Finite Group Databasebeta

← C61×#15078 #29518511

Subgroups — C61×#15078 #29518511

Order 15616.

1434 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
1C2×C61×#388C212yesmaximal
2C2×C61×#388C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
4C61×#1444C211yesmaximal, characteristic, complemented
5C61×#1442C211yesmaximal, characteristic, complemented
6C61×#1441C212yesmaximal, complemented
7C61×#1440C212yesmaximal
8C61×#1439C211yesmaximal, characteristic
9C61×#1439C211yesmaximal, characteristic
10C61×#1434C212yesmaximal, complemented
11C61×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C61C2211yescharacteristic
13C22×C4⋊C4×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2212yes—
15C22×C4⋊C4×C61C2211yescharacteristic
16C22×C22⋊C4×C61C2211yescharacteristic
17C22×C22⋊C4×C61C2212yes—
18C22×C22⋊C4×C61C2212yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C22×C22⋊C4×C61C2211yescharacteristic
21C22×C42×C61C2212yes—
22C22×C42×C61C2211yescharacteristic
23C61×#388—28no—
24C61×#388—24no—
25C4×C4⋊C4×C61—24no—
26C2×#95×C61C2212yes—
27C2×#95×C61C2211yescharacteristic
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2211yescharacteristic
31C2×#95×C61C2212yes—
32C2×#95×C61C2211yescharacteristic
33C2×#95×C61C2211yescharacteristic
34C2×#95×C61C2212yes—
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C2×#95×C61C2212yes—
38C25×C61C2311yescharacteristic
39C23×C4×C61C2312yes—
40C23×C4×C61C2311yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C23×C4×C61C2312yes—
45C23×C4×C61C2312yes—
46C23×C4×C61C2312yes—
47C23×C4×C61C2311yescharacteristic
48C23×C4×C61C2311yescharacteristic
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61—22no—
51C2×C4⋊C4×C61—24no—
52C2×C4⋊C4×C61—24no—
53C2×C4⋊C4×C61—24no—
54C2×C4⋊C4×C61—22no—
55C2×C4⋊C4×C61—24no—
56C2×C4⋊C4×C61—42no—
57C2×C4⋊C4×C61—42no—
58C2×C22⋊C4×C61—24no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61—24no—
61C2×C22⋊C4×C61—24no—
62C2×C22⋊C4×C61—22no—
63C2×C22⋊C4×C61—24no—
64C2×C22⋊C4×C61—22no—
65C2×C22⋊C4×C61—22no—
66C2×C22⋊C4×C61—24no—
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61—42no—
70C2×C22⋊C4×C61—42no—
71C2×C22⋊C4×C61D414yes—
72C2×C42×C61—22no—
73C2×C42×C61—22no—
74C2×C42×C61—24no—
75C2×C42×C61—24no—
76C2×C42×C61—24no—
77C2×C42×C61D414yes—
78#95×C61—24no—
79#95×C61—24no—
80#95×C61—24no—
81#95×C61—22no—
82#95×C61—22no—
83#95×C61—22no—
84#95×C61—24no—
85#95×C61—24no—
86#95×C61—24no—
87#95×C61—22no—
88#95×C61—42no—
89#95×C61—42no—
90C24×C61—21no—
91C24×C61—21no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—21no—
100C24×C61C2×D412yes—
101C24×C61C2411yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
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—22no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61#4114yes—
157C22×C4×C61#4114yes—
158C22×C4×C61#4114yes—
159C22×C4×C61#4112yes—
160C22×C4×C61#4112yes—
161C22×C4×C61#4112yes—
162C4⋊C4×C61—28no—
163C4⋊C4×C61—28no—
164C4⋊C4×C61—48no—
165C4⋊C4×C61—44no—
166C22⋊C4×C61—216no—
167C22⋊C4×C61—28no—
168C22⋊C4×C61—48no—
169C22⋊C4×C61—44no—
170C22⋊C4×C61—44no—
171C42×C61—28no—
172C42×C61—48no—
173C23×C61—22no—
174C23×C61—24no—
175C23×C61—24no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C22×D411yescharacteristic
197C23×C61C22×D411yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C61C2×#4111yescharacteristic
200C23×C61C2×#4111yescharacteristic
201C23×C61C2×#4112yes—
202C23×C61C2×#4112yes—
203C23×C61C2×#4112yes—
204C23×C61H(5,𝔽2)12yes—
205C23×C612−1+411yescharacteristic
206C23×C612−1+411yescharacteristic
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
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—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61#12114yes—
266C2×C4×C61#12314yes—
267#15078C6111yesmaximal, characteristic, direct product factor, Sylow
268C22×C61—42no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61C2×#12111yescharacteristic
280C22×C61C2×#12311yescharacteristic
281C22×C61#52511yescharacteristic
282C22×C61#52511yescharacteristic
283C22×C61#52512yes—
284C22×C61#52912yes—
285C22×C61#52912yes—
286C22×C61#52912yes—
287C22×C61#53211yescharacteristic
288C22×C61#53511yescharacteristic
289C22×C61#53611yescharacteristic
290C22×C61#54212yes—
291C22×C61#54212yes—
292C22×C61#54612yes—
293C22×C61#54711yescharacteristic
294C22×C61#54711yescharacteristic
295C22×C61#55212yes—
296C22×C61#55211yescharacteristic
297C22×C61#55211yescharacteristic
298C22×C61#55312yes—
299C22×C61#55312yes—
300C22×C61#55512yes—
301C22×C61#55612yes—
302C4×C61—24no—
303C4×C61—24no—
304C4×C61—44no—
305C4×C61—44no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61—44no—
309C4×C61—81no—
310C4×C61—82no—
311C4×C61—81no—
312C2×#388C2×C6111yescharacteristic
313C2×#388C2×C6112yes—
314C2×C4×C4⋊C4C2×C6111yescharacteristic, complemented
315#1444C2×C6111yescharacteristic, complemented
316#1442C2×C6111yescharacteristic, complemented
317#1441C2×C6112yescomplemented
318#1440C2×C6112yes—
319#1439C2×C6111yescharacteristic
320#1439C2×C6111yescharacteristic
321#1434C2×C6112yescomplemented
322#1434C2×C6111yescharacteristic, complemented
323C2×C61—82no—
324C2×C61#239211yescharacteristic
325C2×C61#246812yes—
326C2×C61#247412yes—
327C2×C61#248611yescharacteristic
328C2×C61#250011yescharacteristic
329C2×C61#252512yes—
330C2×C61#252912yes—
331C2×C61#253511yescharacteristic
332C2×C61#260411yescharacteristic
333C2×C61#263211yescharacteristic
334C2×C61#285311yescharacteristic
335C22×C4⋊C4C22×C6111yescharacteristic
336C22×C4⋊C4C22×C6111yescharacteristic
337C22×C4⋊C4C22×C6112yes—
338C22×C4⋊C4C22×C6111yescharacteristic
339C22×C22⋊C4C22×C6111yescharacteristic
340C22×C22⋊C4C22×C6112yes—
341C22×C22⋊C4C22×C6112yes—
342C22×C22⋊C4C22×C6111yescharacteristic
343C22×C22⋊C4C22×C6111yescharacteristic
344C22×C42C22×C6112yes—
345C22×C42C22×C6111yescharacteristic
346#388—28no—
347#388—24no—
348C4×C4⋊C4—24no—
349C2×#95C22×C6111yescharacteristic
350C2×#95C22×C6112yes—
351C2×#95C22×C6112yes—
352C2×#95C22×C6111yescharacteristic
353C2×#95C22×C6111yescharacteristic
354C2×#95C22×C6112yes—
355C2×#95C22×C6112yes—
356C2×#95C22×C6112yes—
357C2×#95C22×C6112yes—
358C2×#95C22×C6111yescharacteristic
359C2×#95C22×C6112yes—
360C2×#95C22×C6112yes—
361C61#1507811yescharacteristic, direct product factor, Sylow
362C25C23×C6111yescharacteristic
363C23×C4C23×C6111yescharacteristic
364C23×C4C23×C6112yes—
365C23×C4C23×C6111yescharacteristic
366C23×C4C23×C6111yescharacteristic
367C23×C4C23×C6111yescharacteristic
368C23×C4C23×C6112yes—
369C23×C4C23×C6112yes—
370C23×C4C23×C6112yes—
371C23×C4C23×C6111yescharacteristic
372C23×C4C23×C6111yescharacteristic
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—42no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4D4×C6114yes—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—24no—
399C2×C42—24no—
400C2×C42—24no—
401C2×C42D4×C6114yes—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—22no—
412#95—42no—
413#95—42no—
414C24—21no—
415C24—22no—
416C24—21no—
417C24—21no—
418C24—21no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—21no—
423C24—21no—
424C24C2×D4×C6112yes—
425C24C24×C6111yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4C2×D4×C6112yes—
476C22×C4C2×D4×C6112yes—
477C22×C4C2×D4×C6112yes—
478C22×C4C2×D4×C6112yes—
479C22×C4C2×D4×C6112yes—
480C22×C4#41×C6114yes—
481C22×C4#41×C6114yes—
482C22×C4#41×C6114yes—
483C22×C4#41×C6112yes—
484C22×C4#41×C6112yes—
485C22×C4#41×C6112yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C42—28no—
496C42—48no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C22×D4×C6111yescharacteristic
521C23C22×D4×C6111yescharacteristic
522C23C2×#41×C6112yes—
523C23C2×#41×C6112yes—
524C23C2×#41×C6112yes—
525C23C2×#41×C6111yescharacteristic
526C23C2×#41×C6111yescharacteristic
527C23C2×#41×C6111yescharacteristic
528C23H(5,𝔽2)×C6112yes—
529C232−1+4×C6111yescharacteristic
530C232−1+4×C6111yescharacteristic
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4#121×C6114yes—
590C2×C4#123×C6114yes—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×#121×C6111yescharacteristic
603C22C2×#123×C6111yescharacteristic
604C22C61×#52511yescharacteristic
605C22C61×#52511yescharacteristic
606C22C61×#52512yes—
607C22C61×#52912yes—
608C22C61×#52912yes—
609C22C61×#52912yes—
610C22C61×#53211yescharacteristic
611C22C61×#53511yescharacteristic
612C22C61×#53611yescharacteristic
613C22C61×#54212yes—
614C22C61×#54212yes—
615C22C61×#54612yes—
616C22C61×#54711yescharacteristic
617C22C61×#54711yescharacteristic
618C22C61×#55211yescharacteristic
619C22C61×#55211yescharacteristic
620C22C61×#55212yes—
621C22C61×#55312yes—
622C22C61×#55312yes—
623C22C61×#55512yes—
624C22C61×#55612yes—
625C4—24no—
626C4—24no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—81no—
635C2—82no—
636C2C61×#239211yescharacteristic
637C2C61×#246812yes—
638C2C61×#247412yes—
639C2C61×#248611yescharacteristic
640C2C61×#250011yescharacteristic
641C2C61×#252512yes—
642C2C61×#252912yes—
643C2C61×#253511yescharacteristic
644C2C61×#260411yescharacteristic
645C2C61×#263211yescharacteristic
646C2C61×#285311yescharacteristic
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
3. C2×C4×C4⋊C4×C61C2semidirect
4. C61×#1444C2semidirect
5. C61×#1442C2semidirect
6. C61×#1441C2semidirect
10. C61×#1434C2semidirect
11. C61×#1434C2semidirect
267. #15078C61direct (N × H)
314. C2×C4×C4⋊C4C2×C61semidirect
315. #1444C2×C61semidirect
316. #1442C2×C61semidirect
317. #1441C2×C61semidirect
321. #1434C2×C61semidirect
322. #1434C2×C61semidirect
361. C61#15078direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#3882
C2×C4×C4⋊C4×C612
C61×#14442
C61×#14422
C61×#14412
C61×#14402
C61×#14392
C61×#14342
#1507861