×C8—21no—
400C4⋊Q8—21no—
401#126—22no—
402#125—22no—
403#124—21no—
404#124—22no—
405#124—22no—
406#124—44no—
407#123—22no—
408#123—22no—
409#123—22no—
410#123—21no—
411#123—22no—
412C22⋊Q8—21no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8D4×C6112yes—
422C22⋊Q8D4×C6111yescharacteristic
423C22⋊Q8C23×C6112yes—
424C22⋊Q8C23×C6111yescharacteristic
425#121—21no—
426#121—22no—
427#121—22no—
428#121—22no—
429#121—22no—
430#121—22no—
431#121—21no—
432#121—21no—
433#121—21no—
434#121—21no—
435#121D4×C6111yescharacteristic
436#121C23×C6111yescharacteristic
437#120—21no—
438#120—22no—
439C4×Q8—21no—
440C4×D4—22no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—22no—
445#117D4×C6111yescharacteristic
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4D4×C6111yescharacteristic
449C2×C4⋊C4D4×C6111yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—22no—
452C2×C22⋊C4—24no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4C23×C6112yes—
455C2×C42—21no—
456#103—24no—
457#103—22no—
458#103—22no—
459C22⋊C8—21no—
460C22⋊C8—21no—
461C22⋊C8C23×C6112yes—
462C22⋊C8C23×C6112yes—
463#95—24no—
464#95C23×C6112yes—
465C24C2×D4×C6111yescharacteristic
466C24C2×D4×C6112yes—
467#41—42no—
468#41—84no—
469#41—82no—
470C2×Q8—24no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—24no—
474C2×Q8—24no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—44no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—44no—
483C2×Q8—44no—
484C2×Q8—41no—
485C2×Q8—41no—
486C2×Q8C2×D4×C6111yescharacteristic
487C2×Q8C2×D4×C6112yes—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—44no—
499C2×D4C2×D4×C6111yescharacteristic
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—44no—
510C22×C4—41no—
511C22×C4—41no—
512C22×C4C2×D4×C6111yescharacteristic
513C22×C4C2×D4×C6111yescharacteristic
514C22×C4C2×D4×C6111yescharacteristic
515C22×C4C2×D4×C6112yes—
516C22×C4C2×D4×C6111yescharacteristic
517C22×C4C2×D4×C6111yescharacteristic
518C22×C4C24×C6111yescharacteristic
519Q16—48no—
520Q16—44no—
521SD16—44no—
522M16—42no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—42no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4C2×D4×C6112yes—
538C4⋊C4C2×D4×C6111yescharacteristic
539C4⋊C4C2×D4×C6111yescharacteristic
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—48no—
555C22⋊C4—42no—
556C42—22no—
557C42—21no—
558C42—42no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—21no—
563C23—24no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23#120×C6111yescharacteristic
568C23#120×C6111yescharacteristic
569C23C22×D4×C6111yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—42no—
573Q8—41no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—44no—
578Q8—42no—
579Q8—82no—
580D4—41no—
581D4—42no—
582D4—44no—
583D4—42no—
584D4—42no—
585D4—81no—
586D4—82no—
587D4—81no—
588D4—82no—
589D4—82no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#120×C6111yescharacteristic
618C2×C4#120×C6111yescharacteristic
619C2×C4C22×D4×C6111yescharacteristic
620C2×C4C22×D4×C6111yescharacteristic
621C8—42no—
622C8—42no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—44no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—84no—
638C22C61×#45512yes—
639C22C2×#120×C6111yescharacteristic
640C22C61×#57812yes—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—82no—
652C4C61×B(4,𝔽2)12yes—
653C2—22no—
654C2—22no—
655C2—41no—
656C2—82no—
657C2—81no—
658C2C61×#301111yescharacteristic
659C2C2×C61×#45511yescharacteristic
660C2C2×C61×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3004C2semidirect
3. C2×C61×#451C2semidirect
4. C61×#2436C2semidirect
5. C61×#1606C2semidirect
6. C61×#1602C2semidirect
7. C61×#1508C2semidirect
8. C61×#1501C2semidirect
9. C61×#1471C2semidirect
292. #18257C61direct (N × H)
324. #3004C2×C61semidirect
325. C2×#451C2×C61semidirect
326. #2436C2×C61semidirect
327. #1606C2×C61semidirect
328. #1602C2×C61semidirect
329. #1508C2×C61semidirect
330. #1501C2×C61semidirect
331. #1471C2×C61semidirect
377. C61#18257direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35362
C61×#30042
C2×C61×#4512
C61×#24362
C61×#16062
C61×#16022
C61×#15082
C61×#15012
C61×#14712
#1825761