Finite Group Databasebeta

← C61×#26750 #29530183

Subgroups — C61×#26750 #29530183

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#2492C211yesmaximal, characteristic
2C61×#2362C211yesmaximal, characteristic
3C61×#2056C211yesmaximal, characteristic, complemented
4C61×#2035C211yesmaximal, characteristic
5C61×#2024C211yesmaximal, characteristic, complemented
6C61×#2001C211yesmaximal, characteristic
7C61×#1987C211yesmaximal, characteristic, complemented
8C61×#1935C211yesmaximal, characteristic, complemented
9C61×#1927C211yesmaximal, characteristic, complemented
10C61×#1924C211yesmaximal, characteristic
11C61×#1920C211yesmaximal, characteristic, complemented
12C61×#1878C211yesmaximal, characteristic, complemented
13C61×#1873C211yesmaximal, characteristic
14C61×#1825C211yesmaximal, characteristic
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×Q16×C61C2211yescharacteristic
17C2×C4⋊Q8×C61C2211yescharacteristic
18C2×C22⋊Q8×C61C2211yescharacteristic
19C2×C22⋊Q8×C61C2211yescharacteristic
20C2×C4×Q8×C61C2211yescharacteristic
21C2×#117×C61C2211yescharacteristic
22C2×#106×C61C2211yescharacteristic
23C2×C4⋊C8×C61C2211yescharacteristic
24C2×#103×C61C2211yescharacteristic
25C2×#103×C61C2211yescharacteristic
26C2×#103×C61C2211yescharacteristic
27C2×#103×C61C2211yescharacteristic
28C2×#103×C61C2211yescharacteristic
29C2×#103×C61C2211yescharacteristic
30C2×C22⋊C8×C61C2211yescharacteristic
31C2×C22⋊C8×C61C2211yescharacteristic
32C2×#97×C61C2211yescharacteristic
33C61×#400—21no—
34C61×#393—21no—
35C61×#391—21no—
36C61×#391C2211yescharacteristic
37C61×#391C2211yescharacteristic
38C61×#391C2211yescharacteristic
39C61×#389C2211yescharacteristic
40C61×#388—21no—
41C61×#388—21no—
42C61×#388—21no—
43C61×#387—21no—
44C61×#380C2211yescharacteristic
45C4×C4⋊C4×C61—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C4×C4⋊C4×C61C2211yescharacteristic
48C4×C22⋊C4×C61C2211yescharacteristic
49C61×#340C2211yescharacteristic
50C61×#340C2211yescharacteristic
51C61×#340C2211yescharacteristic
52C61×#340C2211yescharacteristic
53C61×#340C2211yescharacteristic
54C61×#340C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Q8×C61—21no—
60C22×Q8×C61C2311yescharacteristic
61C22×Q8×C61C2311yescharacteristic
62C23×C4×C61C2311yescharacteristic
63C2×Q16×C61—22no—
64C2×Q16×C61—41no—
65C2×Q16×C61—41no—
66C22×C8×C61C2311yescharacteristic
67C22×C8×C61C2311yescharacteristic
68C22×C8×C61C2311yescharacteristic
69C22×C8×C61C2311yescharacteristic
70C4⋊Q8×C61—22no—
71C4⋊Q8×C61—41no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—41no—
76C4×Q8×C61—22no—
77C4×Q8×C61—41no—
78#117×C61—22no—
79#117×C61—41no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61D411yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C2×C42×C61C2311yescharacteristic
103#106×C61—22no—
104C4⋊C8×C61—41no—
105#103×C61—22no—
106#103×C61—22no—
107#103×C61—22no—
108#103×C61—22no—
109#103×C61—41no—
110#103×C61—41no—
111C22⋊C8×C61—41no—
112C22⋊C8×C61—41no—
113#97×C61—41no—
114#95×C61—21no—
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C24×C61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—42no—
139C2×Q8×C61—42no—
140C2×Q8×C61—81no—
141C2×Q8×C61#4112yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162Q16×C61—44no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61—42no—
189C4⋊C4×C61#4112yes—
190C4⋊C4×C61#4112yes—
191C4⋊C4×C61#4112yes—
192C22⋊C4×C61—24no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—42no—
196C22⋊C4×C61—42no—
197C22⋊C4×C61—81no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—42no—
203C42×C61—42no—
204C42×C61—42no—
205C42×C61—81no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61C22×D411yescharacteristic
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—84no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61#13712yes—
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C8×C61—41no—
273C8×C61—41no—
274C8×C61—41no—
275C8×C61—41no—
276#26750C6111yesmaximal, characteristic, direct product factor, Sylow
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61#52511yescharacteristic
285C22×C61#54611yescharacteristic
286C22×C61#54611yescharacteristic
287C22×C61C2×#13711yescharacteristic
288C22×C61#57511yescharacteristic
289C22×C61#57711yescharacteristic
290C22×C61#57811yescharacteristic
291C4×C61—22no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61#52911yescharacteristic
305C4×C61#53811yescharacteristic
306C4×C61#55511yescharacteristic
307C4×C61#55911yescharacteristic
308#2492C2×C6111yescharacteristic
309#2362C2×C6111yescharacteristic
310#2056C2×C6111yescharacteristic, complemented
311#2035C2×C6111yescharacteristic
312#2024C2×C6111yescharacteristic, complemented
313#2001C2×C6111yescharacteristic
314#1987C2×C6111yescharacteristic, complemented
315#1935C2×C6111yescharacteristic, complemented
316#1927C2×C6111yescharacteristic, complemented
317#1924C2×C6111yescharacteristic
318#1920C2×C6111yescharacteristic, complemented
319#1878C2×C6111yescharacteristic, complemented
320#1873C2×C6111yescharacteristic
321#1825C2×C6111yescharacteristic
322#1763C2×C6111yescharacteristic, complemented
323C2×C61—81no—
324C2×C61#245911yescharacteristic
325C2×C61#311611yescharacteristic
326C2×C61#312111yescharacteristic
327C2×C61#331111yescharacteristic
328C2×C61#331211yescharacteristic
329C2×C61#331311yescharacteristic
330C2×C61#333111yescharacteristic
331C22×Q16C22×C6111yescharacteristic
332C2×C4⋊Q8C22×C6111yescharacteristic
333C2×C22⋊Q8C22×C6111yescharacteristic
334C2×C22⋊Q8C22×C6111yescharacteristic
335C2×C4×Q8C22×C6111yescharacteristic
336C2×#117C22×C6111yescharacteristic
337C2×#106C22×C6111yescharacteristic
338C2×C4⋊C8C22×C6111yescharacteristic
339C2×#103C22×C6111yescharacteristic
340C2×#103C22×C6111yescharacteristic
341C2×#103C22×C6111yescharacteristic
342C2×#103C22×C6111yescharacteristic
343C2×#103C22×C6111yescharacteristic
344C2×#103C22×C6111yescharacteristic
345C2×C22⋊C8C22×C6111yescharacteristic
346C2×C22⋊C8C22×C6111yescharacteristic
347C2×#97C22×C6111yescharacteristic
348#400—21no—
349#393—21no—
350#391—21no—
351#391C22×C6111yescharacteristic
352#391C22×C6111yescharacteristic
353#391C22×C6111yescharacteristic
354#389C22×C6111yescharacteristic
355#388—21no—
356#388—21no—
357#388—21no—
358#387—21no—
359#380C22×C6111yescharacteristic
360C4×C4⋊C4—21no—
361C4×C4⋊C4C22×C6111yescharacteristic
362C4×C4⋊C4C22×C6111yescharacteristic
363C4×C22⋊C4C22×C6111yescharacteristic
364#340C22×C6111yescharacteristic
365#340C22×C6111yescharacteristic
366#340C22×C6111yescharacteristic
367#340C22×C6111yescharacteristic
368#340C22×C6111yescharacteristic
369#340C22×C6111yescharacteristic
370#340C22×C6111yescharacteristic
371#340C22×C6111yescharacteristic
372#336C22×C6111yescharacteristic
373#336C22×C6111yescharacteristic
374C61#2675011yescharacteristic, direct product factor, Sylow
375C22×Q8—21no—
376C22×Q8C23×C6111yescharacteristic
377C22×Q8C23×C6111yescharacteristic
378C23×C4C23×C6111yescharacteristic
379C2×Q16—22no—
380C2×Q16—41no—
381C2×Q16—41no—
382C22×C8C23×C6111yescharacteristic
383C22×C8C23×C6111yescharacteristic
384C22×C8C23×C6111yescharacteristic
385C22×C8C23×C6111yescharacteristic
386C4⋊Q8—22no—
387C4⋊Q8—41no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—41no—
392C4×Q8—22no—
393C4×Q8—41no—
394#117—22no—
395#117—41no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4C23×C6111yescharacteristic
402C2×C4⋊C4C23×C6111yescharacteristic
403C2×C4⋊C4C23×C6111yescharacteristic
404C2×C4⋊C4C23×C6111yescharacteristic
405C2×C4⋊C4C23×C6111yescharacteristic
406C2×C4⋊C4C23×C6111yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4D4×C6111yescharacteristic
412C2×C22⋊C4D4×C6111yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C23×C6111yescharacteristic
418C2×C42C23×C6111yescharacteristic
419#106—22no—
420C4⋊C8—41no—
421#103—22no—
422#103—22no—
423#103—22no—
424#103—22no—
425#103—41no—
426#103—41no—
427C22⋊C8—41no—
428C22⋊C8—41no—
429#97—41no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95D4×C6111yescharacteristic
441#95D4×C6111yescharacteristic
442C24C2×D4×C6111yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8#41×C6112yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4C2×D4×C6111yescharacteristic
473C22×C4C2×D4×C6111yescharacteristic
474C22×C4C2×D4×C6111yescharacteristic
475C22×C4C2×D4×C6111yescharacteristic
476C22×C4C2×D4×C6111yescharacteristic
477C22×C4C24×C6111yescharacteristic
478Q16—44no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4#41×C6112yes—
506C4⋊C4#41×C6112yes—
507C4⋊C4#41×C6112yes—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—81no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23C22×D4×C6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—84no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4#137×C6112yes—
582C2×C4C2×#41×C6111yescharacteristic
583C2×C4C2×#41×C6111yescharacteristic
584C2×C4C2×#41×C6111yescharacteristic
585C2×C4C2×#41×C6111yescharacteristic
586C2×C4H(5,𝔽2)×C6111yescharacteristic
587C2×C4H(5,𝔽2)×C6111yescharacteristic
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22C61×#52511yescharacteristic
600C22C61×#54611yescharacteristic
601C22C61×#54611yescharacteristic
602C22C2×#137×C6111yescharacteristic
603C22C61×#57511yescharacteristic
604C22C61×#57711yescharacteristic
605C22C61×#57811yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4C61×#52911yescharacteristic
620C4C61×#53811yescharacteristic
621C4C61×#55511yescharacteristic
622C4C61×#55911yescharacteristic
623C2—81no—
624C2C61×#245911yescharacteristic
625C2C61×#311611yescharacteristic
626C2C61×#312111yescharacteristic
627C2C61×#331111yescharacteristic
628C2C61×#331211yescharacteristic
629C2C61×#331311yescharacteristic
630C2C61×#333111yescharacteristic
subgroupG/H
center, socleC23×C61C22×D4
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC24×C61C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C61 ◁ C4×C22⋊C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C61×#2056C2semidirect
5. C61×#2024C2semidirect
7. C61×#1987C2semidirect
8. C61×#1935C2semidirect
9. C61×#1927C2semidirect
11. C61×#1920C2semidirect
12. C61×#1878C2semidirect
15. C61×#1763C2semidirect
276. #26750C61direct (N × H)
310. #2056C2×C61semidirect
312. #2024C2×C61semidirect
314. #1987C2×C61semidirect
315. #1935C2×C61semidirect
316. #1927C2×C61semidirect
318. #1920C2×C61semidirect
319. #1878C2×C61semidirect
322. #1763C2×C61semidirect
374. C61#26750direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#24922
C61×#23622
C61×#20562
C61×#20352
C61×#20242
C61×#20012
C61×#19872
C61×#19352
C61×#19272
C61×#19242
C61×#19202
C61×#18782
C61×#18732
C61×#18252
C61×#17632
#2675061