Finite Group Databasebeta

← C59×#23642 #28435751

Subgroups — C59×#23642 #28435751

Order 15104.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#428C211yesmaximal, characteristic, complemented
2C2×C59×#407C211yesmaximal, characteristic
3C59×#2280C211yesmaximal, characteristic
4C2×C59×#380C211yesmaximal, characteristic
5C59×#2077C212yesmaximal, complemented
6C59×#2077C212yesmaximal, complemented
7C59×#2076C212yesmaximal, complemented
8C59×#2075C212yesmaximal, complemented
9C59×#1736C211yesmaximal, characteristic, complemented
10C59×#1736C211yesmaximal, characteristic, complemented
11C59×#1736C211yesmaximal, characteristic, complemented
12C24×C4×C59C2211yescharacteristic
13C22×M16×C59C2211yescharacteristic
14C22×M16×C59C2211yescharacteristic
15C2×#117×C59C2211yescharacteristic
16C2×#117×C59C2211yescharacteristic
17C22×C4⋊C4×C59C2211yescharacteristic
18C22×C4⋊C4×C59C2211yescharacteristic
19C22×C22⋊C4×C59—21no—
20C59×#428—24no—
21C2×#107×C59C2212yescomplemented
22C2×#106×C59C2212yescomplemented
23C59×#407—24no—
24C2×C22⋊C8×C59C2212yes—
25C2×C22⋊C8×C59C2212yes—
26C59×#381—21no—
27C59×#380—21no—
28C59×#380—21no—
29C59×#380C2212yes—
30C59×#380C2212yes—
31C59×#380C2211yescharacteristic
32C59×#380C2211yescharacteristic
33C59×#380C2211yescharacteristic
34C59×#380C2211yescharacteristic
35C59×#379—21no—
36C4×C22⋊C4×C59—21no—
37C4×C22⋊C4×C59—21no—
38C2×#95×C59—21no—
39C59×#340C2212yescomplemented
40C59×#340C2212yescomplemented
41C59×#340C2212yescomplemented
42C59×#340C2212yescomplemented
43C59×#340C2212yescomplemented
44C59×#340C2212yescomplemented
45C25×C59D411yescharacteristic
46C23×C4×C59—22no—
47C23×C4×C59—22no—
48C23×C4×C59—22no—
49C23×C4×C59—22no—
50C23×C4×C59—22no—
51C23×C4×C59—22no—
52C23×C4×C59—21no—
53C23×C4×C59—21no—
54C23×C4×C59D411yescharacteristic
55C23×C4×C59D411yescharacteristic
56C23×C4×C59D411yescharacteristic
57C23×C4×C59C2311yescharacteristic
58C23×C4×C59C2311yescharacteristic
59C23×C4×C59C2311yescharacteristic
60C2×M16×C59—22no—
61C2×M16×C59—22no—
62C2×M16×C59—22no—
63C2×M16×C59—22no—
64C2×M16×C59—22no—
65C2×M16×C59D414yes—
66C22×C8×C59C2312yes—
67C22×C8×C59C2312yes—
68#117×C59—22no—
69#117×C59—22no—
70#117×C59—22no—
71#117×C59—22no—
72C2×C4⋊C4×C59—22no—
73C2×C4⋊C4×C59—22no—
74C2×C4⋊C4×C59—22no—
75C2×C4⋊C4×C59—22no—
76C2×C4⋊C4×C59—21no—
77C2×C4⋊C4×C59—21no—
78C2×C4⋊C4×C59C2312yes—
79C2×C4⋊C4×C59C2311yescharacteristic
80C2×C4⋊C4×C59C2311yescharacteristic
81C2×C4⋊C4×C59C2311yescharacteristic
82C2×C4⋊C4×C59C2312yes—
83C2×C4⋊C4×C59C2311yescharacteristic
84C2×C22⋊C4×C59—21no—
85C2×C22⋊C4×C59—21no—
86C2×C22⋊C4×C59—21no—
87C2×C22⋊C4×C59—21no—
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—22no—
91C2×C22⋊C4×C59—22no—
92C2×C22⋊C4×C59—44no—
93C2×C42×C59—21no—
94C2×C42×C59—21no—
95#107×C59—24no—
96#106×C59—24no—
97C22⋊C8×C59—24no—
98C22⋊C8×C59—24no—
99#95×C59—22no—
100#95×C59—22no—
101#95×C59—21no—
102#95×C59—21no—
103#95×C59—21no—
104#95×C59—21no—
105C24×C59—22no—
106C24×C59—22no—
107C24×C59—22no—
108C24×C59—22no—
109C24×C59—22no—
110C24×C59—41no—
111C24×C59—41no—
112C24×C59C2×D411yescharacteristic
113C24×C59C2×D411yescharacteristic
114C24×C59C2×D411yescharacteristic
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—44no—
143C22×C4×C59—44no—
144C22×C4×C59—44no—
145C22×C4×C59—44no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D412yes—
156C22×C4×C59#4112yes—
157C22×C4×C59#4112yes—
158C22×C4×C59#4112yes—
159C22×C4×C59#4112yes—
160C22×C4×C59#4112yes—
161C22×C4×C59C2411yescharacteristic
162M16×C59—28no—
163M16×C59—44no—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59C2×D414yes—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—42no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C22⋊C4×C59—44no—
187C22⋊C4×C59—42no—
188C22⋊C4×C59—42no—
189C22⋊C4×C59—42no—
190C22⋊C4×C59—44no—
191C22⋊C4×C59—42no—
192C42×C59—42no—
193C42×C59—42no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—28no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—44no—
212C23×C59—44no—
213C23×C59—44no—
214C23×C59—44no—
215C23×C59—44no—
216C23×C59—44no—
217C23×C59#12112yes—
218C23×C59#12312yes—
219C23×C59#12312yes—
220C23×C59AGL(1,ℤ/8)12yes—
221C23×C59#13712yes—
222C23×C59C22×D411yescharacteristic
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—44no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—44no—
239C2×C4×C59—44no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—42no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—42no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—42no—
257C2×C4×C59—42no—
258C2×C4×C59—84no—
259C2×C4×C59—84no—
260C2×C4×C59#12112yes—
261C2×C4×C59#12314yes—
262C2×C4×C59#12312yes—
263C2×C4×C59#12312yes—
264C2×C4×C59#12414yes—
265C2×C4×C59#12614yes—
266C2×C4×C59#12614yes—
267C2×C4×C59C22×D411yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59C2×#4111yescharacteristic
271C2×C4×C59C2×#4111yescharacteristic
272C2×C4×C59C2×#4111yescharacteristic
273C8×C59—24no—
274C8×C59—42no—
275#23642C5911yesmaximal, characteristic, direct product factor, Sylow
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—416no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59#39714yes—
298C22×C59C2×#12111yescharacteristic
299C22×C59C2×#12311yescharacteristic
300C22×C59C2×#12311yescharacteristic
301C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C59C2×#13711yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57511yescharacteristic
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—81no—
308C4×C59—81no—
309C4×C59—82no—
310C4×C59—81no—
311C4×C59—82no—
312C4×C59—81no—
313C4×C59#39714yes—
314C4×C59C2×#12311yescharacteristic
315C4×C59C2×#12411yescharacteristic
316C4×C59C2×#12611yescharacteristic
317C4×C59C2×#12611yescharacteristic
318C2×#428C2×C5911yescharacteristic, complemented
319C2×#407C2×C5911yescharacteristic
320#2280C2×C5911yescharacteristic
321C2×#380C2×C5911yescharacteristic
322#2077C2×C5912yescomplemented
323#2077C2×C5912yescomplemented
324#2076C2×C5912yescomplemented
325#2075C2×C5912yescomplemented
326#1736C2×C5911yescharacteristic, complemented
327#1736C2×C5911yescharacteristic, complemented
328#1736C2×C5911yescharacteristic, complemented
329C2×C59—24no—
330C2×C59—42no—
331C2×C59—42no—
332C2×C59C2×#39711yescharacteristic
333C2×C59#305511yescharacteristic
334C2×C59#305611yescharacteristic
335C2×C59#309211yescharacteristic
336C2×C59#309311yescharacteristic
337C2×C59#309411yescharacteristic
338C2×C59#309511yescharacteristic
339C24×C4C22×C5911yescharacteristic
340C22×M16C22×C5911yescharacteristic
341C22×M16C22×C5911yescharacteristic
342C2×#117C22×C5911yescharacteristic
343C2×#117C22×C5911yescharacteristic
344C22×C4⋊C4C22×C5911yescharacteristic
345C22×C4⋊C4C22×C5911yescharacteristic
346C22×C22⋊C4—21no—
347#428—24no—
348C2×#107C22×C5912yescomplemented
349C2×#106C22×C5912yescomplemented
350#407—24no—
351C2×C22⋊C8C22×C5912yes—
352C2×C22⋊C8C22×C5912yes—
353#381—21no—
354#380—21no—
355#380—21no—
356#380C22×C5911yescharacteristic
357#380C22×C5912yes—
358#380C22×C5911yescharacteristic
359#380C22×C5912yes—
360#380C22×C5911yescharacteristic
361#380C22×C5911yescharacteristic
362#379—21no—
363C4×C22⋊C4—21no—
364C4×C22⋊C4—21no—
365C2×#95—21no—
366#340C22×C5912yescomplemented
367#340C22×C5912yescomplemented
368#340C22×C5912yescomplemented
369#340C22×C5912yescomplemented
370#340C22×C5912yescomplemented
371#340C22×C5912yescomplemented
372C59#2364211yescharacteristic, direct product factor, Sylow
373C25D4×C5911yescharacteristic
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—21no—
381C23×C4—21no—
382C23×C4D4×C5911yescharacteristic
383C23×C4D4×C5911yescharacteristic
384C23×C4D4×C5911yescharacteristic
385C23×C4C23×C5911yescharacteristic
386C23×C4C23×C5911yescharacteristic
387C23×C4C23×C5911yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—22no—
392C2×M16—22no—
393C2×M16D4×C5914yes—
394C22×C8C23×C5912yes—
395C22×C8C23×C5912yes—
396#117—22no—
397#117—22no—
398#117—22no—
399#117—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4C23×C5912yes—
407C2×C4⋊C4C23×C5912yes—
408C2×C4⋊C4C23×C5911yescharacteristic
409C2×C4⋊C4C23×C5911yescharacteristic
410C2×C4⋊C4C23×C5911yescharacteristic
411C2×C4⋊C4C23×C5911yescharacteristic
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—44no—
421C2×C42—21no—
422C2×C42—21no—
423#107—24no—
424#106—24no—
425C22⋊C8—24no—
426C22⋊C8—24no—
427#95—22no—
428#95—22no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—41no—
439C24—41no—
440C24C2×D4×C5911yescharacteristic
441C24C2×D4×C5911yescharacteristic
442C24C2×D4×C5911yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C2×D4×C5911yescharacteristic
481C22×C4C2×D4×C5911yescharacteristic
482C22×C4C2×D4×C5911yescharacteristic
483C22×C4C2×D4×C5912yes—
484C22×C4#41×C5912yes—
485C22×C4#41×C5912yes—
486C22×C4#41×C5912yes—
487C22×C4#41×C5912yes—
488C22×C4#41×C5912yes—
489C22×C4C24×C5911yescharacteristic
490M16—28no—
491M16—44no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8C2×D4×C5914yes—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C42—42no—
521C42—42no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#121×C5912yes—
546C23#123×C5912yes—
547C23#123×C5912yes—
548C23AGL(1,ℤ/8)×C5912yes—
549C23#137×C5912yes—
550C23C22×D4×C5911yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#121×C5912yes—
589C2×C4#123×C5912yes—
590C2×C4#123×C5912yes—
591C2×C4#123×C5914yes—
592C2×C4#124×C5914yes—
593C2×C4#126×C5914yes—
594C2×C4#126×C5914yes—
595C2×C4C22×D4×C5911yescharacteristic
596C2×C4C2×#41×C5911yescharacteristic
597C2×C4C2×#41×C5911yescharacteristic
598C2×C4C2×#41×C5911yescharacteristic
599C2×C4C2×#41×C5911yescharacteristic
600C2×C4C2×#41×C5911yescharacteristic
601C8—24no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22C59×#39714yes—
625C22C2×#121×C5911yescharacteristic
626C22C2×#123×C5911yescharacteristic
627C22C2×#123×C5911yescharacteristic
628C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
629C22C2×#137×C5911yescharacteristic
630C22C59×#57511yescharacteristic
631C22C59×#57511yescharacteristic
632C4—41no—
633C4—41no—
634C4—82no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—82no—
640C4C59×#39714yes—
641C4C2×#123×C5911yescharacteristic
642C4C2×#124×C5911yescharacteristic
643C4C2×#126×C5911yescharacteristic
644C4C2×#126×C5911yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×C59×#39711yescharacteristic
649C2C59×#305511yescharacteristic
650C2C59×#305611yescharacteristic
651C2C59×#309211yescharacteristic
652C2C59×#309311yescharacteristic
653C2C59×#309411yescharacteristic
654C2C59×#309511yescharacteristic
subgroupG/H
center, socleC23×C59C22×D4
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC25×C59D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C24×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C59×#428C2semidirect
5. C59×#2077C2semidirect
6. C59×#2077C2semidirect
7. C59×#2076C2semidirect
8. C59×#2075C2semidirect
9. C59×#1736C2semidirect
10. C59×#1736C2semidirect
11. C59×#1736C2semidirect
21. C2×#107×C59C22semidirect
22. C2×#106×C59C22semidirect
39. C59×#340C22semidirect
40. C59×#340C22semidirect
41. C59×#340C22semidirect
42. C59×#340C22semidirect
43. C59×#340C22semidirect
44. C59×#340C22semidirect
275. #23642C59direct (N × H)
318. C2×#428C2×C59semidirect
322. #2077C2×C59semidirect
323. #2077C2×C59semidirect
324. #2076C2×C59semidirect
325. #2075C2×C59semidirect
326. #1736C2×C59semidirect
327. #1736C2×C59semidirect
328. #1736C2×C59semidirect
348. C2×#107C22×C59semidirect
349. C2×#106C22×C59semidirect
366. #340C22×C59semidirect
367. #340C22×C59semidirect
368. #340C22×C59semidirect
369. #340C22×C59semidirect
370. #340C22×C59semidirect
371. #340C22×C59semidirect
372. C59#23642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#4282
C2×C59×#4072
C59×#22802
C2×C59×#3802
C59×#20772
C59×#20762
C59×#20752
C59×#17362
#2364259