Finite Group Databasebeta

← C61×#12690 #29516123

Subgroups — C61×#12690 #29516123

Order 15616.

1228 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#534C211yesmaximal, characteristic
2C2×C61×#352C212yesmaximal, complemented
3C2×C61×AΣL(1,𝔽16)C212yesmaximal, complemented
4C61×#1783C211yesmaximal, characteristic, complemented
5C61×#1781C211yesmaximal, characteristic, complemented
6C23×D4×C61C411yescharacteristic, complemented
7C23×D4×C61C2211yescharacteristic
8C61×#534—24no—
9C61×#534—24no—
10C2×#123×C61—21no—
11C2×#123×C61C411yescharacteristic, complemented
12C2×#123×C61C2211yescharacteristic
13C2×#121×C61—21no—
14C2×#121×C61—21no—
15C2×#120×C61—21no—
16C2×#120×C61C411yescharacteristic, complemented
17C2×#120×C61C2211yescharacteristic
18C22×C22⋊C4×C61C411yescharacteristic, complemented
19C2×#100×C61—21no—
20C2×#100×C61C2212yes—
21C2×#99×C61—21no—
22C2×#99×C61C2212yes—
23C61×#407—21no—
24C61×#379—21no—
25C61×#352—24no—
26C61×AΣL(1,𝔽16)—24no—
27C25×C61C2×C411yescharacteristic
28C25×C61C2×C411yescharacteristic
29C25×C61D411yescharacteristic
30C22×D4×C61—21no—
31C22×D4×C61—21no—
32C22×D4×C61—22no—
33C22×D4×C61—21no—
34C22×D4×C61—21no—
35C22×D4×C61—22no—
36C22×D4×C61—22no—
37C22×D4×C61—22no—
38C22×D4×C61—42no—
39C22×D4×C61—42no—
40C22×D4×C61—42no—
41C22×D4×C61—42no—
42C22×D4×C61C2×C411yescharacteristic
43C22×D4×C61D411yescharacteristic
44C22×D4×C61D411yescharacteristic
45C22×D4×C61C2311yescharacteristic
46C23×C4×C61—21no—
47C23×C4×C61C2×C411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C2×M16×C61—22no—
50#123×C61—28no—
51#123×C61—22no—
52#123×C61—22no—
53#123×C61—22no—
54#123×C61D414yes—
55#121×C61—28no—
56#121×C61—28no—
57#120×C61—28no—
58#120×C61—22no—
59#120×C61—22no—
60#120×C61—22no—
61#120×C61D414yes—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—22no—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61—42no—
73C2×C22⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61C2×C411yescharacteristic
75#100×C61—22no—
76#100×C61—22no—
77#100×C61—42no—
78#99×C61—22no—
79#99×C61—22no—
80#99×C61—42no—
81C22⋊C8×C61—42no—
82C24×C61—22no—
83C24×C61—22no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—22no—
88C24×C61—21no—
89C24×C61—22no—
90C24×C61—41no—
91C24×C61—41no—
92C24×C61—41no—
93C24×C61—82no—
94C24×C61—82no—
95C24×C61—82no—
96C24×C61C22⋊C411yescharacteristic
97C24×C61C22⋊C411yescharacteristic
98C24×C61C22⋊C411yescharacteristic
99C24×C61C22⋊C412yescomplemented
100C24×C61C22×C411yescharacteristic
101C24×C61C2×D411yescharacteristic
102C2×D4×C61—24no—
103C2×D4×C61—24no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—24no—
110C2×D4×C61—24no—
111C2×D4×C61—22no—
112C2×D4×C61—41no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—48no—
119C2×D4×C61—48no—
120C2×D4×C61—48no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—48no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—82no—
127C2×D4×C61—82no—
128C2×D4×C61C2×D412yes—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61—41no—
140C22×C4×C61—42no—
141C22×C4×C61—82no—
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C412yescomplemented
144C22×C4×C61C2×D411yescharacteristic
145M16×C61—42no—
146M16×C61—42no—
147C2×C8×C61—42no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—42no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—84no—
160C22⋊C4×C61C22⋊C412yescomplemented
161C22⋊C4×C61C22⋊C412yescomplemented
162C23×C61—24no—
163C23×C61—24no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—42no—
177C23×C61—44no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—44no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—82no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61—84no—
193C23×C61—82no—
194C23×C61—82no—
195C23×C61—82no—
196C23×C61—84no—
197C23×C61—82no—
198C23×C61—84no—
199C23×C61—84no—
200C23×C61—84no—
201C23×C61#9912yes—
202C23×C61#9912yes—
203C23×C61C2×C22⋊C411yescharacteristic
204C23×C61C2×C22⋊C411yescharacteristic
205C23×C61C2×C22⋊C411yescharacteristic
206C23×C61#12012yes—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—44no—
210D4×C61—44no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—48no—
214D4×C61—48no—
215D4×C61—84no—
216D4×C61—84no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—44no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—84no—
233C2×C4×C61—82no—
234C2×C4×C61—84no—
235C2×C4×C61#12012yes—
236C8×C61—82no—
237#12690C6111yesmaximal, characteristic, direct product factor, Sylow
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—42no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—41no—
262C22×C61—42no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—82no—
266C22×C61—84no—
267C22×C61—84no—
268C22×C61—88no—
269C22×C61—84no—
270C22×C61—84no—
271C22×C61—82no—
272C22×C61#37911yescharacteristic
273C22×C61C2×#9911yescharacteristic
274C22×C61C2×#9911yescharacteristic
275C4×C61—22no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61—162no—
281C2×#534C2×C6111yescharacteristic
282C2×#352C2×C6112yescomplemented
283C2×AΣL(1,𝔽16)C2×C6112yescomplemented
284#1783C2×C6111yescharacteristic, complemented
285#1781C2×C6111yescharacteristic, complemented
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—42no—
289C2×C61—42no—
290C2×C61—42no—
291C2×C61—41no—
292C2×C61—41no—
293C2×C61—81no—
294C2×C61—82no—
295C2×C61#178111yescharacteristic
296C2×C61#212112yes—
297C23×D4C4×C6111yescharacteristic, complemented
298C23×D4C22×C6111yescharacteristic
299#534—24no—
300#534—24no—
301C2×#123—21no—
302C2×#123C4×C6111yescharacteristic, complemented
303C2×#123C22×C6111yescharacteristic
304C2×#121—21no—
305C2×#121—21no—
306C2×#120—21no—
307C2×#120C4×C6111yescharacteristic, complemented
308C2×#120C22×C6111yescharacteristic
309C22×C22⋊C4C4×C6111yescharacteristic, complemented
310C2×#100—21no—
311C2×#100C22×C6112yes—
312C2×#99—21no—
313C2×#99C22×C6112yes—
314#407—21no—
315#379—21no—
316#352—24no—
317AΣL(1,𝔽16)—24no—
318C61#1269011yescharacteristic, direct product factor, Sylow
319C25C2×C4×C6111yescharacteristic
320C25C2×C4×C6111yescharacteristic
321C25D4×C6111yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4—42no—
334C22×D4C2×C4×C6111yescharacteristic
335C22×D4D4×C6111yescharacteristic
336C22×D4D4×C6111yescharacteristic
337C22×D4C23×C6111yescharacteristic
338C23×C4—21no—
339C23×C4C2×C4×C6111yescharacteristic
340C23×C4D4×C6111yescharacteristic
341C2×M16—22no—
342#123—28no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123D4×C6114yes—
347#121—28no—
348#121—28no—
349#120—28no—
350#120—22no—
351#120—22no—
352#120—22no—
353#120D4×C6114yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4C2×C4×C6111yescharacteristic
366C2×C22⋊C4C2×C4×C6111yescharacteristic
367#100—22no—
368#100—22no—
369#100—42no—
370#99—22no—
371#99—22no—
372#99—42no—
373C22⋊C8—42no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24C22⋊C4×C6111yescharacteristic
389C24C22⋊C4×C6111yescharacteristic
390C24C22⋊C4×C6111yescharacteristic
391C24C22⋊C4×C6112yescomplemented
392C24C22×C4×C6111yescharacteristic
393C24C2×D4×C6111yescharacteristic
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×D4×C6112yes—
421C2×D4C2×D4×C6112yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4C22⋊C4×C6111yescharacteristic
435C22×C4C22⋊C4×C6112yescomplemented
436C22×C4C2×D4×C6111yescharacteristic
437M16—42no—
438M16—42no—
439C2×C8—42no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—84no—
452C22⋊C4C22⋊C4×C6112yescomplemented
453C22⋊C4C22⋊C4×C6112yescomplemented
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23#99×C6112yes—
494C23#99×C6112yes—
495C23C2×C22⋊C4×C6111yescharacteristic
496C23C2×C22⋊C4×C6111yescharacteristic
497C23C2×C22⋊C4×C6111yescharacteristic
498C23#120×C6112yes—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—48no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4#120×C6112yes—
528C8—82no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—88no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22C61×#37911yescharacteristic
564C22C2×#99×C6111yescharacteristic
565C22C2×#99×C6111yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—162no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—41no—
577C2—41no—
578C2—42no—
579C2—81no—
580C2—82no—
581C2C61×#178111yescharacteristic
582C2C61×#212112yes—
subgroupG/H
center, socleC22×C61C2×#99
commutatorC24C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC2×C61×#534C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C24×C61 ◁ C23×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#352C2semidirect
3. C2×C61×AΣL(1,𝔽16)C2semidirect
4. C61×#1783C2semidirect
5. C61×#1781C2semidirect
6. C23×D4×C61C4semidirect
11. C2×#123×C61C4semidirect
16. C2×#120×C61C4semidirect
18. C22×C22⋊C4×C61C4semidirect
99. C24×C61C22⋊C4semidirect
143. C22×C4×C61C22⋊C4semidirect
160. C22⋊C4×C61C22⋊C4semidirect
161. C22⋊C4×C61C22⋊C4semidirect
237. #12690C61direct (N × H)
282. C2×#352C2×C61semidirect
283. C2×AΣL(1,𝔽16)C2×C61semidirect
284. #1783C2×C61semidirect
285. #1781C2×C61semidirect
297. C23×D4C4×C61semidirect
302. C2×#123C4×C61semidirect
307. C2×#120C4×C61semidirect
309. C22×C22⋊C4C4×C61semidirect
318. C61#12690direct (N × H)
391. C24C22⋊C4×C61semidirect
435. C22×C4C22⋊C4×C61semidirect
452. C22⋊C4C22⋊C4×C61semidirect
453. C22⋊C4C22⋊C4×C61semidirect

Maximal subgroups

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

subgroupindex
C2×C61×#5342
C2×C61×#3522
C2×C61×AΣL(1,𝔽16)2
C61×#17832
C61×#17812
#1269061