Finite Group Databasebeta

← C61×#12758 #29516191

Subgroups — C61×#12758 #29516191

Order 15616.

758 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×#3540C211yesmaximal, characteristic
2C61×#2134C211yesmaximal, characteristic, complemented
3C61×#2123C211yesmaximal, characteristic, complemented
4C61×#2120C211yesmaximal, characteristic, complemented
5C61×#1915C211yesmaximal, characteristic, complemented
6C61×#1910C211yesmaximal, characteristic, complemented
7C61×#1891C211yesmaximal, characteristic, complemented
8C61×#568C2211yescharacteristic
9C61×H(3,𝔽4)—21no—
10C61×#560C412yescomplemented
11C61×#559C412yescomplemented
12C61×#556—21no—
13C61×#555—21no—
14C61×#551—21no—
15C61×#548—21no—
16C61×#546—21no—
17C61×#542—22no—
18C61×#542—21no—
19C61×#538—22no—
20C61×#536C2211yescharacteristic
21C61×#535C2211yescharacteristic
22C61×#525—21no—
23C61×#525C2211yescharacteristic
24C61×#418—21no—
25C61×#413C2211yescharacteristic
26C61×#413C2211yescharacteristic
27C2×#101×C61—21no—
28C61×#410C2211yescharacteristic
29C61×#407—21no—
30C61×#355—22no—
31C61×#352—21no—
32C61×#352—22no—
33C61×AΣL(1,𝔽16)—21no—
34C61×#344—21no—
35C61×#344—21no—
36C61×H(3,ℤ/4)—21no—
37C2×#41×C61—21no—
38C2×#41×C61C2311yescharacteristic
39C22×Q8×C61D411yescharacteristic
40C22×D4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42#131×C61—42no—
43C2×M16×C61—21no—
44C2×M16×C61—21no—
45C2×M16×C61—21no—
46C2×M16×C61—21no—
47C2×M16×C61—21no—
48C4⋊Q8×C61—21no—
49C4⋊Q8×C61—21no—
50#127×C61C2×C412yes—
51#126×C61—22no—
52#126×C61—22no—
53#126×C61—22no—
54#124×C61—22no—
55#124×C61—21no—
56#124×C61—22no—
57#124×C61—22no—
58#124×C61—21no—
59#124×C61—22no—
60#124×C61—22no—
61#124×C61D412yes—
62#123×C61—22no—
63#123×C61—22no—
64#123×C61—22no—
65#123×C61C2×C412yes—
66#123×C61C2×C411yescharacteristic
67#123×C61D412yes—
68#123×C61D411yescharacteristic
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75#121×C61—21no—
76#121×C61—22no—
77#121×C61—21no—
78#121×C61—22no—
79#121×C61—21no—
80#121×C61—21no—
81#121×C61—22no—
82#120×C61—22no—
83#120×C61—22no—
84#120×C61—21no—
85#120×C61—21no—
86#120×C61C2×C411yescharacteristic
87#120×C61D411yescharacteristic
88C4×Q8×C61—21no—
89C4×Q8×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—21no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96#117×C61—22no—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61D411yescharacteristic
100#117×C61D411yescharacteristic
101#117×C61D411yescharacteristic
102C2×C22⋊C4×C61—22no—
103#102×C61—42no—
104#101×C61—21no—
105#101×C61—21no—
106#101×C61—42no—
107#100×C61—21no—
108#100×C61—21no—
109#99×C61—21no—
110#99×C61—21no—
111C22⋊C8×C61—42no—
112C24×C61—21no—
113C24×C61C22⋊C411yescharacteristic
114#41×C61—41no—
115#41×C61—41no—
116#41×C61—81no—
117#41×C61—81no—
118C2×Q8×C61—22no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—41no—
122C2×Q8×C61—41no—
123C2×Q8×C61C2×D411yescharacteristic
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—41no—
132C2×D4×C61—41no—
133C2×D4×C61C22⋊C412yes—
134C2×D4×C61C22×C411yescharacteristic
135C2×D4×C61C2×D411yescharacteristic
136C2×D4×C61C2×D411yescharacteristic
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22⋊C412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148M16×C61—22no—
149M16×C61—41no—
150M16×C61—41no—
151M16×C61—41no—
152M16×C61—41no—
153M16×C61—41no—
154M16×C61—41no—
155M16×C61—41no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—41no—
159C2×C8×C61—41no—
160C2×C8×C61—41no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—41no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61C22⋊C411yescharacteristic
189C22⋊C4×C61C22⋊C411yescharacteristic
190C22⋊C4×C61C22⋊C412yes—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—41no—
196C42×C61C22⋊C412yes—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61—81no—
201C23×C61—81no—
202C23×C61—82no—
203C23×C61C2×C22⋊C411yescharacteristic
204C23×C61C2×C22⋊C411yescharacteristic
205C23×C61#12011yescharacteristic
206Q8×C61—41no—
207Q8×C61—41no—
208Q8×C61—41no—
209Q8×C61—81no—
210Q8×C61—82no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—42no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—81no—
218D4×C61—82no—
219D4×C61—82no—
220D4×C61—82no—
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—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—81no—
240C2×C4×C61—82no—
241C2×C4×C61—82no—
242C2×C4×C61—81no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C8×C61—42no—
248C8×C61—81no—
249C8×C61—81no—
250#12758C6111yesmaximal, 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—42no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—42no—
264C22×C61—81no—
265C22×C61—162no—
266C22×C61#37911yescharacteristic
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—161no—
277C4×C61B(4,𝔽2)11yescharacteristic
278C4×C61#45811yescharacteristic
279#3540C2×C6111yescharacteristic
280#2134C2×C6111yescharacteristic, complemented
281#2123C2×C6111yescharacteristic, complemented
282#2120C2×C6111yescharacteristic, complemented
283#1915C2×C6111yescharacteristic, complemented
284#1910C2×C6111yescharacteristic, complemented
285#1891C2×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
294H(3,𝔽4)—21no—
295#560C4×C6112yescomplemented
296#559C4×C6112yescomplemented
297#556—21no—
298#555—21no—
299#551—21no—
300#548—21no—
301#546—21no—
302#542—22no—
303#542—21no—
304#538—22no—
305#536C22×C6111yescharacteristic
306#535C22×C6111yescharacteristic
307#525—21no—
308#525C22×C6111yescharacteristic
309#418—21no—
310#413C22×C6111yescharacteristic
311#413C22×C6111yescharacteristic
312C2×#101—21no—
313#410C22×C6111yescharacteristic
314#407—21no—
315#355—22no—
316#352—21no—
317#352—22no—
318AΣL(1,𝔽16)—21no—
319#344—21no—
320#344—21no—
321H(3,ℤ/4)—21no—
322C61#1275811yescharacteristic, direct product factor, Sylow
323C2×#41—21no—
324C2×#41C23×C6111yescharacteristic
325C22×Q8D4×C6111yescharacteristic
326C22×D4D4×C6111yescharacteristic
327C23×C4D4×C6111yescharacteristic
328#131—42no—
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336#127C2×C4×C6112yes—
337#126—22no—
338#126—22no—
339#126—22no—
340#124—22no—
341#124—21no—
342#124—22no—
343#124—22no—
344#124—21no—
345#124—22no—
346#124—22no—
347#124D4×C6112yes—
348#123—22no—
349#123—22no—
350#123—22no—
351#123C2×C4×C6111yescharacteristic
352#123C2×C4×C6112yes—
353#123D4×C6111yescharacteristic
354#123D4×C6112yes—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361#121—21no—
362#121—22no—
363#121—21no—
364#121—22no—
365#121—21no—
366#121—21no—
367#121—22no—
368#120—22no—
369#120—22no—
370#120—21no—
371#120—21no—
372#120C2×C4×C6111yescharacteristic
373#120D4×C6111yescharacteristic
374C4×Q8—21no—
375C4×Q8—22no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382#117—21no—
383#117—22no—
384#117—21no—
385#117D4×C6111yescharacteristic
386#117D4×C6111yescharacteristic
387#117D4×C6111yescharacteristic
388C2×C22⋊C4—22no—
389#102—42no—
390#101—21no—
391#101—21no—
392#101—42no—
393#100—21no—
394#100—21no—
395#99—21no—
396#99—21no—
397C22⋊C8—42no—
398C24—21no—
399C24C22⋊C4×C6111yescharacteristic
400#41—41no—
401#41—41no—
402#41—81no—
403#41—81no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8C2×D4×C6111yescharacteristic
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4C22⋊C4×C6112yes—
420C2×D4C22×C4×C6111yescharacteristic
421C2×D4C2×D4×C6111yescharacteristic
422C2×D4C2×D4×C6111yescharacteristic
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4C22⋊C4×C6111yescharacteristic
430C22×C4C22⋊C4×C6112yes—
431C22×C4C2×D4×C6111yescharacteristic
432C22×C4C2×D4×C6111yescharacteristic
433C22×C4C2×D4×C6111yescharacteristic
434M16—22no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—21no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—41no—
474C22⋊C4C22⋊C4×C6111yescharacteristic
475C22⋊C4C22⋊C4×C6111yescharacteristic
476C22⋊C4C22⋊C4×C6112yes—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42C22⋊C4×C6112yes—
483C23—21no—
484C23—22no—
485C23—41no—
486C23—81no—
487C23—81no—
488C23—82no—
489C23C2×C22⋊C4×C6111yescharacteristic
490C23C2×C22⋊C4×C6111yescharacteristic
491C23#120×C6111yescharacteristic
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—81no—
496Q8—82no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—41no—
502D4—41no—
503D4—81no—
504D4—82no—
505D4—82no—
506D4—82no—
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—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4C2×C22⋊C4×C6111yescharacteristic
530C2×C4#120×C6111yescharacteristic
531C2×C4#120×C6111yescharacteristic
532C2×C4#120×C6111yescharacteristic
533C8—42no—
534C8—81no—
535C8—81no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—42no—
549C22—81no—
550C22—162no—
551C22C61×#37911yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—161no—
562C4C61×B(4,𝔽2)11yescharacteristic
563C4C61×#45811yescharacteristic
564C2—21no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—81no—
569C2—82no—
570C2C61×#179011yescharacteristic
subgroupG/H
center, socleC2×C61#1790
commutatorC2×D4C22×C4×C61
Frattini, cosocleC2×#41C23×C61
prime radicalC61×#3540C2
LCSG ▷ C2×D4 ▷ 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×#2134C2semidirect
3. C61×#2123C2semidirect
4. C61×#2120C2semidirect
5. C61×#1915C2semidirect
6. C61×#1910C2semidirect
7. C61×#1891C2semidirect
10. C61×#560C4semidirect
11. C61×#559C4semidirect
250. #12758C61direct (N × H)
280. #2134C2×C61semidirect
281. #2123C2×C61semidirect
282. #2120C2×C61semidirect
283. #1915C2×C61semidirect
284. #1910C2×C61semidirect
285. #1891C2×C61semidirect
295. #560C4×C61semidirect
296. #559C4×C61semidirect
322. C61#12758direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35402
C61×#21342
C61×#21232
C61×#21202
C61×#19152
C61×#19102
C61×#18912
#1275861