Finite Group Databasebeta

← C61×#24551 #29527984

Subgroups — C61×#24551 #29527984

Order 15616.

1188 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#483C211yesmaximal, characteristic, complemented
2C22×C22⋊C8×C61C211yesmaximal, characteristic
3C61×#2848C211yesmaximal, characteristic
4C2×C61×#380C211yesmaximal, characteristic
5C61×#2077C212yesmaximal, complemented
6C61×#2035C212yesmaximal, complemented
7C61×#1892C212yesmaximal, complemented
8C61×#1872C211yesmaximal, characteristic, complemented
9C61×#1853C211yesmaximal, characteristic, complemented
10C61×#1823C211yesmaximal, characteristic, complemented
11C61×#1807C212yesmaximal, complemented
12C24×C4×C61C2211yescharacteristic
13C23×C8×C61C2212yes—
14C2×C22⋊Q8×C61C2211yescharacteristic
15C2×C22⋊Q8×C61C2211yescharacteristic
16C2×C22⋊Q8×C61C2211yescharacteristic
17C22×C4⋊C4×C61C2211yescharacteristic
18C22×C4⋊C4×C61C2211yescharacteristic
19C22×C22⋊C4×C61—21no—
20C61×#483—24no—
21C2×#106×C61C2212yescomplemented
22C2×#106×C61C2212yescomplemented
23C2×#103×C61C2212yescomplemented
24C2×#103×C61C2212yescomplemented
25C2×C22⋊C8×C61—24no—
26C2×C22⋊C8×C61C2212yes—
27C2×C22⋊C8×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#385—21no—
30C61×#385—21no—
31C61×#385—21no—
32C61×#380—21no—
33C61×#380—21no—
34C61×#380C2212yes—
35C61×#380C2212yes—
36C61×#380C2211yescharacteristic
37C61×#380C2211yescharacteristic
38C61×#380C2211yescharacteristic
39C61×#379—21no—
40C2×#95×C61—21no—
41C61×#340C2212yescomplemented
42C61×#340C2212yescomplemented
43C61×#340C2212yescomplemented
44C61×#340C2212yescomplemented
45C25×C61D411yescharacteristic
46C22×Q8×C61C2311yescharacteristic
47C23×C4×C61—22no—
48C23×C4×C61—22no—
49C23×C4×C61—24no—
50C23×C4×C61—21no—
51C23×C4×C61—21no—
52C23×C4×C61—42no—
53C23×C4×C61D411yescharacteristic
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—24no—
60C22×C8×C61—24no—
61C22×C8×C61—42no—
62C22×C8×C61C2312yes—
63C22×C8×C61C2312yes—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—24no—
67C22⋊Q8×C61—42no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—42no—
76C2×C4⋊C4×C61C2311yescharacteristic
77C2×C4⋊C4×C61C2312yes—
78C2×C4⋊C4×C61C2312yes—
79C2×C4⋊C4×C61C2311yescharacteristic
80C2×C4⋊C4×C61C2311yescharacteristic
81C2×C22⋊C4×C61—21no—
82C2×C22⋊C4×C61—21no—
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—21no—
85C2×C22⋊C4×C61—21no—
86C2×C22⋊C4×C61—21no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—44no—
90#106×C61—24no—
91#106×C61—42no—
92#103×C61—24no—
93#103×C61—42no—
94C22⋊C8×C61—22no—
95C22⋊C8×C61—24no—
96C22⋊C8×C61D414yes—
97#95×C61—22no—
98#95×C61—22no—
99#95×C61—21no—
100#95×C61—21no—
101#95×C61—21no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61—41no—
107C24×C61—41no—
108C24×C61SD1614yes—
109C24×C61C2×D411yescharacteristic
110C24×C61C2×D411yescharacteristic
111C24×C61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61—41no—
140C22×C4×C61—48no—
141C22×C4×C61—48no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4112yes—
153C22×C4×C61#4114yes—
154C22×C4×C61C2411yescharacteristic
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—48no—
161C2×C8×C61C2×D414yes—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61—42no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—44no—
180C22⋊C4×C61—44no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—28no—
198C23×C61—28no—
199C23×C61—44no—
200C23×C61—44no—
201C23×C61—44no—
202C23×C61—44no—
203C23×C61—44no—
204C23×C61#12112yes—
205C23×C61#12312yes—
206C23×C61C2×SD1612yes—
207C23×C61C2×SD1612yes—
208C23×C61C2×SD1612yes—
209C23×C61#13512yes—
210C23×C61#13712yes—
211C23×C61C22×D411yescharacteristic
212Q8×C61—42no—
213Q8×C61—42no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61—44no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—84no—
249C2×C4×C61—84no—
250C2×C4×C61#12112yes—
251C2×C4×C61#12312yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4112yes—
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)12yes—
256C8×C61—24no—
257C8×C61—42no—
258#24551C6111yesmaximal, characteristic, direct product factor, Sylow
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—416no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61#46514yes—
280C22×C61#48314yescomplemented
281C22×C61C2×#12111yescharacteristic
282C22×C61C2×#12311yescharacteristic
283C22×C61#53411yescharacteristic
284C22×C61C22×SD1611yescharacteristic
285C22×C61C2×#13511yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57511yescharacteristic
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61—82no—
294C4×C61—81no—
295C4×C61—82no—
296C4×C61—81no—
297C4×C61#54612yes—
298C4×C61#55112yes—
299C2×#483C2×C6111yescharacteristic, complemented
300C22×C22⋊C8C2×C6111yescharacteristic
301#2848C2×C6111yescharacteristic
302C2×#380C2×C6111yescharacteristic
303#2077C2×C6112yescomplemented
304#2035C2×C6112yescomplemented
305#1892C2×C6112yescomplemented
306#1872C2×C6111yescharacteristic, complemented
307#1853C2×C6111yescharacteristic, complemented
308#1823C2×C6111yescharacteristic, complemented
309#1807C2×C6112yescomplemented
310C2×C61—24no—
311C2×C61—24no—
312C2×C61—42no—
313C2×C61#241211yescharacteristic
314C2×C61C2×#46511yescharacteristic
315C2×C61#305611yescharacteristic
316C2×C61C2×#48311yescharacteristic
317C2×C61#309111yescharacteristic
318C2×C61#318711yescharacteristic
319C2×C61#319011yescharacteristic
320C24×C4C22×C6111yescharacteristic
321C23×C8C22×C6112yes—
322C2×C22⋊Q8C22×C6111yescharacteristic
323C2×C22⋊Q8C22×C6111yescharacteristic
324C2×C22⋊Q8C22×C6111yescharacteristic
325C22×C4⋊C4C22×C6111yescharacteristic
326C22×C4⋊C4C22×C6111yescharacteristic
327C22×C22⋊C4—21no—
328#483—24no—
329C2×#106C22×C6112yescomplemented
330C2×#106C22×C6112yescomplemented
331C2×#103C22×C6112yescomplemented
332C2×#103C22×C6112yescomplemented
333C2×C22⋊C8—24no—
334C2×C22⋊C8C22×C6112yes—
335C2×C22⋊C8C22×C6111yescharacteristic
336C2×C22⋊C8C22×C6111yescharacteristic
337#385—21no—
338#385—21no—
339#385—21no—
340#380—21no—
341#380—21no—
342#380C22×C6111yescharacteristic
343#380C22×C6112yes—
344#380C22×C6111yescharacteristic
345#380C22×C6112yes—
346#380C22×C6111yescharacteristic
347#379—21no—
348C2×#95—21no—
349#340C22×C6112yescomplemented
350#340C22×C6112yescomplemented
351#340C22×C6112yescomplemented
352#340C22×C6112yescomplemented
353C61#2455111yescharacteristic, direct product factor, Sylow
354C25D4×C6111yescharacteristic
355C22×Q8C23×C6111yescharacteristic
356C23×C4—22no—
357C23×C4—22no—
358C23×C4—24no—
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—42no—
362C23×C4D4×C6111yescharacteristic
363C23×C4D4×C6111yescharacteristic
364C23×C4D4×C6111yescharacteristic
365C23×C4C23×C6111yescharacteristic
366C23×C4C23×C6111yescharacteristic
367C23×C4C23×C6111yescharacteristic
368C22×C8—24no—
369C22×C8—24no—
370C22×C8—42no—
371C22×C8C23×C6112yes—
372C22×C8C23×C6112yes—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—24no—
376C22⋊Q8—42no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—42no—
385C2×C4⋊C4C23×C6112yes—
386C2×C4⋊C4C23×C6112yes—
387C2×C4⋊C4C23×C6111yescharacteristic
388C2×C4⋊C4C23×C6111yescharacteristic
389C2×C4⋊C4C23×C6111yescharacteristic
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—44no—
399#106—24no—
400#106—42no—
401#103—24no—
402#103—42no—
403C22⋊C8—22no—
404C22⋊C8—24no—
405C22⋊C8D4×C6114yes—
406#95—22no—
407#95—22no—
408#95—21no—
409#95—21no—
410#95—21no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—41no—
416C24—41no—
417C24SD16×C6114yes—
418C24C2×D4×C6111yescharacteristic
419C24C2×D4×C6111yescharacteristic
420C24C2×D4×C6111yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×D4×C6111yescharacteristic
458C22×C4C2×D4×C6111yescharacteristic
459C22×C4C2×D4×C6112yes—
460C22×C4C2×D4×C6111yescharacteristic
461C22×C4#41×C6112yes—
462C22×C4#41×C6114yes—
463C22×C4C24×C6111yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—48no—
470C2×C8C2×D4×C6114yes—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—28no—
505C23—28no—
506C23—21no—
507C23—21no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23#121×C6112yes—
514C23#123×C6112yes—
515C23C2×SD16×C6112yes—
516C23C2×SD16×C6112yes—
517C23C2×SD16×C6112yes—
518C23#135×C6112yes—
519C23#137×C6112yes—
520C23C22×D4×C6111yescharacteristic
521Q8—42no—
522Q8—42no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4#121×C6112yes—
560C2×C4#123×C6112yes—
561C2×C4C22×D4×C6111yescharacteristic
562C2×C4C2×#41×C6112yes—
563C2×C4C2×#41×C6111yescharacteristic
564C2×C4H(5,𝔽2)×C6112yes—
565C8—24no—
566C8—42no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—416no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C61×#46514yes—
588C22C61×#48314yescomplemented
589C22C2×#121×C6111yescharacteristic
590C22C2×#123×C6111yescharacteristic
591C22C61×#53411yescharacteristic
592C22C22×SD16×C6111yescharacteristic
593C22C2×#135×C6111yescharacteristic
594C22C2×#137×C6111yescharacteristic
595C22C61×#57511yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—82no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—82no—
604C4—81no—
605C4C61×#54612yes—
606C4C61×#55112yes—
607C2—24no—
608C2—24no—
609C2—42no—
610C2C61×#241211yescharacteristic
611C2C2×C61×#46511yescharacteristic
612C2C61×#305611yescharacteristic
613C2C2×C61×#48311yescharacteristic
614C2C61×#309111yescharacteristic
615C2C61×#318711yescharacteristic
616C2C61×#319011yescharacteristic
subgroupG/H
center, socleC23×C61C22×D4
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC25×C61D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C61 ◁ C24×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C61×#483C2semidirect
5. C61×#2077C2semidirect
6. C61×#2035C2semidirect
7. C61×#1892C2semidirect
8. C61×#1872C2semidirect
9. C61×#1853C2semidirect
10. C61×#1823C2semidirect
11. C61×#1807C2semidirect
21. C2×#106×C61C22semidirect
22. C2×#106×C61C22semidirect
23. C2×#103×C61C22semidirect
24. C2×#103×C61C22semidirect
41. C61×#340C22semidirect
42. C61×#340C22semidirect
43. C61×#340C22semidirect
44. C61×#340C22semidirect
258. #24551C61direct (N × H)
280. C22×C61#483semidirect
299. C2×#483C2×C61semidirect
303. #2077C2×C61semidirect
304. #2035C2×C61semidirect
305. #1892C2×C61semidirect
306. #1872C2×C61semidirect
307. #1853C2×C61semidirect
308. #1823C2×C61semidirect
309. #1807C2×C61semidirect
329. C2×#106C22×C61semidirect
330. C2×#106C22×C61semidirect
331. C2×#103C22×C61semidirect
332. C2×#103C22×C61semidirect
349. #340C22×C61semidirect
350. #340C22×C61semidirect
351. #340C22×C61semidirect
352. #340C22×C61semidirect
353. C61#24551direct (N × H)
588. C22C61×#483semidirect

Maximal subgroups

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

subgroupindex
C2×C61×#4832
C22×C22⋊C8×C612
C61×#28482
C2×C61×#3802
C61×#20772
C61×#20352
C61×#18922
C61×#18722
C61×#18532
C61×#18232
C61×#18072
#2455161