Finite Group Databasebeta

← C61×#12780 #29516213

Subgroups — C61×#12780 #29516213

Order 15616.

756 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3551C211yesmaximal, characteristic
2C61×#2129C211yesmaximal, characteristic, complemented
3C61×#2128C211yesmaximal, characteristic, complemented
4C61×#2126C211yesmaximal, characteristic, complemented
5C61×#1910C211yesmaximal, characteristic, complemented
6C61×#1909C211yesmaximal, characteristic, complemented
7C61×#1884C211yesmaximal, characteristic, complemented
8C61×#568C2211yescharacteristic
9C61×#563C412yescomplemented
10C61×#560—21no—
11C61×#555C412yescomplemented
12C61×#553—21no—
13C61×#552—21no—
14C61×#551—21no—
15C61×#548—21no—
16C61×#546—21no—
17C61×#540—22no—
18C61×#538—22no—
19C61×#537—21no—
20C61×#536C2211yescharacteristic
21C61×#535C2211yescharacteristic
22C61×#532C2211yescharacteristic
23C61×#529—21no—
24C61×#421—21no—
25C61×#419—21no—
26C61×#418—21no—
27C61×#410C2211yescharacteristic
28C61×#410C2211yescharacteristic
29C61×#410C2211yescharacteristic
30C61×#354—21no—
31C61×#354—22no—
32C61×#353—21no—
33C61×#353—22no—
34C61×#339—21no—
35C61×H(3,ℤ/4)—21no—
36C61×H(3,ℤ/4)—21no—
37C2×#41×C61—21no—
38C2×#41×C61D411yescharacteristic
39C2×#41×C61C2311yescharacteristic
40C22×Q8×C61D411yescharacteristic
41C22×D4×C61D411yescharacteristic
42#131×C61—42no—
43C2×M16×C61—21no—
44C2×M16×C61—21no—
45C2×M16×C61—21no—
46C4⋊Q8×C61C2×C411yescharacteristic
47C4⋊Q8×C61C2×C412yes—
48#127×C61D411yescharacteristic
49#127×C61D412yes—
50#126×C61—22no—
51#126×C61—22no—
52#126×C61—22no—
53#125×C61—21no—
54#125×C61—22no—
55#124×C61—21no—
56#124×C61—22no—
57#124×C61C2×C411yescharacteristic
58#124×C61C2×C412yes—
59#124×C61D411yescharacteristic
60#124×C61D412yes—
61#123×C61—22no—
62#123×C61—22no—
63#123×C61—22no—
64#123×C61—22no—
65#123×C61—21no—
66#123×C61—21no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72#121×C61—22no—
73#121×C61—22no—
74#121×C61—21no—
75#121×C61—22no—
76#121×C61—22no—
77#121×C61—22no—
78#121×C61—22no—
79#121×C61—21no—
80#121×C61—21no—
81#121×C61—21no—
82#121×C61—21no—
83#120×C61—22no—
84#120×C61—21no—
85#120×C61—21no—
86C4×Q8×C61—21no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61—21no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61—22no—
93#117×C61—22no—
94#117×C61—22no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61—21no—
98#117×C61D411yescharacteristic
99#117×C61D411yescharacteristic
100C2×C4⋊C4×C61—21no—
101C2×C22⋊C4×C61—22no—
102C2×C42×C61D411yescharacteristic
103#104×C61—42no—
104#103×C61—42no—
105#102×C61—42no—
106#99×C61—21no—
107#99×C61—21no—
108#99×C61—21no—
109#99×C61—21no—
110#99×C61—21no—
111#99×C61—21no—
112C24×C61—21no—
113#41×C61—41no—
114#41×C61—41no—
115#41×C61—41no—
116#41×C61—41no—
117#41×C61—81no—
118#41×C61—81no—
119C2×Q8×C61—21no—
120C2×Q8×C61—41no—
121C2×Q8×C61—41no—
122C2×Q8×C61C22⋊C411yescharacteristic
123C2×Q8×C61C22⋊C412yes—
124C2×Q8×C61C22×C411yescharacteristic
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—41no—
134C2×D4×C61—41no—
135C2×D4×C61C22⋊C411yescharacteristic
136C2×D4×C61C22⋊C412yes—
137C2×D4×C61C2×D411yescharacteristic
138C2×D4×C61C2×D411yescharacteristic
139C2×D4×C61C2×D411yescharacteristic
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149M16×C61—22no—
150M16×C61—41no—
151M16×C61—41no—
152M16×C61—41no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—41no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—22no—
189C42×C61—21no—
190C42×C61—41no—
191C42×C61—41no—
192C42×C61—41no—
193C42×C61C22⋊C411yescharacteristic
194C42×C61C22⋊C411yescharacteristic
195C42×C61C22⋊C412yes—
196C42×C61C22⋊C412yes—
197C23×C61—22no—
198C23×C61—21no—
199C23×C61—81no—
200C23×C61—82no—
201C23×C61#12011yescharacteristic
202C23×C61#12011yescharacteristic
203C23×C61#12011yescharacteristic
204Q8×C61—41no—
205Q8×C61—42no—
206Q8×C61—41no—
207Q8×C61—41no—
208Q8×C61—41no—
209D4×C61—41no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—42no—
213D4×C61—41no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—81no—
217D4×C61—82no—
218D4×C61—82no—
219D4×C61—82no—
220D4×C61—81no—
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—22no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—81no—
238C2×C4×C61—82no—
239C2×C4×C61—81no—
240C2×C4×C61—82no—
241C2×C4×C61—82no—
242C2×C4×C61—81no—
243C2×C4×C61—81no—
244C2×C4×C61—81no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C8×C61—42no—
250#12780C6111yesmaximal, characteristic, direct product factor, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—42no—
261C22×C61—41no—
262C22×C61—42no—
263C22×C61—162no—
264C22×C61#37911yescharacteristic
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61—161no—
275C4×C61—161no—
276C4×C61—161no—
277C4×C61B(4,𝔽2)11yescharacteristic
278C4×C61#45811yescharacteristic
279#3551C2×C6111yescharacteristic
280#2129C2×C6111yescharacteristic, complemented
281#2128C2×C6111yescharacteristic, complemented
282#2126C2×C6111yescharacteristic, complemented
283#1910C2×C6111yescharacteristic, complemented
284#1909C2×C6111yescharacteristic, complemented
285#1884C2×C6111yescharacteristic, complemented
286C2×C61—21no—
287C2×C61—41no—
288C2×C61—41no—
289C2×C61—41no—
290C2×C61—81no—
291C2×C61—82no—
292C2×C61#179011yescharacteristic
293#568C22×C6111yescharacteristic
294#563C4×C6112yescomplemented
295#560—21no—
296#555C4×C6112yescomplemented
297#553—21no—
298#552—21no—
299#551—21no—
300#548—21no—
301#546—21no—
302#540—22no—
303#538—22no—
304#537—21no—
305#536C22×C6111yescharacteristic
306#535C22×C6111yescharacteristic
307#532C22×C6111yescharacteristic
308#529—21no—
309#421—21no—
310#419—21no—
311#418—21no—
312#410C22×C6111yescharacteristic
313#410C22×C6111yescharacteristic
314#410C22×C6111yescharacteristic
315#354—21no—
316#354—22no—
317#353—21no—
318#353—22no—
319#339—21no—
320H(3,ℤ/4)—21no—
321H(3,ℤ/4)—21no—
322C61#1278011yescharacteristic, direct product factor, Sylow
323C2×#41—21no—
324C2×#41D4×C6111yescharacteristic
325C2×#41C23×C6111yescharacteristic
326C22×Q8D4×C6111yescharacteristic
327C22×D4D4×C6111yescharacteristic
328#131—42no—
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—21no—
332C4⋊Q8C2×C4×C6111yescharacteristic
333C4⋊Q8C2×C4×C6112yes—
334#127D4×C6111yescharacteristic
335#127D4×C6112yes—
336#126—22no—
337#126—22no—
338#126—22no—
339#125—21no—
340#125—22no—
341#124—21no—
342#124—22no—
343#124C2×C4×C6111yescharacteristic
344#124C2×C4×C6112yes—
345#124D4×C6111yescharacteristic
346#124D4×C6112yes—
347#123—22no—
348#123—22no—
349#123—22no—
350#123—22no—
351#123—21no—
352#123—21no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358#121—22no—
359#121—22no—
360#121—21no—
361#121—22no—
362#121—22no—
363#121—22no—
364#121—22no—
365#121—21no—
366#121—21no—
367#121—21no—
368#121—21no—
369#120—22no—
370#120—21no—
371#120—21no—
372C4×Q8—21no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—22no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—22no—
379#117—22no—
380#117—22no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117D4×C6111yescharacteristic
385#117D4×C6111yescharacteristic
386C2×C4⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C42D4×C6111yescharacteristic
389#104—42no—
390#103—42no—
391#102—42no—
392#99—21no—
393#99—21no—
394#99—21no—
395#99—21no—
396#99—21no—
397#99—21no—
398C24—21no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403#41—81no—
404#41—81no—
405C2×Q8—21no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8C22⋊C4×C6111yescharacteristic
409C2×Q8C22⋊C4×C6112yes—
410C2×Q8C22×C4×C6111yescharacteristic
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4C22⋊C4×C6111yescharacteristic
422C2×D4C22⋊C4×C6112yes—
423C2×D4C2×D4×C6111yescharacteristic
424C2×D4C2×D4×C6111yescharacteristic
425C2×D4C2×D4×C6111yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C2×D4×C6111yescharacteristic
433C22×C4C2×D4×C6111yescharacteristic
434C22×C4C2×D4×C6111yescharacteristic
435M16—22no—
436M16—41no—
437M16—41no—
438M16—41no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—41no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C42—22no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42C22⋊C4×C6111yescharacteristic
480C42C22⋊C4×C6111yescharacteristic
481C42C22⋊C4×C6112yes—
482C42C22⋊C4×C6112yes—
483C23—22no—
484C23—21no—
485C23—81no—
486C23—82no—
487C23#120×C6111yescharacteristic
488C23#120×C6111yescharacteristic
489C23#120×C6111yescharacteristic
490Q8—41no—
491Q8—42no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—42no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—81no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—81no—
507D4—81no—
508D4—82no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4C2×C22⋊C4×C6111yescharacteristic
532C2×C4C2×C22⋊C4×C6111yescharacteristic
533C2×C4C2×C22⋊C4×C6111yescharacteristic
534C2×C4#120×C6111yescharacteristic
535C8—42no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—42no—
548C22—162no—
549C22C61×#37911yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—161no—
560C4—161no—
561C4—161no—
562C4C61×B(4,𝔽2)11yescharacteristic
563C4C61×#45811yescharacteristic
564C2—21no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—82no—
569C2—81no—
570C2C61×#179011yescharacteristic
subgroupG/H
center, socleC2×C61#1790
commutatorC2×Q8C22×C4×C61
Frattini, cosocleC2×#41C23×C61
prime radicalC61×#3551C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C61 ◁ C2×C4×C61 ◁ C2×#41×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2129C2semidirect
3. C61×#2128C2semidirect
4. C61×#2126C2semidirect
5. C61×#1910C2semidirect
6. C61×#1909C2semidirect
7. C61×#1884C2semidirect
9. C61×#563C4semidirect
11. C61×#555C4semidirect
250. #12780C61direct (N × H)
280. #2129C2×C61semidirect
281. #2128C2×C61semidirect
282. #2126C2×C61semidirect
283. #1910C2×C61semidirect
284. #1909C2×C61semidirect
285. #1884C2×C61semidirect
294. #563C4×C61semidirect
296. #555C4×C61semidirect
322. C61#12780direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35512
C61×#21292
C61×#21282
C61×#21262
C61×#19102
C61×#19092
C61×#18842
#1278061