Finite Group Databasebeta

← C61×#15071 #29518504

Subgroups — C61×#15071 #29518504

Order 15616.

1478 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125×C61C211yesmaximal, characteristic
2C2×C61×#387C212yesmaximal
3C2×C4×C4⋊C4×C61C211yesmaximal, characteristic
4C61×#1444C211yesmaximal, characteristic
5C61×#1444C212yesmaximal
6C61×#1442C212yesmaximal
7C61×#1442C211yesmaximal, characteristic
8C61×#1441C211yesmaximal, characteristic
9C61×#1441C211yesmaximal, characteristic
10C61×#1436C212yesmaximal
11C61×#1434C211yesmaximal, characteristic
12C2×#125×C61—28no—
13C2×#125×C61—24no—
14C22×C4⋊C4×C61C2211yescharacteristic
15C22×C4⋊C4×C61C2212yes—
16C22×C4⋊C4×C61C2211yescharacteristic
17C22×C4⋊C4×C61C2211yescharacteristic
18C22×C4⋊C4×C61C2212yes—
19C22×C4⋊C4×C61C2212yes—
20C22×C4⋊C4×C61C2211yescharacteristic
21C22×C42×C61C2211yescharacteristic
22C22×C42×C61C2211yescharacteristic
23C22×C42×C61C2211yescharacteristic
24C61×#387—28no—
25C4×C4⋊C4×C61—24no—
26C2×#95×C61C2211yescharacteristic
27C2×#95×C61C2211yescharacteristic
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2211yescharacteristic
33C2×#95×C61C2211yescharacteristic
34C2×#95×C61C2212yes—
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C2×#95×C61C2212yes—
38C2×#95×C61C2212yes—
39C23×C4×C61C2312yes—
40C23×C4×C61C2312yes—
41C23×C4×C61C2311yescharacteristic
42C23×C4×C61C2312yes—
43C23×C4×C61C2311yescharacteristic
44C23×C4×C61C2311yescharacteristic
45C23×C4×C61C2312yes—
46C23×C4×C61C2311yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C23×C4×C61C2311yescharacteristic
49C23×C4×C61C2311yescharacteristic
50#125×C61—232no—
51C2×C4⋊C4×C61—22no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61—24no—
54C2×C4⋊C4×C61—24no—
55C2×C4⋊C4×C61—24no—
56C2×C4⋊C4×C61—24no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—24no—
59C2×C4⋊C4×C61—22no—
60C2×C4⋊C4×C61—22no—
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—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61D418yes—
69C2×C4⋊C4×C61Q814yes—
70C2×C42×C61—22no—
71C2×C42×C61—24no—
72C2×C42×C61—22no—
73C2×C42×C61—22no—
74C2×C42×C61—22no—
75C2×C42×C61—24no—
76C2×C42×C61D414yes—
77#95×C61—22no—
78#95×C61—22no—
79#95×C61—24no—
80#95×C61—24no—
81#95×C61—24no—
82#95×C61—24no—
83#95×C61—24no—
84#95×C61—22no—
85#95×C61—24no—
86#95×C61—24no—
87#95×C61—24no—
88#95×C61—24no—
89#95×C61—22no—
90C24×C61C2411yescharacteristic
91C22×C4×C61—22no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
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—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D414yes—
144C22×C4×C61C2×D414yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×Q812yes—
148C22×C4×C61C2×Q812yes—
149C22×C4×C61C2×Q812yes—
150C22×C4×C61#4112yes—
151C22×C4×C61#4114yes—
152C22×C4×C61#4112yes—
153C22×C4×C61#4112yes—
154C22×C4×C61#4112yes—
155C22×C4×C61#4112yes—
156C4⋊C4×C61—28no—
157C4⋊C4×C61—216no—
158C4⋊C4×C61—28no—
159C4⋊C4×C61—28no—
160C4⋊C4×C61—216no—
161C4⋊C4×C61—216no—
162C4⋊C4×C61—28no—
163C42×C61—28no—
164C42×C61—28no—
165C42×C61—44no—
166C23×C61C22×D411yescharacteristic
167C23×C61C22×D412yes—
168C23×C61C22×Q811yescharacteristic
169C23×C61C2×#4111yescharacteristic
170C23×C61C2×#4111yescharacteristic
171C23×C61C2×#4111yescharacteristic
172C23×C61C2×#4111yescharacteristic
173C23×C61C2×#4111yescharacteristic
174C23×C61C2×#4112yes—
175C23×C612−1+412yes—
176C23×C612−1+412yes—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61#12014yes—
240C2×C4×C61C22⋊Q814yes—
241C2×C4×C61#12514yescomplemented
242#15071C6111yesmaximal, characteristic, direct product factor, Sylow
243C22×C61C2×#12011yescharacteristic
244C22×C61C2×C22⋊Q811yescharacteristic
245C22×C61#52511yescharacteristic
246C22×C61#52511yescharacteristic
247C22×C61#52512yes—
248C22×C61C2×#12511yescharacteristic
249C22×C61#52911yescharacteristic
250C22×C61#52911yescharacteristic
251C22×C61#53211yescharacteristic
252C22×C61#53212yes—
253C22×C61#53311yescharacteristic
254C22×C61#53611yescharacteristic
255C22×C61#53611yescharacteristic
256C22×C61#54112yes—
257C22×C61#54112yes—
258C22×C61#54712yes—
259C22×C61#54712yes—
260C22×C61#54812yes—
261C22×C61D4×Q812yes—
262C22×C61#55212yes—
263C22×C61#55212yes—
264C22×C61#55212yes—
265C22×C61#55712yes—
266C4×C61—24no—
267C4×C61—24no—
268C4×C61—24no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—44no—
272C4×C61—42no—
273C4×C61—44no—
274C4×C61—42no—
275C4×C61—44no—
276C4×C61—82no—
277C22×#125C2×C6111yescharacteristic
278C2×#387C2×C6112yes—
279C2×C4×C4⋊C4C2×C6111yescharacteristic
280#1444C2×C6111yescharacteristic
281#1444C2×C6112yes—
282#1442C2×C6111yescharacteristic
283#1442C2×C6112yes—
284#1441C2×C6111yescharacteristic
285#1441C2×C6111yescharacteristic
286#1436C2×C6112yes—
287#1434C2×C6111yescharacteristic
288C2×C61#238811yescharacteristic
289C2×C61#240611yescharacteristic
290C2×C61#240911yescharacteristic
291C2×C61#246212yes—
292C2×C61#247512yes—
293C2×C61#250612yes—
294C2×C61#251912yes—
295C2×C61#256511yescharacteristic
296C2×C61#261111yescharacteristic
297C2×C61#263411yescharacteristic
298C2×C61#285111yescharacteristic
299C2×#125—28no—
300C2×#125—24no—
301C22×C4⋊C4C22×C6111yescharacteristic
302C22×C4⋊C4C22×C6111yescharacteristic
303C22×C4⋊C4C22×C6112yes—
304C22×C4⋊C4C22×C6111yescharacteristic
305C22×C4⋊C4C22×C6112yes—
306C22×C4⋊C4C22×C6112yes—
307C22×C4⋊C4C22×C6111yescharacteristic
308C22×C42C22×C6111yescharacteristic
309C22×C42C22×C6111yescharacteristic
310C22×C42C22×C6111yescharacteristic
311#387—28no—
312C4×C4⋊C4—24no—
313C2×#95C22×C6112yes—
314C2×#95C22×C6111yescharacteristic
315C2×#95C22×C6111yescharacteristic
316C2×#95C22×C6112yes—
317C2×#95C22×C6112yes—
318C2×#95C22×C6112yes—
319C2×#95C22×C6112yes—
320C2×#95C22×C6112yes—
321C2×#95C22×C6112yes—
322C2×#95C22×C6112yes—
323C2×#95C22×C6111yescharacteristic
324C2×#95C22×C6111yescharacteristic
325C2×#95C22×C6112yes—
326C61#1507111yescharacteristic, direct product factor, Sylow
327C23×C4C23×C6112yes—
328C23×C4C23×C6112yes—
329C23×C4C23×C6112yes—
330C23×C4C23×C6111yescharacteristic
331C23×C4C23×C6111yescharacteristic
332C23×C4C23×C6111yescharacteristic
333C23×C4C23×C6112yes—
334C23×C4C23×C6111yescharacteristic
335C23×C4C23×C6111yescharacteristic
336C23×C4C23×C6111yescharacteristic
337C23×C4C23×C6111yescharacteristic
338#125—232no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4D4×C6118yes—
357C2×C4⋊C4Q8×C6114yes—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42D4×C6114yes—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378C24C24×C6111yescharacteristic
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4C2×D4×C6114yes—
430C22×C4C2×D4×C6112yes—
431C22×C4C2×D4×C6114yes—
432C22×C4C2×D4×C6114yes—
433C22×C4C2×D4×C6112yes—
434C22×C4C2×D4×C6112yes—
435C22×C4C2×Q8×C6112yes—
436C22×C4C2×Q8×C6112yes—
437C22×C4C2×Q8×C6112yes—
438C22×C4#41×C6112yes—
439C22×C4#41×C6114yes—
440C22×C4#41×C6112yes—
441C22×C4#41×C6112yes—
442C22×C4#41×C6112yes—
443C22×C4#41×C6112yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C42—28no—
452C42—28no—
453C42—44no—
454C23C22×D4×C6111yescharacteristic
455C23C22×D4×C6112yes—
456C23C22×Q8×C6111yescharacteristic
457C23C2×#41×C6111yescharacteristic
458C23C2×#41×C6112yes—
459C23C2×#41×C6111yescharacteristic
460C23C2×#41×C6111yescharacteristic
461C23C2×#41×C6111yescharacteristic
462C23C2×#41×C6111yescharacteristic
463C232−1+4×C6112yes—
464C232−1+4×C6112yes—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4#120×C6114yes—
528C2×C4C22⋊Q8×C6114yes—
529C2×C4#125×C6114yescomplemented
530C22C2×#120×C6111yescharacteristic
531C22C2×C22⋊Q8×C6111yescharacteristic
532C22C61×#52511yescharacteristic
533C22C61×#52511yescharacteristic
534C22C61×#52512yes—
535C22C2×#125×C6111yescharacteristic
536C22C61×#52911yescharacteristic
537C22C61×#52911yescharacteristic
538C22C61×#53212yes—
539C22C61×#53211yescharacteristic
540C22C61×#53311yescharacteristic
541C22C61×#53611yescharacteristic
542C22C61×#53611yescharacteristic
543C22C61×#54112yes—
544C22C61×#54112yes—
545C22C61×#54712yes—
546C22C61×#54712yes—
547C22C61×#54812yes—
548C22D4×Q8×C6112yes—
549C22C61×#55212yes—
550C22C61×#55212yes—
551C22C61×#55212yes—
552C22C61×#55712yes—
553C4—24no—
554C4—24no—
555C4—24no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—42no—
560C4—44no—
561C4—42no—
562C4—44no—
563C4—82no—
564C2C61×#238811yescharacteristic
565C2C61×#240611yescharacteristic
566C2C61×#240911yescharacteristic
567C2C61×#246212yes—
568C2C61×#247512yes—
569C2C61×#250612yes—
570C2C61×#251912yes—
571C2C61×#256511yescharacteristic
572C2C61×#261111yescharacteristic
573C2C61×#263411yescharacteristic
574C2C61×#285111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C61C24
commutator, Frattini, cosocleC24C24×C61
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
241. C2×C4×C61#125semidirect
242. #15071C61direct (N × H)
326. C61#15071direct (N × H)
529. C2×C4#125×C61semidirect

Maximal subgroups

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

subgroupindex
C22×#125×C612
C2×C61×#3872
C2×C4×C4⋊C4×C612
C61×#14442
C61×#14422
C61×#14412
C61×#14362
C61×#14342
#1507161