Finite Group Databasebeta

← C61×#19373 #29522806

Subgroups — C61×#19373 #29522806

Order 15616.

576 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3254C211yesmaximal, characteristic
2C61×#3237C211yesmaximal, characteristic, complemented
3C61×#3182C211yesmaximal, characteristic
4C61×#2962C211yesmaximal, characteristic, complemented
5C61×#1704C211yesmaximal, characteristic, complemented
6C61×#1698C211yesmaximal, characteristic, complemented
7C61×#1696C211yesmaximal, characteristic, complemented
8C61×#1686C211yesmaximal, characteristic, complemented
9C61×#1684C211yesmaximal, characteristic, complemented
10C61×#1678C211yesmaximal, characteristic, complemented
11C61×#1676C211yesmaximal, characteristic, complemented
12C61×#1666C211yesmaximal, characteristic, complemented
13C61×#1569C211yesmaximal, characteristic, complemented
14C61×#1551C211yesmaximal, characteristic
15C61×#1550C211yesmaximal, characteristic, complemented
16C61×#533C2211yescharacteristic
17C61×#533C2211yescharacteristic
18C61×#532C2211yescharacteristic
19C61×#501—21no—
20C61×#500C2211yescharacteristic
21C61×#492C2211yescharacteristic
22C61×#491—21no—
23C61×#487C2211yescharacteristic
24C61×#486C2211yescharacteristic
25C61×#485—21no—
26C61×#485—21no—
27C61×#477C2211yescharacteristic
28C61×#477C2211yescharacteristic
29C61×#477C2211yescharacteristic
30C61×#477C2211yescharacteristic
31C61×#475C2211yescharacteristic
32C61×#475C2211yescharacteristic
33C61×#474—21no—
34C61×#468—21no—
35C61×#468—21no—
36C61×#468—21no—
37C61×#468—21no—
38C61×#461—21no—
39C61×#460C2211yescharacteristic
40C61×#460C2211yescharacteristic
41C61×#446—21no—
42C61×#446—21no—
43C61×#446C2211yescharacteristic
44C61×#446C2211yescharacteristic
45C8×Q8×C61C2211yescharacteristic
46C8×Q8×C61C2211yescharacteristic
47C61×#440—21no—
48C61×#440—21no—
49C61×#432C2211yescharacteristic
50C61×#424—21no—
51C61×#423C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#405—21no—
54C4×M16×C61C2211yescharacteristic
55C61×#335C2211yescharacteristic
56C61×#335C2211yescharacteristic
57C61×#334C2211yescharacteristic
58C61×#334C2211yescharacteristic
59C61×#333C2211yescharacteristic
60C61×#333C2211yescharacteristic
61C61×#333C2211yescharacteristic
62C61×#333C2211yescharacteristic
63C61×#332C2211yescharacteristic
64C61×#332C2211yescharacteristic
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#41×C61—21no—
68C2×SD16×C61—21no—
69C2×SD16×C61—21no—
70C2×SD16×C61—21no—
71C2×SD16×C61—21no—
72C2×M16×C61—21no—
73C2×M16×C61D411yescharacteristic
74C2×M16×C61D411yescharacteristic
75C22×C8×C61—21no—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61C2311yescharacteristic
79C4⋊Q8×C61C2311yescharacteristic
80C4⋊Q8×C61C2311yescharacteristic
81#127×C61C2311yescharacteristic
82#125×C61—21no—
83#125×C61—21no—
84#124×C61—21no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61D411yescharacteristic
89C22⋊Q8×C61D411yescharacteristic
90#121×C61—21no—
91#121×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61C2311yescharacteristic
96C4×Q8×C61C2311yescharacteristic
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—21no—
103#107×C61—21no—
104#107×C61—21no—
105#107×C61—21no—
106#107×C61—21no—
107#107×C61—21no—
108#106×C61—21no—
109#106×C61—21no—
110C4⋊C8×C61—21no—
111C4⋊C8×C61—21no—
112C4⋊C8×C61—21no—
113C4⋊C8×C61—21no—
114C4⋊C8×C61—21no—
115C4⋊C8×C61—21no—
116C4⋊C8×C61C2311yescharacteristic
117C4⋊C8×C61C2311yescharacteristic
118C4⋊C8×C61C2311yescharacteristic
119C4⋊C8×C61C2311yescharacteristic
120C4⋊C8×C61C2311yescharacteristic
121C4⋊C8×C61C2311yescharacteristic
122#103×C61—21no—
123#103×C61—21no—
124#103×C61—21no—
125#103×C61—21no—
126#103×C61—21no—
127#103×C61—21no—
128#103×C61—21no—
129#103×C61—21no—
130#102×C61—21no—
131#102×C61—21no—
132#102×C61—21no—
133#102×C61—21no—
134C22⋊C8×C61—21no—
135#97×C61—21no—
136#97×C61—21no—
137#97×C61—21no—
138#97×C61D411yescharacteristic
139#97×C61D411yescharacteristic
140C4×C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C22×C4×C61—41no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4111yescharacteristic
159SD16×C61—42no—
160SD16×C61—42no—
161SD16×C61—81no—
162SD16×C61—81no—
163M16×C61—22no—
164M16×C61—41no—
165M16×C61—81no—
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—42no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
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—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—41no—
191C4⋊C4×C61—41no—
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—41no—
197C22⋊C4×C61—41no—
198C22⋊C4×C61—41no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—41no—
202C42×C61C2×D411yescharacteristic
203C42×C61C2×D411yescharacteristic
204C42×C61#4111yescharacteristic
205C42×C61C2411yescharacteristic
206C23×C61—41no—
207C23×C61#12111yescharacteristic
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215Q8×C61—81no—
216Q8×C61—81no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—81no—
222D4×C61—82no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—82no—
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—21no—
245C8×C61—41no—
246C8×C61—41no—
247C8×C61—41no—
248C8×C61—41no—
249C8×C61—41no—
250C8×C61—41no—
251C8×C61#13712yes—
252#19373C6111yesmaximal, characteristic, direct product factor, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61C2×#12111yescharacteristic
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C2×#13711yescharacteristic
272C4×C61#57511yescharacteristic
273C4×C61#57711yescharacteristic
274C4×C61#57811yescharacteristic
275#3254C2×C6111yescharacteristic
276#3237C2×C6111yescharacteristic, complemented
277#3182C2×C6111yescharacteristic
278#2962C2×C6111yescharacteristic, complemented
279#1704C2×C6111yescharacteristic, complemented
280#1698C2×C6111yescharacteristic, complemented
281#1696C2×C6111yescharacteristic, complemented
282#1686C2×C6111yescharacteristic, complemented
283#1684C2×C6111yescharacteristic, complemented
284#1678C2×C6111yescharacteristic, complemented
285#1676C2×C6111yescharacteristic, complemented
286#1666C2×C6111yescharacteristic, complemented
287#1569C2×C6111yescharacteristic, complemented
288#1551C2×C6111yescharacteristic
289#1550C2×C6111yescharacteristic, complemented
290C2×C61—41no—
291C2×C61—161no—
292C2×C61#303711yescharacteristic
293C2×C61#306311yescharacteristic
294C2×C61#306811yescharacteristic
295#533C22×C6111yescharacteristic
296#533C22×C6111yescharacteristic
297#532C22×C6111yescharacteristic
298#501—21no—
299#500C22×C6111yescharacteristic
300#492C22×C6111yescharacteristic
301#491—21no—
302#487C22×C6111yescharacteristic
303#486C22×C6111yescharacteristic
304#485—21no—
305#485—21no—
306#477C22×C6111yescharacteristic
307#477C22×C6111yescharacteristic
308#477C22×C6111yescharacteristic
309#477C22×C6111yescharacteristic
310#475C22×C6111yescharacteristic
311#475C22×C6111yescharacteristic
312#474—21no—
313#468—21no—
314#468—21no—
315#468—21no—
316#468—21no—
317#461—21no—
318#460C22×C6111yescharacteristic
319#460C22×C6111yescharacteristic
320#446—21no—
321#446—21no—
322#446C22×C6111yescharacteristic
323#446C22×C6111yescharacteristic
324C8×Q8C22×C6111yescharacteristic
325C8×Q8C22×C6111yescharacteristic
326#440—21no—
327#440—21no—
328#432C22×C6111yescharacteristic
329#424—21no—
330#423C22×C6111yescharacteristic
331C2×C4⋊C8C22×C6111yescharacteristic
332#405—21no—
333C4×M16C22×C6111yescharacteristic
334#335C22×C6111yescharacteristic
335#335C22×C6111yescharacteristic
336#334C22×C6111yescharacteristic
337#334C22×C6111yescharacteristic
338#333C22×C6111yescharacteristic
339#333C22×C6111yescharacteristic
340#333C22×C6111yescharacteristic
341#333C22×C6111yescharacteristic
342#332C22×C6111yescharacteristic
343#332C22×C6111yescharacteristic
344#331C22×C6111yescharacteristic
345#331C22×C6111yescharacteristic
346C61#1937311yescharacteristic, direct product factor, Sylow
347C2×#41—21no—
348C2×SD16—21no—
349C2×SD16—21no—
350C2×SD16—21no—
351C2×SD16—21no—
352C2×M16—21no—
353C2×M16D4×C6111yescharacteristic
354C2×M16D4×C6111yescharacteristic
355C22×C8—21no—
356C4⋊Q8D4×C6111yescharacteristic
357C4⋊Q8D4×C6111yescharacteristic
358C4⋊Q8C23×C6111yescharacteristic
359C4⋊Q8C23×C6111yescharacteristic
360C4⋊Q8C23×C6111yescharacteristic
361#127C23×C6111yescharacteristic
362#125—21no—
363#125—21no—
364#124—21no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8D4×C6111yescharacteristic
369C22⋊Q8D4×C6111yescharacteristic
370#121—21no—
371#121—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8C23×C6111yescharacteristic
376C4×Q8C23×C6111yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379#117—21no—
380#117—21no—
381C2×C42C23×C6111yescharacteristic
382#107—21no—
383#107—21no—
384#107—21no—
385#107—21no—
386#107—21no—
387#107—21no—
388#106—21no—
389#106—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8C23×C6111yescharacteristic
397C4⋊C8C23×C6111yescharacteristic
398C4⋊C8C23×C6111yescharacteristic
399C4⋊C8C23×C6111yescharacteristic
400C4⋊C8C23×C6111yescharacteristic
401C4⋊C8C23×C6111yescharacteristic
402#103—21no—
403#103—21no—
404#103—21no—
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—21no—
410#102—21no—
411#102—21no—
412#102—21no—
413#102—21no—
414C22⋊C8—21no—
415#97—21no—
416#97—21no—
417#97—21no—
418#97D4×C6111yescharacteristic
419#97D4×C6111yescharacteristic
420C4×C8—21no—
421C4×C8—21no—
422C4×C8C23×C6111yescharacteristic
423C4×C8C23×C6111yescharacteristic
424#41—82no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8C2×D4×C6111yescharacteristic
430C2×Q8C2×D4×C6111yescharacteristic
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—41no—
435C22×C4—41no—
436C22×C4C2×D4×C6111yescharacteristic
437C22×C4C2×D4×C6111yescharacteristic
438C22×C4#41×C6111yescharacteristic
439SD16—42no—
440SD16—42no—
441SD16—81no—
442SD16—81no—
443M16—22no—
444M16—41no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8C2×D4×C6111yescharacteristic
454C2×C8C2×D4×C6111yescharacteristic
455C2×C8C2×D4×C6111yescharacteristic
456C2×C8C2×D4×C6111yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4C2×D4×C6111yescharacteristic
473C4⋊C4C2×D4×C6111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42C2×D4×C6111yescharacteristic
483C42C2×D4×C6111yescharacteristic
484C42#41×C6111yescharacteristic
485C42C24×C6111yescharacteristic
486C23—41no—
487C23#121×C6111yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—41no—
493Q8—41no—
494Q8—81no—
495Q8—81no—
496Q8—81no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—81no—
502D4—82no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4#121×C6111yescharacteristic
517C2×C4#121×C6111yescharacteristic
518C2×C4#121×C6111yescharacteristic
519C2×C4C22×D4×C6111yescharacteristic
520C2×C4C22×D4×C6111yescharacteristic
521C2×C4C2×#41×C6111yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8#137×C6112yes—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22C2×#121×C6111yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4C2×#137×C6111yescharacteristic
551C4C61×#57511yescharacteristic
552C4C61×#57711yescharacteristic
553C4C61×#57811yescharacteristic
554C2—41no—
555C2—161no—
556C2C61×#303711yescharacteristic
557C2C61×#306311yescharacteristic
558C2C61×#306811yescharacteristic
subgroupG/H
center, socleC22×C61C2×#121
commutator, Frattini, cosocleC42C24×C61
prime radicalC61×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3237C2semidirect
4. C61×#2962C2semidirect
5. C61×#1704C2semidirect
6. C61×#1698C2semidirect
7. C61×#1696C2semidirect
8. C61×#1686C2semidirect
9. C61×#1684C2semidirect
10. C61×#1678C2semidirect
11. C61×#1676C2semidirect
12. C61×#1666C2semidirect
13. C61×#1569C2semidirect
15. C61×#1550C2semidirect
252. #19373C61direct (N × H)
276. #3237C2×C61semidirect
278. #2962C2×C61semidirect
279. #1704C2×C61semidirect
280. #1698C2×C61semidirect
281. #1696C2×C61semidirect
282. #1686C2×C61semidirect
283. #1684C2×C61semidirect
284. #1678C2×C61semidirect
285. #1676C2×C61semidirect
286. #1666C2×C61semidirect
287. #1569C2×C61semidirect
289. #1550C2×C61semidirect
346. C61#19373direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#32542
C61×#32372
C61×#31822
C61×#29622
C61×#17042
C61×#16982
C61×#16962
C61×#16862
C61×#16842
C61×#16782
C61×#16762
C61×#16662
C61×#15692
C61×#15512
C61×#15502
#1937361