Finite Group Databasebeta

← C61×#15342 #29518775

Subgroups — C61×#15342 #29518775

Order 15616.

1838 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#397C212yesmaximal, complemented
2C2×C61×#390C211yesmaximal, characteristic
3C2×C61×#386C211yesmaximal, characteristic
4C2×C61×#385C212yesmaximal, complemented
5C61×#1444C212yesmaximal, complemented
6C61×#1440C212yesmaximal, complemented
7C61×#1439C212yesmaximal, complemented
8C61×#1438C211yesmaximal, characteristic
9C61×#1434C212yesmaximal, complemented
10C23×D4×C61C2211yescharacteristic
11C24×C4×C61C2211yescharacteristic
12C22×C4⋊C4×C61C2212yes—
13C22×C4⋊C4×C61C2212yes—
14C22×C4⋊C4×C61C2211yescharacteristic
15C22×C22⋊C4×C61C2212yes—
16C22×C22⋊C4×C61C2211yescharacteristic
17C22×C22⋊C4×C61C2212yes—
18C22×C22⋊C4×C61C2212yes—
19C22×C22⋊C4×C61C2212yes—
20C22×C22⋊C4×C61C2211yescharacteristic
21C22×C22⋊C4×C61C2211yescharacteristic
22C22×C42×C61C2211yescharacteristic
23C61×#397—28no—
24C61×#390—24no—
25C61×#386—24no—
26C61×#385—28no—
27C2×#95×C61C2212yes—
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2212yes—
33C2×#95×C61C2212yes—
34C2×#95×C61C2212yes—
35C25×C61C2311yescharacteristic
36C25×C61C2311yescharacteristic
37C22×D4×C61—24no—
38C22×D4×C61—24no—
39C22×D4×C61—41no—
40C22×D4×C61—41no—
41C22×D4×C61D414yes—
42C23×C4×C61—22no—
43C23×C4×C61—24no—
44C23×C4×C61—42no—
45C23×C4×C61—41no—
46C23×C4×C61—41no—
47C23×C4×C61C2312yes—
48C23×C4×C61C2312yes—
49C23×C4×C61C2312yes—
50C23×C4×C61C2312yes—
51C23×C4×C61C2312yes—
52C23×C4×C61C2312yes—
53C23×C4×C61C2311yescharacteristic
54C2×C4⋊C4×C61—24no—
55C2×C4⋊C4×C61—24no—
56C2×C4⋊C4×C61—22no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—42no—
59C2×C4⋊C4×C61—41no—
60C2×C4⋊C4×C61—41no—
61C2×C4⋊C4×C61—42no—
62C2×C4⋊C4×C61—42no—
63C2×C22⋊C4×C61—24no—
64C2×C22⋊C4×C61—24no—
65C2×C22⋊C4×C61—24no—
66C2×C22⋊C4×C61—24no—
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61—24no—
69C2×C22⋊C4×C61—24no—
70C2×C22⋊C4×C61—24no—
71C2×C22⋊C4×C61—24no—
72C2×C22⋊C4×C61—24no—
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—41no—
77C2×C22⋊C4×C61—41no—
78C2×C22⋊C4×C61—42no—
79C2×C22⋊C4×C61—42no—
80C2×C22⋊C4×C61—42no—
81C2×C42×C61—22no—
82C2×C42×C61—24no—
83#95×C61—24no—
84#95×C61—24no—
85#95×C61—24no—
86#95×C61—24no—
87#95×C61—24no—
88#95×C61—24no—
89#95×C61—42no—
90#95×C61D418yescomplemented
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—21no—
104C24×C61—21no—
105C24×C61—22no—
106C24×C61C2×D412yes—
107C24×C61C2×D412yes—
108C24×C61#4114yes—
109C24×C61C2411yescharacteristic
110C2×D4×C61—28no—
111C2×D4×C61—28no—
112C2×D4×C61—44no—
113C2×D4×C61—48no—
114C2×D4×C61—48no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
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—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×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—22no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—44no—
157C22×C4×C61—48no—
158C22×C4×C61—48no—
159C22×C4×C61—44no—
160C22×C4×C61—44no—
161C22×C4×C61C2×D414yes—
162C22×C4×C61C2×D414yes—
163C22×C4×C61C2×D414yes—
164C22×C4×C61C2×D412yes—
165C22×C4×C61#4114yes—
166C22×C4×C61#4114yes—
167C4⋊C4×C61—48no—
168C4⋊C4×C61—44no—
169C4⋊C4×C61—48no—
170C22⋊C4×C61—216no—
171C22⋊C4×C61—44no—
172C22⋊C4×C61—44no—
173C22⋊C4×C61—48no—
174C22⋊C4×C61—48no—
175C22⋊C4×C61—48no—
176C22⋊C4×C61—44no—
177C42×C61—44no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—22no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—24no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—42no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—41no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—41no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61#12314yes—
220C23×C61C22×D411yescharacteristic
221C23×C61C22×D412yes—
222C23×C61C2×#4112yes—
223C23×C61C2×#4112yes—
224C23×C61C2×#4112yes—
225C23×C61H(5,𝔽2)11yescharacteristic
226C23×C61H(5,𝔽2)11yescharacteristic
227C23×C61H(5,𝔽2)12yes—
228C23×C612−1+412yes—
229D4×C61—416no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
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—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
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—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#12314yes—
278#15342C6111yesmaximal, characteristic, direct product factor, Sylow
279C22×C61—28no—
280C22×C61—24no—
281C22×C61—44no—
282C22×C61—42no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—42no—
289C22×C61—44no—
290C22×C61—44no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61—82no—
297C22×C61C2×#12311yescharacteristic
298C22×C61C2×#12311yescharacteristic
299C22×C61#52512yes—
300C22×C61#53212yes—
301C22×C61#53411yescharacteristic
302C22×C61#53512yes—
303C22×C61#53912yes—
304C22×C61#54212yes—
305C22×C61#54212yes—
306C22×C61D4212yes—
307C22×C61D4211yescharacteristic
308C22×C61#54712yes—
309C22×C61#54712yes—
310C22×C61#54812yes—
311C22×C61#55111yescharacteristic
312C22×C61#55112yes—
313C22×C61#55112yes—
314C22×C61#55312yes—
315C22×C61#55312yes—
316C22×C61#56011yescharacteristic
317C22×C61#56311yescharacteristic
318C4×C61—24no—
319C4×C61—44no—
320C4×C61—44no—
321C4×C61—44no—
322C4×C61—44no—
323C4×C61—82no—
324C4×C61—82no—
325C2×#397C2×C6112yescomplemented
326C2×#390C2×C6111yescharacteristic
327C2×#386C2×C6111yescharacteristic
328C2×#385C2×C6112yescomplemented
329#1444C2×C6112yescomplemented
330#1440C2×C6112yescomplemented
331#1439C2×C6112yescomplemented
332#1438C2×C6111yescharacteristic
333#1434C2×C6112yescomplemented
334C2×C61—44no—
335C2×C61—82no—
336C2×C61#242212yes—
337C2×C61#244312yes—
338C2×C61#246112yes—
339C2×C61#248311yescharacteristic
340C2×C61#260312yes—
341C2×C61#266911yescharacteristic
342C2×C61#271711yescharacteristic
343C2×C61#272312yes—
344C2×C61#275712yes—
345C23×D4C22×C6111yescharacteristic
346C24×C4C22×C6111yescharacteristic
347C22×C4⋊C4C22×C6112yes—
348C22×C4⋊C4C22×C6112yes—
349C22×C4⋊C4C22×C6111yescharacteristic
350C22×C22⋊C4C22×C6112yes—
351C22×C22⋊C4C22×C6111yescharacteristic
352C22×C22⋊C4C22×C6111yescharacteristic
353C22×C22⋊C4C22×C6112yes—
354C22×C22⋊C4C22×C6112yes—
355C22×C22⋊C4C22×C6111yescharacteristic
356C22×C22⋊C4C22×C6112yes—
357C22×C42C22×C6111yescharacteristic
358#397—28no—
359#390—24no—
360#386—24no—
361#385—28no—
362C2×#95C22×C6112yes—
363C2×#95C22×C6112yes—
364C2×#95C22×C6112yes—
365C2×#95C22×C6112yes—
366C2×#95C22×C6112yes—
367C2×#95C22×C6112yes—
368C2×#95C22×C6112yes—
369C2×#95C22×C6112yes—
370C61#1534211yescharacteristic, direct product factor, Sylow
371C25C23×C6111yescharacteristic
372C25C23×C6111yescharacteristic
373C22×D4—24no—
374C22×D4—24no—
375C22×D4—41no—
376C22×D4—41no—
377C22×D4D4×C6114yes—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—42no—
381C23×C4—41no—
382C23×C4—41no—
383C23×C4C23×C6112yes—
384C23×C4C23×C6112yes—
385C23×C4C23×C6112yes—
386C23×C4C23×C6112yes—
387C23×C4C23×C6112yes—
388C23×C4C23×C6112yes—
389C23×C4C23×C6111yescharacteristic
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—42no—
395C2×C4⋊C4—41no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—42no—
398C2×C4⋊C4—42no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—41no—
413C2×C22⋊C4—41no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C42—22no—
418C2×C42—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—42no—
426#95D4×C6118yescomplemented
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—21no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—21no—
440C24—21no—
441C24—22no—
442C24C2×D4×C6112yes—
443C24C2×D4×C6112yes—
444C24#41×C6114yes—
445C24C24×C6111yescharacteristic
446C2×D4—28no—
447C2×D4—28no—
448C2×D4—44no—
449C2×D4—48no—
450C2×D4—48no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
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—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
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—22no—
491C22×C4—22no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4C2×D4×C6114yes—
498C22×C4C2×D4×C6114yes—
499C22×C4C2×D4×C6114yes—
500C22×C4C2×D4×C6112yes—
501C22×C4#41×C6114yes—
502C22×C4#41×C6114yes—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C22⋊C4—216no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C42—44no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23#123×C6114yes—
556C23C22×D4×C6112yes—
557C23C22×D4×C6111yescharacteristic
558C23C2×#41×C6112yes—
559C23C2×#41×C6112yes—
560C23C2×#41×C6112yes—
561C23H(5,𝔽2)×C6111yescharacteristic
562C23H(5,𝔽2)×C6111yescharacteristic
563C23H(5,𝔽2)×C6112yes—
564C232−1+4×C6112yes—
565D4—416no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
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—42no—
603C2×C4—42no—
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#123×C6114yes—
614C22—28no—
615C22—24no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×#123×C6111yescharacteristic
633C22C2×#123×C6111yescharacteristic
634C22C61×#52512yes—
635C22C61×#53212yes—
636C22C61×#53411yescharacteristic
637C22C61×#53512yes—
638C22C61×#53912yes—
639C22C61×#54212yes—
640C22C61×#54212yes—
641C22D42×C6112yes—
642C22D42×C6111yescharacteristic
643C22C61×#54712yes—
644C22C61×#54712yes—
645C22C61×#54812yes—
646C22C61×#55111yescharacteristic
647C22C61×#55112yes—
648C22C61×#55112yes—
649C22C61×#55312yes—
650C22C61×#55312yes—
651C22C61×#56011yescharacteristic
652C22C61×#56311yescharacteristic
653C4—24no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—82no—
660C2—44no—
661C2—82no—
662C2C61×#242212yes—
663C2C61×#244312yes—
664C2C61×#246112yes—
665C2C61×#248311yescharacteristic
666C2C61×#260312yes—
667C2C61×#266911yescharacteristic
668C2C61×#271711yescharacteristic
669C2C61×#272312yes—
670C2C61×#275712yes—
subgroupG/H
center, socleC24×C61C24
commutator, Frattini, cosocleC24C24×C61
prime radicalC23×D4×C61C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C61×#397C2semidirect
4. C2×C61×#385C2semidirect
5. C61×#1444C2semidirect
6. C61×#1440C2semidirect
7. C61×#1439C2semidirect
9. C61×#1434C2semidirect
90. #95×C61D4semidirect
278. #15342C61direct (N × H)
325. C2×#397C2×C61semidirect
328. C2×#385C2×C61semidirect
329. #1444C2×C61semidirect
330. #1440C2×C61semidirect
331. #1439C2×C61semidirect
333. #1434C2×C61semidirect
370. C61#15342direct (N × H)
426. #95D4×C61semidirect

Maximal subgroups

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

subgroupindex
C2×C61×#3972
C2×C61×#3902
C2×C61×#3862
C2×C61×#3852
C61×#14442
C61×#14402
C61×#14392
C61×#14382
C61×#14342
#1534261