Finite Group Databasebeta

← C61×#15001 #29518434

Subgroups — C61×#15001 #29518434

Order 15616.

1846 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C61C211yesmaximal, characteristic
2C2×C61×#387C212yesmaximal, complemented
3C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
4C61×#1446C211yesmaximal, characteristic, complemented
5C61×#1446C211yesmaximal, characteristic, complemented
6C61×#1444C211yesmaximal, characteristic, complemented
7C61×#1442C211yesmaximal, characteristic, complemented
8C61×#1440C212yesmaximal
9C61×#1439C212yesmaximal
10C61×#1436C212yesmaximal, complemented
11C61×C22⋊#95C211yesmaximal, characteristic
12C23×Q8×C61C2211yescharacteristic
13C24×C4×C61C2211yescharacteristic
14C2×C22⋊Q8×C61—24no—
15C2×C22⋊Q8×C61—28no—
16C22×C4⋊C4×C61C2212yes—
17C22×C4⋊C4×C61C2211yescharacteristic
18C22×C4⋊C4×C61C2212yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C22×C22⋊C4×C61C2212yes—
21C22×C22⋊C4×C61C2212yes—
22C22×C22⋊C4×C61C2211yescharacteristic
23C22×C42×C61C2211yescharacteristic
24C22×C42×C61C2211yescharacteristic
25C61×#387—28no—
26C4×C22⋊C4×C61—24no—
27C2×#95×C61C2211yescharacteristic
28C2×#95×C61C2211yescharacteristic
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2212yes—
33C2×#95×C61C2211yescharacteristic
34C2×#95×C61C2211yescharacteristic
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C2×#95×C61C2212yes—
38C2×#95×C61C2212yes—
39C25×C61C2311yescharacteristic
40C22×Q8×C61—24no—
41C22×Q8×C61—22no—
42C22×Q8×C61—22no—
43C22×Q8×C61—22no—
44C22×Q8×C61—42no—
45C23×C4×C61—24no—
46C23×C4×C61—22no—
47C23×C4×C61—22no—
48C23×C4×C61—42no—
49C23×C4×C61D414yes—
50C23×C4×C61C2312yes—
51C23×C4×C61C2312yes—
52C23×C4×C61C2312yes—
53C23×C4×C61C2311yescharacteristic
54C23×C4×C61C2311yescharacteristic
55C23×C4×C61C2311yescharacteristic
56C23×C4×C61C2312yes—
57C23×C4×C61C2311yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C22⋊Q8×C61—232no—
61C2×C4⋊C4×C61—24no—
62C2×C4⋊C4×C61—24no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61—24no—
65C2×C4⋊C4×C61—24no—
66C2×C4⋊C4×C61—24no—
67C2×C4⋊C4×C61—24no—
68C2×C4⋊C4×C61D418yes—
69C2×C22⋊C4×C61—22no—
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—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61—24no—
79C2×C22⋊C4×C61—42no—
80C2×C22⋊C4×C61Q814yescomplemented
81C2×C42×C61—22no—
82C2×C42×C61—22no—
83C2×C42×C61—24no—
84C2×C42×C61—42no—
85#95×C61—22no—
86#95×C61—22no—
87#95×C61—24no—
88#95×C61—24no—
89#95×C61—24no—
90#95×C61—24no—
91#95×C61—24no—
92#95×C61—22no—
93#95×C61—24no—
94#95×C61—22no—
95#95×C61—42no—
96#95×C61—42no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—21no—
100C24×C61—22no—
101C24×C61—21no—
102C24×C61—21no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61C2×D412yes—
106C24×C61C2×Q812yes—
107C24×C61#4112yes—
108C24×C61C2411yescharacteristic
109C2×Q8×C61—28no—
110C2×Q8×C61—48no—
111C2×Q8×C61—48no—
112C2×Q8×C61—44no—
113C2×Q8×C61—44no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—28no—
154C22×C4×C61—28no—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—44no—
163C22×C4×C61—48no—
164C22×C4×C61—48no—
165C22×C4×C61C2×D414yes—
166C22×C4×C61C2×D414yes—
167C22×C4×C61C2×D414yes—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D412yes—
170C22×C4×C61C2×Q812yes—
171C22×C4×C61C2×Q812yes—
172C22×C4×C61#4112yes—
173C22×C4×C61#4114yes—
174C22×C4×C61#4112yes—
175C22×C4×C61#4112yes—
176C22×C4×C61#4112yes—
177C4⋊C4×C61—216no—
178C4⋊C4×C61—28no—
179C4⋊C4×C61—216no—
180C22⋊C4×C61—28no—
181C22⋊C4×C61—216no—
182C22⋊C4×C61—28no—
183C22⋊C4×C61—48no—
184C42×C61—44no—
185C42×C61—44no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—24no—
197C23×C61—24no—
198C23×C61—24no—
199C23×C61—24no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61C22⋊Q814yes—
209C23×C61C22×D411yescharacteristic
210C23×C61C22×D412yes—
211C23×C61C22×Q811yescharacteristic
212C23×C61C2×#4111yescharacteristic
213C23×C61C2×#4111yescharacteristic
214C23×C61C2×#4111yescharacteristic
215C23×C61C2×#4111yescharacteristic
216C23×C61C2×#4111yescharacteristic
217C23×C61C2×#4112yes—
218C23×C61H(5,𝔽2)12yes—
219C23×C612−1+412yes—
220Q8×C61—416no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—416no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—41no—
257C2×C4×C61—42no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—42no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—42no—
274C2×C4×C61—42no—
275C2×C4×C61—42no—
276C2×C4×C61—42no—
277C2×C4×C61—42no—
278C2×C4×C61#12014yes—
279C2×C4×C61#12314yes—
280#15001C6111yesmaximal, characteristic, direct product factor, Sylow
281C22×C61—24no—
282C22×C61—24no—
283C22×C61—24no—
284C22×C61—44no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—42no—
291C22×C61—44no—
292C22×C61C2×#12011yescharacteristic
293C22×C61C2×C22⋊Q811yescharacteristic
294C22×C61C2×#12311yescharacteristic
295C22×C61#52511yescharacteristic
296C22×C61#52512yes—
297C22×C61#52911yescharacteristic
298C22×C61#52911yescharacteristic
299C22×C61#53212yes—
300C22×C61#53211yescharacteristic
301C22×C61#53311yescharacteristic
302C22×C61#53311yescharacteristic
303C22×C61#53411yescharacteristic
304C22×C61#53611yescharacteristic
305C22×C61#53912yes—
306C22×C61#54212yes—
307C22×C61#54612yes—
308C22×C61#54612yes—
309C22×C61#54812yes—
310C22×C61D4×Q812yes—
311C22×C61#55112yes—
312C22×C61#55212yes—
313C22×C61#55412yes—
314C22×C61#55612yes—
315C4×C61—24no—
316C4×C61—24no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—44no—
320C4×C61—42no—
321C4×C61—44no—
322C4×C61—42no—
323C4×C61—44no—
324C4×C61—82no—
325C22×C22⋊Q8C2×C6111yescharacteristic
326C2×#387C2×C6112yescomplemented
327C2×C4×C22⋊C4C2×C6111yescharacteristic
328#1446C2×C6111yescharacteristic, complemented
329#1446C2×C6111yescharacteristic, complemented
330#1444C2×C6111yescharacteristic, complemented
331#1442C2×C6111yescharacteristic, complemented
332#1440C2×C6112yes—
333#1439C2×C6112yes—
334#1436C2×C6112yescomplemented
335C22⋊#95C2×C6111yescharacteristic
336C2×C61—44no—
337C2×C61#238811yescharacteristic
338C2×C61#240511yescharacteristic
339C2×C61#240911yescharacteristic
340C2×C61#246512yes—
341C2×C61#247712yes—
342C2×C61#249911yescharacteristic
343C2×C61#255612yes—
344C2×C61#256012yes—
345C2×C61#256411yescharacteristic
346C2×C61#263811yescharacteristic
347C2×C61#285511yescharacteristic
348C23×Q8C22×C6111yescharacteristic
349C24×C4C22×C6111yescharacteristic
350C2×C22⋊Q8—24no—
351C2×C22⋊Q8—28no—
352C22×C4⋊C4C22×C6111yescharacteristic
353C22×C4⋊C4C22×C6112yes—
354C22×C4⋊C4C22×C6112yes—
355C22×C22⋊C4C22×C6112yes—
356C22×C22⋊C4C22×C6111yescharacteristic
357C22×C22⋊C4C22×C6112yes—
358C22×C22⋊C4C22×C6111yescharacteristic
359C22×C42C22×C6111yescharacteristic
360C22×C42C22×C6111yescharacteristic
361#387—28no—
362C4×C22⋊C4—24no—
363C2×#95C22×C6112yes—
364C2×#95C22×C6111yescharacteristic
365C2×#95C22×C6111yescharacteristic
366C2×#95C22×C6112yes—
367C2×#95C22×C6112yes—
368C2×#95C22×C6112yes—
369C2×#95C22×C6112yes—
370C2×#95C22×C6112yes—
371C2×#95C22×C6111yescharacteristic
372C2×#95C22×C6111yescharacteristic
373C2×#95C22×C6112yes—
374C2×#95C22×C6112yes—
375C61#1500111yescharacteristic, direct product factor, Sylow
376C25C23×C6111yescharacteristic
377C22×Q8—24no—
378C22×Q8—22no—
379C22×Q8—22no—
380C22×Q8—22no—
381C22×Q8—42no—
382C23×C4—24no—
383C23×C4—22no—
384C23×C4—22no—
385C23×C4—42no—
386C23×C4D4×C6114yes—
387C23×C4C23×C6112yes—
388C23×C4C23×C6112yes—
389C23×C4C23×C6112yes—
390C23×C4C23×C6111yescharacteristic
391C23×C4C23×C6111yescharacteristic
392C23×C4C23×C6111yescharacteristic
393C23×C4C23×C6112yes—
394C23×C4C23×C6111yescharacteristic
395C23×C4C23×C6111yescharacteristic
396C23×C4C23×C6111yescharacteristic
397C22⋊Q8—232no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4D4×C6118yes—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4Q8×C6114yescomplemented
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—24no—
421C2×C42—42no—
422#95—22no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—24no—
431#95—22no—
432#95—42no—
433#95—42no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—21no—
439C24—21no—
440C24—22no—
441C24—22no—
442C24C2×D4×C6112yes—
443C24C2×Q8×C6112yes—
444C24#41×C6112yes—
445C24C24×C6111yescharacteristic
446C2×Q8—28no—
447C2×Q8—48no—
448C2×Q8—48no—
449C2×Q8—44no—
450C2×Q8—44no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—28no—
491C22×C4—28no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4C2×D4×C6114yes—
503C22×C4C2×D4×C6114yes—
504C22×C4C2×D4×C6114yes—
505C22×C4C2×D4×C6112yes—
506C22×C4C2×D4×C6112yes—
507C22×C4C2×Q8×C6112yes—
508C22×C4C2×Q8×C6112yes—
509C22×C4#41×C6112yes—
510C22×C4#41×C6114yes—
511C22×C4#41×C6112yes—
512C22×C4#41×C6112yes—
513C22×C4#41×C6112yes—
514C4⋊C4—28no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C42—44no—
522C42—44no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C22⋊Q8×C6114yes—
546C23C22×D4×C6111yescharacteristic
547C23C22×D4×C6112yes—
548C23C22×Q8×C6111yescharacteristic
549C23C2×#41×C6111yescharacteristic
550C23C2×#41×C6112yes—
551C23C2×#41×C6111yescharacteristic
552C23C2×#41×C6111yescharacteristic
553C23C2×#41×C6111yescharacteristic
554C23C2×#41×C6111yescharacteristic
555C23H(5,𝔽2)×C6112yes—
556C232−1+4×C6112yes—
557Q8—416no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—416no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4#120×C6114yes—
616C2×C4#123×C6114yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22C2×#120×C6111yescharacteristic
629C22C2×C22⋊Q8×C6111yescharacteristic
630C22C2×#123×C6111yescharacteristic
631C22C61×#52511yescharacteristic
632C22C61×#52512yes—
633C22C61×#52911yescharacteristic
634C22C61×#52911yescharacteristic
635C22C61×#53212yes—
636C22C61×#53211yescharacteristic
637C22C61×#53311yescharacteristic
638C22C61×#53311yescharacteristic
639C22C61×#53411yescharacteristic
640C22C61×#53611yescharacteristic
641C22C61×#53912yes—
642C22C61×#54212yes—
643C22C61×#54612yes—
644C22C61×#54612yes—
645C22C61×#54812yes—
646C22D4×Q8×C6112yes—
647C22C61×#55112yes—
648C22C61×#55212yes—
649C22C61×#55412yes—
650C22C61×#55612yes—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—82no—
661C2—44no—
662C2C61×#238811yescharacteristic
663C2C61×#240511yescharacteristic
664C2C61×#240911yescharacteristic
665C2C61×#246512yes—
666C2C61×#247712yes—
667C2C61×#249911yescharacteristic
668C2C61×#255612yes—
669C2C61×#256012yes—
670C2C61×#256411yescharacteristic
671C2C61×#263811yescharacteristic
672C2C61×#285511yescharacteristic
subgroupG/H
center, socleC24×C61C24
commutator, Frattini, cosocleC24C24×C61
prime radicalC25×C61C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#387C2semidirect
4. C61×#1446C2semidirect
5. C61×#1446C2semidirect
6. C61×#1444C2semidirect
7. C61×#1442C2semidirect
10. C61×#1436C2semidirect
80. C2×C22⋊C4×C61Q8semidirect
280. #15001C61direct (N × H)
326. C2×#387C2×C61semidirect
328. #1446C2×C61semidirect
329. #1446C2×C61semidirect
330. #1444C2×C61semidirect
331. #1442C2×C61semidirect
334. #1436C2×C61semidirect
375. C61#15001direct (N × H)
417. C2×C22⋊C4Q8×C61semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C612
C2×C61×#3872
C2×C4×C22⋊C4×C612
C61×#14462
C61×#14442
C61×#14422
C61×#14402
C61×#14392
C61×#14362
C61×C22⋊#952
#1500161