Finite Group Databasebeta

← C61×#15233 #29518666

Subgroups — C61×#15233 #29518666

Order 15616.

1542 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#382C211yesmaximal, characteristic, complemented
2C2×C61×#380C211yesmaximal, characteristic
3C2×C61×#380C211yesmaximal, characteristic
4C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
5C61×#1443C214yesmaximal, complemented
6C61×#1440C214yesmaximal
7C61×C4⋊#95C211yesmaximal, characteristic, complemented
8C61×C4⋊#95C211yesmaximal, characteristic, complemented
9C61×C4⋊#95C211yesmaximal, characteristic, complemented
10C24×C4×C61C2211yescharacteristic
11C22×C4⋊C4×C61C2211yescharacteristic
12C22×C4⋊C4×C61C2211yescharacteristic
13C22×C4⋊C4×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2211yescharacteristic
15C22×C22⋊C4×C61C2212yes—
16C22×C22⋊C4×C61C2212yes—
17C22×C22⋊C4×C61C2212yes—
18C22×C42×C61C2211yescharacteristic
19C22×C42×C61C2211yescharacteristic
20C61×#382—24no—
21C61×#380—24no—
22C61×#380—24no—
23C4×C22⋊C4×C61—24no—
24C2×#95×C61C2212yes—
25C2×#95×C61C2212yes—
26C2×#95×C61C2212yes—
27C2×#95×C61C2214yes—
28C2×#95×C61C2214yes—
29C2×#95×C61C2214yes—
30C2×#95×C61C2214yes—
31C25×C61C2311yescharacteristic
32C23×C4×C61—22no—
33C23×C4×C61—22no—
34C23×C4×C61—22no—
35C23×C4×C61—22no—
36C23×C4×C61—22no—
37C23×C4×C61—22no—
38C23×C4×C61D414yes—
39C23×C4×C61C2312yes—
40C23×C4×C61C2312yes—
41C23×C4×C61C2312yes—
42C23×C4×C61C2312yes—
43C23×C4×C61C2311yescharacteristic
44C23×C4×C61C2312yes—
45C23×C4×C61C2312yes—
46C23×C4×C61C2311yescharacteristic
47C2×C4⋊C4×C61—22no—
48C2×C4⋊C4×C61—22no—
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61—22no—
51C2×C4⋊C4×C61—22no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61—22no—
54C2×C4⋊C4×C61—22no—
55C2×C4⋊C4×C61—22no—
56C2×C4⋊C4×C61—22no—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61—22no—
59C2×C22⋊C4×C61—24no—
60C2×C22⋊C4×C61—24no—
61C2×C22⋊C4×C61—24no—
62C2×C22⋊C4×C61—24no—
63C2×C22⋊C4×C61—24no—
64C2×C22⋊C4×C61—28no—
65C2×C22⋊C4×C61—44no—
66C2×C42×C61—22no—
67C2×C42×C61—22no—
68C2×C42×C61—22no—
69C2×C42×C61—22no—
70C2×C42×C61—22no—
71C2×C42×C61D414yes—
72#95×C61—24no—
73#95×C61—24no—
74#95×C61—24no—
75#95×C61—28no—
76#95×C61—28no—
77#95×C61—44no—
78#95×C61—44no—
79C24×C61—21no—
80C24×C61—22no—
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—22no—
87C24×C61—22no—
88C24×C61C2×D412yes—
89C24×C61#4114yes—
90C24×C61C2411yescharacteristic
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
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—28no—
110C22×C4×C61—28no—
111C22×C4×C61—28no—
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—24no—
119C22×C4×C61—24no—
120C22×C4×C61—24no—
121C22×C4×C61—24no—
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—24no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—44no—
133C22×C4×C61—44no—
134C22×C4×C61—44no—
135C22×C4×C61—44no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D414yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61#4114yes—
141C22×C4×C61#4112yes—
142C22×C4×C61#4112yes—
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C4⋊C4×C61—44no—
147C4⋊C4×C61—44no—
148C4⋊C4×C61—44no—
149C4⋊C4×C61—44no—
150C22⋊C4×C61—216no—
151C22⋊C4×C61—48no—
152C22⋊C4×C61—48no—
153C42×C61—28no—
154C42×C61—44no—
155C23×C61—22no—
156C23×C61—22no—
157C23×C61—22no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—24no—
164C23×C61—24no—
165C23×C61—24no—
166C23×C61—24no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61#12414yes—
178C23×C61C22×D411yescharacteristic
179C23×C61C22×D411yescharacteristic
180C23×C61C2×#4111yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4111yescharacteristic
183C23×C61C2×#4111yescharacteristic
184C23×C61C2×#4111yescharacteristic
185C23×C61C2×#4112yes—
186C23×C61C2×#4112yes—
187C23×C61H(5,𝔽2)12yes—
188C23×C612−1+412yes—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—416no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—44no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—44no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—44no—
231C2×C4×C61#12114yes—
232C2×C4×C61#12314yes—
233C2×C4×C61#12314yes—
234C2×C4×C61#12314yes—
235C2×C4×C61#12414yes—
236C2×C4×C61#12614yes—
237C2×C4×C61#12614yes—
238#15233C6111yesmaximal, characteristic, direct product factor, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—44no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—44no—
246C22×C61—44no—
247C22×C61—44no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61C2×#12111yescharacteristic
251C22×C61C2×#12311yescharacteristic
252C22×C61C2×#12311yescharacteristic
253C22×C61C2×#12311yescharacteristic
254C22×C61#52512yes—
255C22×C61C2×#12411yescharacteristic
256C22×C61C2×#12411yescharacteristic
257C22×C61C2×#12611yescharacteristic
258C22×C61C2×#12611yescharacteristic
259C22×C61#52912yes—
260C22×C61#53211yescharacteristic
261C22×C61#53511yescharacteristic
262C22×C61#53611yescharacteristic
263C22×C61#53712yes—
264C22×C61#54212yes—
265C22×C61#55212yes—
266C22×C61#55212yes—
267C22×C61#55212yes—
268C22×C61#55312yes—
269C22×C61#55312yes—
270C22×C61#55312yes—
271C22×C61#55512yes—
272C22×C61#55612yes—
273C4×C61—44no—
274C4×C61—42no—
275C4×C61—44no—
276C4×C61—82no—
277C4×C61—82no—
278C4×C61—82no—
279C4×C61—82no—
280C4×C61#39718yes—
281C2×#382C2×C6111yescharacteristic, complemented
282C2×#380C2×C6111yescharacteristic
283C2×#380C2×C6111yescharacteristic
284C2×C4×C22⋊C4C2×C6111yescharacteristic
285#1443C2×C6114yescomplemented
286#1440C2×C6114yes—
287C4⋊#95C2×C6111yescharacteristic, complemented
288C4⋊#95C2×C6111yescharacteristic, complemented
289C4⋊#95C2×C6111yescharacteristic, complemented
290C2×C61—44no—
291C2×C61C2×#39711yescharacteristic
292C2×C61#248512yes—
293C2×C61#249412yes—
294C2×C61#249512yes—
295C2×C61#249911yescharacteristic
296C2×C61#250011yescharacteristic
297C2×C61#250311yescharacteristic
298C2×C61#250411yescharacteristic
299C2×C61#256212yes—
300C2×C61#262011yescharacteristic
301C2×C61#285011yescharacteristic
302C24×C4C22×C6111yescharacteristic
303C22×C4⋊C4C22×C6111yescharacteristic
304C22×C4⋊C4C22×C6111yescharacteristic
305C22×C4⋊C4C22×C6111yescharacteristic
306C22×C4⋊C4C22×C6111yescharacteristic
307C22×C22⋊C4C22×C6112yes—
308C22×C22⋊C4C22×C6112yes—
309C22×C22⋊C4C22×C6112yes—
310C22×C42C22×C6111yescharacteristic
311C22×C42C22×C6111yescharacteristic
312#382—24no—
313#380—24no—
314#380—24no—
315C4×C22⋊C4—24no—
316C2×#95C22×C6114yes—
317C2×#95C22×C6114yes—
318C2×#95C22×C6112yes—
319C2×#95C22×C6112yes—
320C2×#95C22×C6112yes—
321C2×#95C22×C6114yes—
322C2×#95C22×C6114yes—
323C61#1523311yescharacteristic, direct product factor, Sylow
324C25C23×C6111yescharacteristic
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4D4×C6114yes—
332C23×C4C23×C6112yes—
333C23×C4C23×C6112yes—
334C23×C4C23×C6112yes—
335C23×C4C23×C6112yes—
336C23×C4C23×C6112yes—
337C23×C4C23×C6111yescharacteristic
338C23×C4C23×C6112yes—
339C23×C4C23×C6111yescharacteristic
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—44no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42D4×C6114yes—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—28no—
369#95—28no—
370#95—44no—
371#95—44no—
372C24—21no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24C2×D4×C6112yes—
382C24#41×C6114yes—
383C24C24×C6111yescharacteristic
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4C2×D4×C6112yes—
430C22×C4C2×D4×C6114yes—
431C22×C4C2×D4×C6112yes—
432C22×C4C2×D4×C6112yes—
433C22×C4#41×C6114yes—
434C22×C4#41×C6112yes—
435C22×C4#41×C6112yes—
436C22×C4#41×C6112yes—
437C22×C4#41×C6112yes—
438C22×C4#41×C6112yes—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C22⋊C4—216no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C42—28no—
447C42—44no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23#124×C6114yes—
471C23C22×D4×C6111yescharacteristic
472C23C22×D4×C6111yescharacteristic
473C23C2×#41×C6112yes—
474C23C2×#41×C6111yescharacteristic
475C23C2×#41×C6111yescharacteristic
476C23C2×#41×C6111yescharacteristic
477C23C2×#41×C6112yes—
478C23C2×#41×C6111yescharacteristic
479C23C2×#41×C6111yescharacteristic
480C23H(5,𝔽2)×C6112yes—
481C232−1+4×C6112yes—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—416no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4#121×C6114yes—
525C2×C4#123×C6114yes—
526C2×C4#123×C6114yes—
527C2×C4#123×C6114yes—
528C2×C4#124×C6114yes—
529C2×C4#126×C6114yes—
530C2×C4#126×C6114yes—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22C2×#121×C6111yescharacteristic
543C22C2×#123×C6111yescharacteristic
544C22C2×#123×C6111yescharacteristic
545C22C2×#123×C6111yescharacteristic
546C22C61×#52512yes—
547C22C2×#124×C6111yescharacteristic
548C22C2×#124×C6111yescharacteristic
549C22C2×#126×C6111yescharacteristic
550C22C2×#126×C6111yescharacteristic
551C22C61×#52912yes—
552C22C61×#53211yescharacteristic
553C22C61×#53511yescharacteristic
554C22C61×#53611yescharacteristic
555C22C61×#53712yes—
556C22C61×#54212yes—
557C22C61×#55212yes—
558C22C61×#55212yes—
559C22C61×#55212yes—
560C22C61×#55312yes—
561C22C61×#55312yes—
562C22C61×#55312yes—
563C22C61×#55512yes—
564C22C61×#55612yes—
565C4—44no—
566C4—42no—
567C4—44no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4C61×#39718yes—
573C2—44no—
574C2C2×C61×#39711yescharacteristic
575C2C61×#248512yes—
576C2C61×#249412yes—
577C2C61×#249512yes—
578C2C61×#249911yescharacteristic
579C2C61×#250011yescharacteristic
580C2C61×#250311yescharacteristic
581C2C61×#250411yescharacteristic
582C2C61×#256212yes—
583C2C61×#262011yescharacteristic
584C2C61×#285011yescharacteristic
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
1. C2×C61×#382C2semidirect
5. C61×#1443C2semidirect
7. C61×C4⋊#95C2semidirect
8. C61×C4⋊#95C2semidirect
9. C61×C4⋊#95C2semidirect
238. #15233C61direct (N × H)
281. C2×#382C2×C61semidirect
285. #1443C2×C61semidirect
287. C4⋊#95C2×C61semidirect
288. C4⋊#95C2×C61semidirect
289. C4⋊#95C2×C61semidirect
323. C61#15233direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#3822
C2×C61×#3802
C2×C4×C22⋊C4×C612
C61×#14432
C61×#14402
C61×C4⋊#952
#1523361