Finite Group Databasebeta

← C61×#29611 #29533044

Subgroups — C61×#29611 #29533044

Order 15616.

1282 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#536C211yesmaximal, characteristic, complemented
2C61×#3094C211yesmaximal, characteristic, complemented
3C61×#3006C211yesmaximal, characteristic
4C61×#2436C211yesmaximal, characteristic, complemented
5C61×#2038C214yesmaximal, complemented
6C61×#2003C214yesmaximal, complemented
7C61×#1826C211yesmaximal, characteristic, complemented
8C61×#1789C211yesmaximal, characteristic, complemented
9C61×#1784C211yesmaximal, characteristic, complemented
10C22×#41×C61C2211yescharacteristic
11C23×Q8×C61C2211yescharacteristic
12C2×AGL(1,ℤ/8)×C61C2212yes—
13C61×#536—22no—
14C61×#536—22no—
15C61×#536—22no—
16C61×#536C2214yes—
17C2×C4⋊Q8×C61—21no—
18C2×#124×C61—21no—
19C2×#124×C61C2212yescomplemented
20C2×#123×C61—22no—
21C2×C22⋊Q8×C61—21no—
22C2×C22⋊Q8×C61—21no—
23C2×#121×C61C2211yescharacteristic
24C2×#117×C61C2211yescharacteristic
25C61×#482—22no—
26C61×#481—21no—
27C61×#480—21no—
28C61×#449—22no—
29C61×#448—21no—
30C61×#447—21no—
31C61×#428C2212yescomplemented
32C2×#104×C61C2214yes—
33C61×#418C2212yescomplemented
34C61×#417C2212yes—
35C2×#101×C61C2214yescomplemented
36C61×#407C2211yescharacteristic
37C61×#407C2211yescharacteristic
38C61×#393—22no—
39C61×#391C2212yescomplemented
40C61×#386—22no—
41C4×C22⋊C4×C61C2211yescharacteristic
42C61×#339C2214yescomplemented
43C2×#41×C61—21no—
44C2×#41×C61—21no—
45C2×#41×C61—44no—
46C2×#41×C61D412yes—
47C2×#41×C61C2312yes—
48C22×Q8×C61—21no—
49C22×Q8×C61—21no—
50C22×Q8×C61—21no—
51C22×Q8×C61—21no—
52C22×Q8×C61—44no—
53C22×Q8×C61D411yescharacteristic
54C22×Q8×C61D412yes—
55C22×Q8×C61C2312yes—
56C22×D4×C61—21no—
57C22×D4×C61—21no—
58C22×D4×C61D411yescharacteristic
59C22×D4×C61C2311yescharacteristic
60C23×C4×C61—21no—
61C23×C4×C61D412yes—
62C23×C4×C61C2311yescharacteristic
63AGL(1,ℤ/8)×C61—44no—
64C2×SD16×C61—22no—
65C2×D8×C61—22no—
66C2×M16×C61C2312yes—
67C2×M16×C61C2312yes—
68C4⋊Q8×C61—24no—
69C4⋊Q8×C61—24no—
70#124×C61—24no—
71#124×C61—24no—
72#124×C61—44no—
73#123×C61—28no—
74#123×C61—28no—
75C22⋊Q8×C61—24no—
76C22⋊Q8×C61—24no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79#121×C61—21no—
80#121×C61—21no—
81#121×C61—21no—
82#121×C61—21no—
83#117×C61—21no—
84#117×C61—21no—
85#117×C61D412yes—
86#117×C61C2312yes—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61D411yescharacteristic
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—24no—
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61C2312yes—
100#107×C61—22no—
101#106×C61—22no—
102#104×C61—44no—
103#103×C61—22no—
104#102×C61—22no—
105#102×C61—22no—
106#102×C61—22no—
107#101×C61—44no—
108C22⋊C8×C61—21no—
109C22⋊C8×C61—21no—
110C22⋊C8×C61—21no—
111C22⋊C8×C61—21no—
112#95×C61—21no—
113#95×C61—24no—
114C24×C61—21no—
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—44no—
122#41×C61—44no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—24no—
129C2×Q8×C61—24no—
130C2×Q8×C61—44no—
131C2×Q8×C61—42no—
132C2×Q8×C61—44no—
133C2×Q8×C61—44no—
134C2×Q8×C61—44no—
135C2×Q8×C61—44no—
136C2×Q8×C61C2×D412yes—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—24no—
143C2×D4×C61—42no—
144C2×D4×C61—44no—
145C2×D4×C61—44no—
146C2×D4×C61—41no—
147C2×D4×C61—41no—
148C2×D4×C61C2×D412yes—
149C22×C4×C61—24no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—21no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—44no—
160C22×C4×C61—44no—
161C22×C4×C61C2×D414yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2411yescharacteristic
168SD16×C61—44no—
169D8×C61—44no—
170M16×C61—24no—
171M16×C61—24no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—28no—
175C4⋊C4×C61—28no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—28no—
183C22⋊C4×C61—28no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—48no—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—24no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—41no—
203C23×C61—42no—
204C23×C61#12011yescharacteristic
205C23×C61#12011yescharacteristic
206C23×C61C22×D411yescharacteristic
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—44no—
210Q8×C61—44no—
211Q8×C61—42no—
212Q8×C61—44no—
213Q8×C61—44no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—42no—
217D4×C61—44no—
218D4×C61—44no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—82no—
222D4×C61—82no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61—42no—
240C2×C4×C61—44no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)14yes—
251C8×C61—42no—
252C8×C61—42no—
253#29611C6111yesmaximal, characteristic, direct product factor, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—44no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—84no—
266C22×C61—84no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61#45412yes—
271C22×C61C2×#12011yescharacteristic
272C22×C61#53412yes—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—44no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61—81no—
281C4×C61#53412yes—
282C4×C61#53512yes—
283C2×#536C2×C6111yescharacteristic, complemented
284#3094C2×C6111yescharacteristic, complemented
285#3006C2×C6111yescharacteristic
286#2436C2×C6111yescharacteristic, complemented
287#2038C2×C6114yescomplemented
288#2003C2×C6114yescomplemented
289#1826C2×C6111yescharacteristic, complemented
290#1789C2×C6111yescharacteristic, complemented
291#1784C2×C6111yescharacteristic, complemented
292C2×C61—22no—
293C2×C61—22no—
294C2×C61—41no—
295C2×C61—82no—
296C2×C61—161no—
297C2×C61#261311yescharacteristic
298C2×C61C2×#45411yescharacteristic
299C2×C61#302011yescharacteristic
300C22×#41C22×C6111yescharacteristic
301C23×Q8C22×C6111yescharacteristic
302C2×AGL(1,ℤ/8)C22×C6112yes—
303#536—22no—
304#536—22no—
305#536—22no—
306#536C22×C6114yes—
307C2×C4⋊Q8—21no—
308C2×#124—21no—
309C2×#124C22×C6112yescomplemented
310C2×#123—22no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—21no—
313C2×#121C22×C6111yescharacteristic
314C2×#117C22×C6111yescharacteristic
315#482—22no—
316#481—21no—
317#480—21no—
318#449—22no—
319#448—21no—
320#447—21no—
321#428C22×C6112yescomplemented
322C2×#104C22×C6114yes—
323#418C22×C6112yescomplemented
324#417C22×C6112yes—
325C2×#101C22×C6114yescomplemented
326#407C22×C6111yescharacteristic
327#407C22×C6111yescharacteristic
328#393—22no—
329#391C22×C6112yescomplemented
330#386—22no—
331C4×C22⋊C4C22×C6111yescharacteristic
332#339C22×C6114yescomplemented
333C61#2961111yescharacteristic, direct product factor, Sylow
334C2×#41—21no—
335C2×#41—21no—
336C2×#41—44no—
337C2×#41D4×C6112yes—
338C2×#41C23×C6112yes—
339C22×Q8—21no—
340C22×Q8—21no—
341C22×Q8—21no—
342C22×Q8—21no—
343C22×Q8—44no—
344C22×Q8D4×C6111yescharacteristic
345C22×Q8D4×C6112yes—
346C22×Q8C23×C6112yes—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4D4×C6111yescharacteristic
350C22×D4C23×C6111yescharacteristic
351C23×C4—21no—
352C23×C4D4×C6112yes—
353C23×C4C23×C6111yescharacteristic
354AGL(1,ℤ/8)—44no—
355C2×SD16—22no—
356C2×D8—22no—
357C2×M16C23×C6112yes—
358C2×M16C23×C6112yes—
359C4⋊Q8—24no—
360C4⋊Q8—24no—
361#124—24no—
362#124—24no—
363#124—44no—
364#123—28no—
365#123—28no—
366C22⋊Q8—24no—
367C22⋊Q8—24no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370#121—21no—
371#121—21no—
372#121—21no—
373#121—21no—
374#117—21no—
375#117—21no—
376#117D4×C6112yes—
377#117C23×C6112yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4D4×C6111yescharacteristic
381C2×C4⋊C4D4×C6111yescharacteristic
382C2×C4⋊C4C23×C6111yescharacteristic
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—24no—
389C2×C42—21no—
390C2×C42C23×C6112yes—
391#107—22no—
392#106—22no—
393#104—44no—
394#103—22no—
395#102—22no—
396#102—22no—
397#102—22no—
398#101—44no—
399C22⋊C8—21no—
400C22⋊C8—21no—
401C22⋊C8—21no—
402C22⋊C8—21no—
403#95—21no—
404#95—24no—
405C24—21no—
406C24—21no—
407C24C2×D4×C6111yescharacteristic
408#41—24no—
409#41—24no—
410#41—42no—
411#41—42no—
412#41—44no—
413#41—44no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8C2×D4×C6112yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—24no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4C2×D4×C6112yes—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C2×D4×C6114yes—
453C22×C4C2×D4×C6111yescharacteristic
454C22×C4C2×D4×C6111yescharacteristic
455C22×C4C2×D4×C6111yescharacteristic
456C22×C4C2×D4×C6111yescharacteristic
457C22×C4C2×D4×C6111yescharacteristic
458C22×C4C24×C6111yescharacteristic
459SD16—44no—
460D8—44no—
461M16—24no—
462M16—24no—
463C2×C8—22no—
464C2×C8—22no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4C2×D4×C6112yes—
472C4⋊C4C2×D4×C6112yes—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—48no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C42—22no—
482C42—22no—
483C42—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23#120×C6111yescharacteristic
496C23#120×C6111yescharacteristic
497C23C22×D4×C6111yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—44no—
501Q8—44no—
502Q8—42no—
503Q8—44no—
504Q8—44no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—44no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—82no—
513D4—82no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4#120×C6111yescharacteristic
538C2×C4#120×C6111yescharacteristic
539C2×C4C22×D4×C6111yescharacteristic
540C2×C4C22×D4×C6111yescharacteristic
541C2×C4H(5,𝔽2)×C6114yes—
542C8—42no—
543C8—42no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—84no—
556C22—84no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C61×#45412yes—
561C22C2×#120×C6111yescharacteristic
562C22C61×#53412yes—
563C4—21no—
564C4—21no—
565C4—44no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—81no—
571C4C61×#53412yes—
572C4C61×#53512yes—
573C2—22no—
574C2—22no—
575C2—41no—
576C2—82no—
577C2—161no—
578C2C61×#261311yescharacteristic
579C2C2×C61×#45411yescharacteristic
580C2C61×#302011yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C61×#536C2semidirect
2. C61×#3094C2semidirect
4. C61×#2436C2semidirect
5. C61×#2038C2semidirect
6. C61×#2003C2semidirect
7. C61×#1826C2semidirect
8. C61×#1789C2semidirect
9. C61×#1784C2semidirect
19. C2×#124×C61C22semidirect
31. C61×#428C22semidirect
33. C61×#418C22semidirect
35. C2×#101×C61C22semidirect
39. C61×#391C22semidirect
42. C61×#339C22semidirect
253. #29611C61direct (N × H)
283. C2×#536C2×C61semidirect
284. #3094C2×C61semidirect
286. #2436C2×C61semidirect
287. #2038C2×C61semidirect
288. #2003C2×C61semidirect
289. #1826C2×C61semidirect
290. #1789C2×C61semidirect
291. #1784C2×C61semidirect
309. C2×#124C22×C61semidirect
321. #428C22×C61semidirect
323. #418C22×C61semidirect
325. C2×#101C22×C61semidirect
329. #391C22×C61semidirect
332. #339C22×C61semidirect
333. C61#29611direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5362
C61×#30942
C61×#30062
C61×#24362
C61×#20382
C61×#20032
C61×#18262
C61×#17892
C61×#17842
#2961161