Finite Group Databasebeta

← C61×#15110 #29518543

Subgroups — C61×#15110 #29518543

Order 15616.

1334 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#388C211yesmaximal, characteristic
2C2×C4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
3C61×#1443C212yesmaximal, complemented
4C61×#1442C211yesmaximal, characteristic, complemented
5C61×#1442C211yesmaximal, characteristic, complemented
6C61×#1440C212yesmaximal
7C61×#1439C211yesmaximal, characteristic
8C61×#1439C211yesmaximal, characteristic
9C61×#1437C212yesmaximal
10C61×#1436C212yesmaximal, complemented
11C61×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C61C2211yescharacteristic
13C22×C4⋊C4×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2211yescharacteristic
15C22×C22⋊C4×C61C2211yescharacteristic
16C22×C22⋊C4×C61C2211yescharacteristic
17C22×C22⋊C4×C61C2211yescharacteristic
18C22×C22⋊C4×C61C2212yes—
19C22×C22⋊C4×C61C2212yes—
20C22×C42×C61C2211yescharacteristic
21C22×C42×C61C2212yes—
22C61×#388—24no—
23C4×C4⋊C4×C61—24no—
24C2×#95×C61C2212yes—
25C2×#95×C61C2211yescharacteristic
26C2×#95×C61C2211yescharacteristic
27C2×#95×C61C2212yes—
28C2×#95×C61C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C2×#95×C61C2212yes—
33C2×#95×C61C2212yes—
34C2×#95×C61C2212yes—
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C25×C61C2311yescharacteristic
38C23×C4×C61C2311yescharacteristic
39C23×C4×C61C2312yes—
40C23×C4×C61C2312yes—
41C23×C4×C61C2311yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C23×C4×C61C2312yes—
45C23×C4×C61C2312yes—
46C23×C4×C61C2311yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C2×C4⋊C4×C61—22no—
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61—24no—
51C2×C4⋊C4×C61—24no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61—24no—
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61—22no—
56C2×C22⋊C4×C61—22no—
57C2×C22⋊C4×C61—24no—
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61—24no—
61C2×C22⋊C4×C61—24no—
62C2×C22⋊C4×C61—22no—
63C2×C22⋊C4×C61—22no—
64C2×C22⋊C4×C61—24no—
65C2×C22⋊C4×C61—42no—
66C2×C22⋊C4×C61—42no—
67C2×C22⋊C4×C61—42no—
68C2×C42×C61—22no—
69C2×C42×C61—22no—
70C2×C42×C61—24no—
71C2×C42×C61—42no—
72C2×C42×C61—42no—
73C2×C42×C61D414yes—
74#95×C61—24no—
75#95×C61—22no—
76#95×C61—22no—
77#95×C61—24no—
78#95×C61—24no—
79#95×C61—24no—
80#95×C61—22no—
81#95×C61—22no—
82#95×C61—24no—
83#95×C61—42no—
84#95×C61—42no—
85#95×C61—42no—
86#95×C61—42no—
87C24×C61—21no—
88C24×C61—21no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61#4112yes—
98C24×C61C2411yescharacteristic
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
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—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—21no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61#4112yes—
154C22×C4×C61#4114yes—
155C22×C4×C61#4114yes—
156C22×C4×C61#4112yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4114yes—
159C4⋊C4×C61—28no—
160C4⋊C4×C61—28no—
161C4⋊C4×C61—44no—
162C22⋊C4×C61—28no—
163C22⋊C4×C61—48no—
164C22⋊C4×C61—48no—
165C22⋊C4×C61—44no—
166C22⋊C4×C61—44no—
167C42×C61—28no—
168C42×C61—48no—
169C23×C61—22no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—41no—
189C23×C61—42no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61C22×D411yescharacteristic
193C23×C61C2×#4111yescharacteristic
194C23×C61C2×#4111yescharacteristic
195C23×C61C2×#4112yes—
196C23×C61C2×#4111yescharacteristic
197C23×C61C2×#4112yes—
198C23×C61C2×#4111yescharacteristic
199C23×C61C2×#4112yes—
200C23×C61H(5,𝔽2)12yes—
201C23×C61H(5,𝔽2)11yescharacteristic
202C23×C612−1+411yescharacteristic
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—42no—
223C2×C4×C61—41no—
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—41no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
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#12314yes—
262C2×C4×C61#12414yes—
263#15110C6111yesmaximal, characteristic, direct product factor, Sylow
264C22×C61—44no—
265C22×C61—44no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×#12311yescharacteristic
276C22×C61#52511yescharacteristic
277C22×C61#52511yescharacteristic
278C22×C61#52511yescharacteristic
279C22×C61C2×#12411yescharacteristic
280C22×C61#52912yes—
281C22×C61#52912yes—
282C22×C61#52912yes—
283C22×C61#52912yes—
284C22×C61#53511yescharacteristic
285C22×C61#53711yescharacteristic
286C22×C61#53812yes—
287C22×C61#53912yes—
288C22×C61#55112yes—
289C22×C61#55211yescharacteristic
290C22×C61#55312yes—
291C22×C61#55312yes—
292C22×C61#55312yes—
293C22×C61#55311yescharacteristic
294C22×C61#55511yescharacteristic
295C22×C61#55612yes—
296C22×C61#55611yescharacteristic
297C22×C61#55912yes—
298C4×C61—24no—
299C4×C61—24no—
300C4×C61—44no—
301C4×C61—44no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—81no—
305C4×C61—82no—
306C4×C61—82no—
307C4×C61—81no—
308C2×#388C2×C6111yescharacteristic
309C2×C4×C4⋊C4C2×C6111yescharacteristic, complemented
310#1443C2×C6112yescomplemented
311#1442C2×C6111yescharacteristic, complemented
312#1442C2×C6111yescharacteristic, complemented
313#1440C2×C6112yes—
314#1439C2×C6111yescharacteristic
315#1439C2×C6111yescharacteristic
316#1437C2×C6112yes—
317#1436C2×C6112yescomplemented
318#1434C2×C6111yescharacteristic, complemented
319C2×C61—82no—
320C2×C61#239811yescharacteristic
321C2×C61#248511yescharacteristic
322C2×C61#250411yescharacteristic
323C2×C61#257312yes—
324C2×C61#259211yescharacteristic
325C2×C61#259312yes—
326C2×C61#259412yes—
327C2×C61#259512yes—
328C2×C61#259811yescharacteristic
329C2×C61#263611yescharacteristic
330C2×C61#285311yescharacteristic
331C22×C4⋊C4C22×C6111yescharacteristic
332C22×C4⋊C4C22×C6111yescharacteristic
333C22×C4⋊C4C22×C6111yescharacteristic
334C22×C22⋊C4C22×C6111yescharacteristic
335C22×C22⋊C4C22×C6111yescharacteristic
336C22×C22⋊C4C22×C6111yescharacteristic
337C22×C22⋊C4C22×C6112yes—
338C22×C22⋊C4C22×C6112yes—
339C22×C42C22×C6111yescharacteristic
340C22×C42C22×C6112yes—
341#388—24no—
342C4×C4⋊C4—24no—
343C2×#95C22×C6112yes—
344C2×#95C22×C6111yescharacteristic
345C2×#95C22×C6111yescharacteristic
346C2×#95C22×C6112yes—
347C2×#95C22×C6112yes—
348C2×#95C22×C6112yes—
349C2×#95C22×C6112yes—
350C2×#95C22×C6111yescharacteristic
351C2×#95C22×C6111yescharacteristic
352C2×#95C22×C6112yes—
353C2×#95C22×C6112yes—
354C2×#95C22×C6112yes—
355C2×#95C22×C6112yes—
356C61#1511011yescharacteristic, direct product factor, Sylow
357C25C23×C6111yescharacteristic
358C23×C4C23×C6111yescharacteristic
359C23×C4C23×C6111yescharacteristic
360C23×C4C23×C6112yes—
361C23×C4C23×C6112yes—
362C23×C4C23×C6111yescharacteristic
363C23×C4C23×C6111yescharacteristic
364C23×C4C23×C6112yes—
365C23×C4C23×C6112yes—
366C23×C4C23×C6111yescharacteristic
367C23×C4C23×C6111yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—24no—
391C2×C42—42no—
392C2×C42—42no—
393C2×C42D4×C6114yes—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—42no—
404#95—42no—
405#95—42no—
406#95—42no—
407C24—21no—
408C24—22no—
409C24—21no—
410C24—21no—
411C24—21no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—21no—
416C24—21no—
417C24#41×C6112yes—
418C24C24×C6111yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
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—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4C2×D4×C6112yes—
471C22×C4C2×D4×C6112yes—
472C22×C4C2×D4×C6112yes—
473C22×C4#41×C6112yes—
474C22×C4#41×C6114yes—
475C22×C4#41×C6114yes—
476C22×C4#41×C6112yes—
477C22×C4#41×C6112yes—
478C22×C4#41×C6114yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—44no—
482C22⋊C4—28no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C42—28no—
488C42—48no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23C22×D4×C6111yescharacteristic
513C23C2×#41×C6112yes—
514C23C2×#41×C6111yescharacteristic
515C23C2×#41×C6112yes—
516C23C2×#41×C6111yescharacteristic
517C23C2×#41×C6111yescharacteristic
518C23C2×#41×C6111yescharacteristic
519C23C2×#41×C6112yes—
520C23H(5,𝔽2)×C6112yes—
521C23H(5,𝔽2)×C6111yescharacteristic
522C232−1+4×C6111yescharacteristic
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
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—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4#123×C6114yes—
582C2×C4#124×C6114yes—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—82no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C2×#123×C6111yescharacteristic
595C22C61×#52511yescharacteristic
596C22C61×#52511yescharacteristic
597C22C61×#52511yescharacteristic
598C22C2×#124×C6111yescharacteristic
599C22C61×#52912yes—
600C22C61×#52912yes—
601C22C61×#52912yes—
602C22C61×#52912yes—
603C22C61×#53511yescharacteristic
604C22C61×#53711yescharacteristic
605C22C61×#53812yes—
606C22C61×#53912yes—
607C22C61×#55112yes—
608C22C61×#55211yescharacteristic
609C22C61×#55312yes—
610C22C61×#55312yes—
611C22C61×#55312yes—
612C22C61×#55311yescharacteristic
613C22C61×#55511yescharacteristic
614C22C61×#55612yes—
615C22C61×#55611yescharacteristic
616C22C61×#55912yes—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—81no—
624C4—82no—
625C4—82no—
626C4—81no—
627C2—82no—
628C2C61×#239811yescharacteristic
629C2C61×#248511yescharacteristic
630C2C61×#250411yescharacteristic
631C2C61×#257312yes—
632C2C61×#259211yescharacteristic
633C2C61×#259312yes—
634C2C61×#259412yes—
635C2C61×#259512yes—
636C2C61×#259811yescharacteristic
637C2C61×#263611yescharacteristic
638C2C61×#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
2. C2×C4×C4⋊C4×C61C2semidirect
3. C61×#1443C2semidirect
4. C61×#1442C2semidirect
5. C61×#1442C2semidirect
10. C61×#1436C2semidirect
11. C61×#1434C2semidirect
263. #15110C61direct (N × H)
309. C2×C4×C4⋊C4C2×C61semidirect
310. #1443C2×C61semidirect
311. #1442C2×C61semidirect
312. #1442C2×C61semidirect
317. #1436C2×C61semidirect
318. #1434C2×C61semidirect
356. C61#15110direct (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×#14432
C61×#14422
C61×#14402
C61×#14392
C61×#14372
C61×#14362
C61×#14342
#1511061