Finite Group Databasebeta

← C61×#13877 #29517310

Subgroups — C61×#13877 #29517310

Order 15616.

4046 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C4×C61C211yesmaximal, characteristic
2C23×C22⋊C4×C61C211yesmaximal, characteristic
3C2×C61×#385C212yesmaximal, complemented
4C2×C61×#380C212yesmaximal, complemented
5C2×C61×#379C211yesmaximal, characteristic
6C61×#1439C214yesmaximal, complemented
7C61×C22⋊#95C212yesmaximal, complemented
8C61×C22⋊#95C212yesmaximal, complemented
9C26×C61C2211yescharacteristic
10C24×C4×C61C2212yes—
11C24×C4×C61C2212yes—
12C22×C4⋊C4×C61C2214yescomplemented
13C22×C22⋊C4×C61—24no—
14C22×C22⋊C4×C61—28no—
15C22×C22⋊C4×C61—28no—
16C22×C22⋊C4×C61C418yescomplemented
17C22×C22⋊C4×C61C2211yescharacteristic
18C22×C22⋊C4×C61C2212yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C22×C22⋊C4×C61C2212yes—
21C22×C22⋊C4×C61C2211yescharacteristic
22C22×C22⋊C4×C61C2212yes—
23C22×C22⋊C4×C61C2212yes—
24C22×C22⋊C4×C61C2212yes—
25C22×C22⋊C4×C61C2211yescharacteristic
26C61×#385—28no—
27C61×#380—28no—
28C61×#379—24no—
29C2×#95×C61C2214yescomplemented
30C2×#95×C61C2214yescomplemented
31C2×#95×C61C2214yescomplemented
32C25×C61—24no—
33C25×C61—24no—
34C25×C61—24no—
35C25×C61—24no—
36C25×C61—44no—
37C25×C61C2×C414yes—
38C25×C61D414yes—
39C25×C61Q814yes—
40C25×C61C2311yescharacteristic
41C25×C61C2311yescharacteristic
42C25×C61C2311yescharacteristic
43C23×C4×C61—24no—
44C23×C4×C61—24no—
45C23×C4×C61—24no—
46C23×C4×C61—28no—
47C23×C4×C61—28no—
48C23×C4×C61—24no—
49C23×C4×C61—24no—
50C23×C4×C61—44no—
51C23×C4×C61—44no—
52C23×C4×C61C2×C418yescomplemented
53C23×C4×C61C2312yes—
54C23×C4×C61C2312yes—
55C23×C4×C61C2312yes—
56C23×C4×C61C2314yes—
57C23×C4×C61C2312yes—
58C2×C4⋊C4×C61—28no—
59C2×C4⋊C4×C61—28no—
60C2×C4⋊C4×C61—44no—
61C2×C22⋊C4×C61—232no—
62C2×C22⋊C4×C61—24no—
63C2×C22⋊C4×C61—24no—
64C2×C22⋊C4×C61—24no—
65C2×C22⋊C4×C61—24no—
66C2×C22⋊C4×C61—28no—
67C2×C22⋊C4×C61—28no—
68C2×C22⋊C4×C61—232no—
69C2×C22⋊C4×C61—24no—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—24no—
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—44no—
78C2×C22⋊C4×C61C2×C414yes—
79C2×C22⋊C4×C61C2×C414yes—
80C2×C22⋊C4×C61C2×C418yescomplemented
81C2×C22⋊C4×C61D418yes—
82C2×C22⋊C4×C61D418yes—
83#95×C61—28no—
84#95×C61—28no—
85#95×C61—44no—
86C24×C61—22no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—24no—
91C24×C61—24no—
92C24×C61—28no—
93C24×C61—28no—
94C24×C61—216no—
95C24×C61—28no—
96C24×C61—216no—
97C24×C61—216no—
98C24×C61—216no—
99C24×C61—28no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61—24no—
105C24×C61—48no—
106C24×C61—48no—
107C24×C61—48no—
108C24×C61—416no—
109C24×C61—416no—
110C24×C61—416no—
111C24×C61—416no—
112C24×C61C4⋊C4116yescomplemented
113C24×C61C22×C412yes—
114C24×C61C22×C412yes—
115C24×C61C22×C412yes—
116C24×C61C2×D412yes—
117C24×C61C2×D414yes—
118C24×C61C2×D412yes—
119C24×C61C2×D412yes—
120C24×C61C2×D414yes—
121C24×C61C2×Q812yes—
122C24×C61C2×Q812yes—
123C24×C61C2×Q812yes—
124C24×C61#4114yes—
125C24×C61#4114yes—
126C24×C61C2411yescharacteristic
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—216no—
131C22×C4×C61—216no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—24no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—24no—
142C22×C4×C61—216no—
143C22×C4×C61—22no—
144C22×C4×C61—24no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—24no—
151C22×C4×C61—24no—
152C22×C4×C61—24no—
153C22×C4×C61—48no—
154C22×C4×C61—416no—
155C22×C4×C61—416no—
156C22×C4×C61—416no—
157C22×C4×C61—416no—
158C22×C4×C61—48no—
159C22×C4×C61—48no—
160C22×C4×C61C22×C414yes—
161C22×C4×C61C22×C414yes—
162C22×C4×C61C2×D418yes—
163C22×C4×C61C2×D418yes—
164C4⋊C4×C61—416no—
165C22⋊C4×C61—216no—
166C22⋊C4×C61—28no—
167C22⋊C4×C61—216no—
168C22⋊C4×C61—216no—
169C22⋊C4×C61—48no—
170C22⋊C4×C61—44no—
171C22⋊C4×C61—48no—
172C22⋊C4×C61—44no—
173C22⋊C4×C61C22⋊C4116yescomplemented
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—24no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—28no—
189C23×C61—28no—
190C23×C61—28no—
191C23×C61—28no—
192C23×C61—28no—
193C23×C61—28no—
194C23×C61—28no—
195C23×C61—28no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—24no—
199C23×C61—24no—
200C23×C61—22no—
201C23×C61—24no—
202C23×C61—24no—
203C23×C61—24no—
204C23×C61—22no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—232no—
208C23×C61—232no—
209C23×C61—416no—
210C23×C61—432no—
211C23×C61—432no—
212C23×C61—432no—
213C23×C61—432no—
214C23×C61—464no—
215C23×C61—44no—
216C23×C61—44no—
217C23×C61—44no—
218C23×C61—44no—
219C23×C61C2×C22⋊C414yes—
220C23×C61C2×C4⋊C414yes—
221C23×C61C2×C4⋊C414yes—
222C23×C61C2×C4⋊C414yes—
223C23×C61#11714yes—
224C23×C61C4×D418yes—
225C23×C61#12014yes—
226C23×C61#12118yes—
227C23×C61C22⋊Q818yes—
228C23×C61C22⋊Q818yes—
229C23×C61#12314yes—
230C23×C61C23×C411yescharacteristic
231C23×C61C22×D412yes—
232C23×C61C22×D411yescharacteristic
233C23×C61C22×D412yes—
234C23×C61C22×Q811yescharacteristic
235C23×C61C2×#4112yes—
236C23×C61C2×#4112yes—
237C23×C61H(5,𝔽2)14yes—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—28no—
241C2×C4×C61—28no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—28no—
246C2×C4×C61—28no—
247C2×C4×C61—28no—
248C2×C4×C61—432no—
249C2×C4×C61—432no—
250C2×C4×C61—44no—
251C2×C4×C61—44no—
252C2×C4×C61—44no—
253C2×C4×C61—44no—
254C2×C4×C61—44no—
255C2×C4×C61—44no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—44no—
263C2×C4×C61—44no—
264C2×C4×C61—44no—
265C2×C4×C61—44no—
266C2×C4×C61C2×C22⋊C418yescomplemented
267#13877C6111yesmaximal, characteristic, direct product factor, Sylow
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—28no—
271C22×C61—28no—
272C22×C61—28no—
273C22×C61—28no—
274C22×C61—28no—
275C22×C61—28no—
276C22×C61—28no—
277C22×C61—28no—
278C22×C61—216no—
279C22×C61—216no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—24no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—464no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—48no—
291C22×C61#38018yescomplemented
292C22×C61#38518yescomplemented
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61C22×C4⋊C411yescharacteristic
295C22×C61C2×#11711yescharacteristic
296C22×C61C2×C4×D412yes—
297C22×C61#51812yes—
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×#12112yes—
300C22×C61C2×C22⋊Q812yes—
301C22×C61C2×C22⋊Q812yes—
302C22×C61C2×#12311yescharacteristic
303C22×C61#53412yes—
304C22×C61#54312yes—
305C22×C61D4214yes—
306C22×C61#54614yes—
307C22×C61#54614yes—
308C22×C61#55114yes—
309C4×C61—28no—
310C4×C61—44no—
311C4×C61—48no—
312C4×C61—82no—
313C4×C61—82no—
314C23×C22⋊C4C2×C6111yescharacteristic
315C23×C22⋊C4C2×C6111yescharacteristic
316C2×#385C2×C6112yescomplemented
317C2×#380C2×C6112yescomplemented
318C2×#379C2×C6111yescharacteristic
319#1439C2×C6114yescomplemented
320C22⋊#95C2×C6112yescomplemented
321C22⋊#95C2×C6112yescomplemented
322C2×C61—28no—
323C2×C61—28no—
324C2×C61—44no—
325C2×C61C2×#38011yescharacteristic
326C2×C61C2×#38511yescharacteristic
327C2×C61#231511yescharacteristic
328C2×C61D4×C22⋊C412yes—
329C2×C61#234212yes—
330C2×C61#241012yes—
331C2×C61#241212yes—
332C2×C61#243714yes—
333C26C22×C6111yescharacteristic
334C24×C4C22×C6112yes—
335C24×C4C22×C6112yes—
336C22×C4⋊C4C22×C6114yescomplemented
337C22×C22⋊C4—24no—
338C22×C22⋊C4—28no—
339C22×C22⋊C4—28no—
340C22×C22⋊C4C4×C6118yescomplemented
341C22×C22⋊C4C22×C6112yes—
342C22×C22⋊C4C22×C6111yescharacteristic
343C22×C22⋊C4C22×C6112yes—
344C22×C22⋊C4C22×C6111yescharacteristic
345C22×C22⋊C4C22×C6111yescharacteristic
346C22×C22⋊C4C22×C6112yes—
347C22×C22⋊C4C22×C6112yes—
348C22×C22⋊C4C22×C6112yes—
349C22×C22⋊C4C22×C6111yescharacteristic
350#385—28no—
351#380—28no—
352#379—24no—
353C2×#95C22×C6114yescomplemented
354C2×#95C22×C6114yescomplemented
355C2×#95C22×C6114yescomplemented
356C61#1387711yescharacteristic, direct product factor, Sylow
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25C2×C4×C6114yes—
363C25D4×C6114yes—
364C25Q8×C6114yes—
365C25C23×C6111yescharacteristic
366C25C23×C6111yescharacteristic
367C25C23×C6111yescharacteristic
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—28no—
372C23×C4—28no—
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—44no—
376C23×C4—44no—
377C23×C4C2×C4×C6118yescomplemented
378C23×C4C23×C6112yes—
379C23×C4C23×C6112yes—
380C23×C4C23×C6112yes—
381C23×C4C23×C6114yes—
382C23×C4C23×C6112yes—
383C2×C4⋊C4—28no—
384C2×C4⋊C4—28no—
385C2×C4⋊C4—44no—
386C2×C22⋊C4—232no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—28no—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—232no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—44no—
403C2×C22⋊C4C2×C4×C6114yes—
404C2×C22⋊C4C2×C4×C6114yes—
405C2×C22⋊C4C2×C4×C6118yescomplemented
406C2×C22⋊C4D4×C6118yes—
407C2×C22⋊C4D4×C6118yes—
408#95—28no—
409#95—28no—
410#95—44no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—28no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4⋊C4×C61116yescomplemented
438C24C22×C4×C6112yes—
439C24C22×C4×C6112yes—
440C24C22×C4×C6112yes—
441C24C2×D4×C6112yes—
442C24C2×D4×C6114yes—
443C24C2×D4×C6112yes—
444C24C2×D4×C6112yes—
445C24C2×D4×C6114yes—
446C24C2×Q8×C6112yes—
447C24C2×Q8×C6112yes—
448C24C2×Q8×C6112yes—
449C24#41×C6114yes—
450C24#41×C6114yes—
451C24C24×C6111yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4C22×C4×C6114yes—
486C22×C4C22×C4×C6114yes—
487C22×C4C2×D4×C6118yes—
488C22×C4C2×D4×C6118yes—
489C4⋊C4—416no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4C22⋊C4×C61116yescomplemented
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—232no—
524C23—232no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—416no—
535C23—432no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—464no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊C4×C6114yes—
545C23C2×C4⋊C4×C6114yes—
546C23C2×C4⋊C4×C6114yes—
547C23C2×C4⋊C4×C6114yes—
548C23#117×C6114yes—
549C23C4×D4×C6118yes—
550C23#120×C6114yes—
551C23#121×C6118yes—
552C23C22⋊Q8×C6118yes—
553C23C22⋊Q8×C6118yes—
554C23#123×C6114yes—
555C23C23×C4×C6111yescharacteristic
556C23C22×D4×C6111yescharacteristic
557C23C22×D4×C6112yes—
558C23C22×D4×C6112yes—
559C23C22×Q8×C6111yescharacteristic
560C23C2×#41×C6112yes—
561C23C2×#41×C6112yes—
562C23H(5,𝔽2)×C6114yes—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4C2×C22⋊C4×C6118yescomplemented
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—216no—
605C22—216no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22C61×#38018yescomplemented
616C22C61×#38518yescomplemented
617C22C22×C22⋊C4×C6111yescharacteristic
618C22C22×C4⋊C4×C6111yescharacteristic
619C22C2×#117×C6111yescharacteristic
620C22C2×C4×D4×C6112yes—
621C22C61×#51812yes—
622C22C2×#120×C6111yescharacteristic
623C22C2×#121×C6112yes—
624C22C2×C22⋊Q8×C6112yes—
625C22C2×C22⋊Q8×C6112yes—
626C22C2×#123×C6111yescharacteristic
627C22C61×#53412yes—
628C22C61×#54312yes—
629C22D42×C6114yes—
630C22C61×#54614yes—
631C22C61×#54614yes—
632C22C61×#55114yes—
633C4—28no—
634C4—44no—
635C4—48no—
636C4—82no—
637C4—82no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×C61×#38011yescharacteristic
642C2C2×C61×#38511yescharacteristic
643C2C61×#231511yescharacteristic
644C2D4×C22⋊C4×C6112yes—
645C2C61×#234212yes—
646C2C61×#241012yes—
647C2C61×#241212yes—
648C2C61×#243714yes—
subgroupG/H
center, socleC24×C61C24
commutatorC23C23×C4×C61
Frattini, cosocleC24C24×C61
prime radicalC26×C61C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C61×#385C2semidirect
4. C2×C61×#380C2semidirect
6. C61×#1439C2semidirect
7. C61×C22⋊#95C2semidirect
8. C61×C22⋊#95C2semidirect
12. C22×C4⋊C4×C61C22semidirect
16. C22×C22⋊C4×C61C4semidirect
29. C2×#95×C61C22semidirect
30. C2×#95×C61C22semidirect
31. C2×#95×C61C22semidirect
52. C23×C4×C61C2×C4semidirect
80. C2×C22⋊C4×C61C2×C4semidirect
112. C24×C61C4⋊C4semidirect
173. C22⋊C4×C61C22⋊C4semidirect
266. C2×C4×C61C2×C22⋊C4semidirect
267. #13877C61direct (N × H)
291. C22×C61#380semidirect
292. C22×C61#385semidirect
316. C2×#385C2×C61semidirect
317. C2×#380C2×C61semidirect
319. #1439C2×C61semidirect
320. C22⋊#95C2×C61semidirect
321. C22⋊#95C2×C61semidirect
336. C22×C4⋊C4C22×C61semidirect
340. C22×C22⋊C4C4×C61semidirect
353. C2×#95C22×C61semidirect
354. C2×#95C22×C61semidirect
355. C2×#95C22×C61semidirect
356. C61#13877direct (N × H)
377. C23×C4C2×C4×C61semidirect
405. C2×C22⋊C4C2×C4×C61semidirect
437. C24C4⋊C4×C61semidirect
498. C22⋊C4C22⋊C4×C61semidirect
591. C2×C4C2×C22⋊C4×C61semidirect
615. C22C61×#380semidirect
616. C22C61×#385semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C612
C2×C61×#3852
C2×C61×#3802
C2×C61×#3792
C61×#14392
C61×C22⋊#952
#1387761