Finite Group Databasebeta

← C59×#14630 #28426739

Subgroups — C59×#14630 #28426739

Order 15104.

1326 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#397C211yesmaximal, characteristic
2C2×C59×#389C211yesmaximal, characteristic, complemented
3C2×C59×#388C211yesmaximal, characteristic
4C59×#1445C211yesmaximal, characteristic, complemented
5C59×#1444C211yesmaximal, characteristic, complemented
6C59×#1444C212yesmaximal, complemented
7C59×#1443C212yesmaximal, complemented
8C59×#1439C212yesmaximal
9C59×#1439C211yesmaximal, characteristic
10C59×#1439C212yesmaximal
11C59×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C59C2211yescharacteristic
13C22×C4⋊C4×C59C2211yescharacteristic
14C22×C4⋊C4×C59C2211yescharacteristic
15C22×C4⋊C4×C59C2212yes—
16C22×C4⋊C4×C59C2212yes—
17C22×C22⋊C4×C59C2211yescharacteristic
18C22×C22⋊C4×C59C2211yescharacteristic
19C22×C22⋊C4×C59C2211yescharacteristic
20C22×C22⋊C4×C59C2212yes—
21C22×C22⋊C4×C59C2212yes—
22C22×C42×C59C2211yescharacteristic
23C59×#397C418yescomplemented
24C59×#389—24no—
25C59×#388—24no—
26C2×#95×C59C2211yescharacteristic
27C2×#95×C59C2211yescharacteristic
28C2×#95×C59C2212yes—
29C2×#95×C59C2211yescharacteristic
30C2×#95×C59C2212yes—
31C2×#95×C59C2212yes—
32C2×#95×C59C2212yes—
33C2×#95×C59C2212yes—
34C2×#95×C59C2212yes—
35C2×#95×C59C2212yes—
36C2×#95×C59C2211yescharacteristic
37C2×#95×C59C2212yes—
38C25×C59C2311yescharacteristic
39C23×C4×C59C2311yescharacteristic
40C23×C4×C59C2312yes—
41C23×C4×C59C2311yescharacteristic
42C23×C4×C59C2312yes—
43C23×C4×C59C2311yescharacteristic
44C23×C4×C59C2312yes—
45C23×C4×C59C2312yes—
46C23×C4×C59C2311yescharacteristic
47C23×C4×C59C2311yescharacteristic
48C23×C4×C59C2311yescharacteristic
49C2×C4⋊C4×C59—22no—
50C2×C4⋊C4×C59—22no—
51C2×C4⋊C4×C59—22no—
52C2×C4⋊C4×C59—24no—
53C2×C4⋊C4×C59—22no—
54C2×C4⋊C4×C59—42no—
55C2×C4⋊C4×C59—42no—
56C2×C4⋊C4×C59—42no—
57C2×C4⋊C4×C59—42no—
58C2×C4⋊C4×C59—42no—
59C2×C4⋊C4×C59—42no—
60C2×C4⋊C4×C59—42no—
61C2×C4⋊C4×C59C2×C414yes—
62C2×C22⋊C4×C59—22no—
63C2×C22⋊C4×C59—22no—
64C2×C22⋊C4×C59—22no—
65C2×C22⋊C4×C59—22no—
66C2×C22⋊C4×C59—42no—
67C2×C22⋊C4×C59—42no—
68C2×C22⋊C4×C59—42no—
69C2×C22⋊C4×C59—42no—
70C2×C22⋊C4×C59—42no—
71C2×C22⋊C4×C59—42no—
72C2×C22⋊C4×C59—42no—
73C2×C22⋊C4×C59C2×C414yes—
74C2×C22⋊C4×C59C2×C418yes—
75C2×C42×C59—42no—
76C2×C42×C59D414yes—
77#95×C59—22no—
78#95×C59—22no—
79#95×C59—24no—
80#95×C59—22no—
81#95×C59—42no—
82#95×C59—42no—
83#95×C59—42no—
84#95×C59—42no—
85#95×C59—42no—
86#95×C59—42no—
87#95×C59C2×C418yes—
88#95×C59C2×C414yes—
89C24×C59—21no—
90C24×C59—21no—
91C24×C59—21no—
92C24×C59—21no—
93C24×C59—22no—
94C24×C59—22no—
95C24×C59—21no—
96C24×C59—22no—
97C24×C59—22no—
98C24×C59—21no—
99C24×C59C22×C412yes—
100C24×C59C2411yescharacteristic
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—21no—
104C22×C4×C59—22no—
105C22×C4×C59—21no—
106C22×C4×C59—22no—
107C22×C4×C59—21no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—21no—
132C22×C4×C59—22no—
133C22×C4×C59—21no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—21no—
147C22×C4×C59—22no—
148C22×C4×C59—21no—
149C22×C4×C59—22no—
150C22×C4×C59—22no—
151C22×C4×C59C22×C412yes—
152C22×C4×C59C22×C414yes—
153C22×C4×C59C22×C414yes—
154C22×C4×C59C22×C412yes—
155C22×C4×C59C2×D412yes—
156C22×C4×C59C2×D412yes—
157C22×C4×C59C2×D412yes—
158C22×C4×C59#4112yes—
159C4⋊C4×C59—28no—
160C4⋊C4×C59—44no—
161C4⋊C4×C59—44no—
162C4⋊C4×C59—48no—
163C4⋊C4×C59—48no—
164C22⋊C4×C59—28no—
165C22⋊C4×C59—48no—
166C22⋊C4×C59—44no—
167C22⋊C4×C59—44no—
168C22⋊C4×C59—48no—
169C42×C59—44no—
170C23×C59—22no—
171C23×C59—22no—
172C23×C59—24no—
173C23×C59—22no—
174C23×C59—24no—
175C23×C59—41no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59—42no—
182C23×C59—41no—
183C23×C59—41no—
184C23×C59—42no—
185C23×C59—42no—
186C23×C59—42no—
187C23×C59—42no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—42no—
191C23×C59—42no—
192C23×C59—42no—
193C23×C59C23×C411yescharacteristic
194C23×C59C22×D411yescharacteristic
195C23×C59C2×#4111yescharacteristic
196C23×C59H(5,𝔽2)11yescharacteristic
197C23×C59H(5,𝔽2)11yescharacteristic
198C23×C59H(5,𝔽2)12yes—
199C23×C59H(5,𝔽2)11yescharacteristic
200C23×C59H(5,𝔽2)12yes—
201C23×C592−1+412yes—
202C23×C592−1+412yes—
203C23×C592−1+411yescharacteristic
204C2×C4×C59—24no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—24no—
208C2×C4×C59—22no—
209C2×C4×C59—24no—
210C2×C4×C59—24no—
211C2×C4×C59—24no—
212C2×C4×C59—24no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—24no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—42no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—41no—
238C2×C4×C59—42no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—42no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—42no—
257C2×C4×C59—42no—
258C2×C4×C59—42no—
259C2×C4×C59—42no—
260C2×C4×C59—42no—
261C2×C4×C59C4×D414yes—
262#14630C5911yesmaximal, characteristic, direct product factor, Sylow
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—44no—
267C22×C59—44no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59—82no—
271C22×C59—82no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59C2×C4×D411yescharacteristic
275C22×C59#51811yescharacteristic
276C22×C59#52012yes—
277C22×C59#52011yescharacteristic
278C22×C59#52012yes—
279C22×C59#53411yescharacteristic
280C22×C59#53511yescharacteristic
281C22×C59#53511yescharacteristic
282C22×C59#53611yescharacteristic
283C22×C59#53611yescharacteristic
284C22×C59#53711yescharacteristic
285C22×C59#53912yes—
286C22×C59#54011yescharacteristic
287C22×C59#54211yescharacteristic
288C22×C59#54212yes—
289C22×C59#56012yes—
290C22×C59#56012yes—
291C22×C59#56212yes—
292C22×C59#56212yes—
293C22×C59#56312yes—
294C22×C59#56312yes—
295C22×C59#56312yes—
296C22×C59#56312yes—
297C4×C59—24no—
298C4×C59—44no—
299C4×C59—42no—
300C4×C59—44no—
301C4×C59—42no—
302C4×C59—42no—
303C4×C59—81no—
304C4×C59—82no—
305C4×C59—81no—
306C4×C59—82no—
307C2×#397C2×C5911yescharacteristic
308C2×#389C2×C5911yescharacteristic, complemented
309C2×#388C2×C5911yescharacteristic
310#1445C2×C5911yescharacteristic, complemented
311#1444C2×C5912yescomplemented
312#1444C2×C5911yescharacteristic, complemented
313#1443C2×C5912yescomplemented
314#1439C2×C5911yescharacteristic
315#1439C2×C5912yes—
316#1439C2×C5912yes—
317#1434C2×C5911yescharacteristic, complemented
318C2×C59—82no—
319C2×C59#232211yescharacteristic
320C2×C59#232411yescharacteristic
321C2×C59#232611yescharacteristic
322C2×C59#237612yes—
323C2×C59#238312yes—
324C2×C59#281411yescharacteristic
325C2×C59#281511yescharacteristic
326C2×C59#282011yescharacteristic
327C2×C59#282111yescharacteristic
328C2×C59#283712yes—
329C2×C59#283812yes—
330C22×C4⋊C4C22×C5911yescharacteristic
331C22×C4⋊C4C22×C5911yescharacteristic
332C22×C4⋊C4C22×C5912yes—
333C22×C4⋊C4C22×C5911yescharacteristic
334C22×C4⋊C4C22×C5912yes—
335C22×C22⋊C4C22×C5911yescharacteristic
336C22×C22⋊C4C22×C5911yescharacteristic
337C22×C22⋊C4C22×C5911yescharacteristic
338C22×C22⋊C4C22×C5912yes—
339C22×C22⋊C4C22×C5912yes—
340C22×C42C22×C5911yescharacteristic
341#397C4×C5918yescomplemented
342#389—24no—
343#388—24no—
344C2×#95C22×C5911yescharacteristic
345C2×#95C22×C5912yes—
346C2×#95C22×C5911yescharacteristic
347C2×#95C22×C5911yescharacteristic
348C2×#95C22×C5912yes—
349C2×#95C22×C5912yes—
350C2×#95C22×C5912yes—
351C2×#95C22×C5912yes—
352C2×#95C22×C5912yes—
353C2×#95C22×C5912yes—
354C2×#95C22×C5911yescharacteristic
355C2×#95C22×C5912yes—
356C59#1463011yescharacteristic, direct product factor, Sylow
357C25C23×C5911yescharacteristic
358C23×C4C23×C5911yescharacteristic
359C23×C4C23×C5912yes—
360C23×C4C23×C5911yescharacteristic
361C23×C4C23×C5912yes—
362C23×C4C23×C5912yes—
363C23×C4C23×C5911yescharacteristic
364C23×C4C23×C5912yes—
365C23×C4C23×C5911yescharacteristic
366C23×C4C23×C5911yescharacteristic
367C23×C4C23×C5911yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4C2×C4×C5914yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
392C2×C22⋊C4C2×C4×C5914yes—
393C2×C22⋊C4C2×C4×C5918yes—
394C2×C42—42no—
395C2×C42D4×C5914yes—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—22no—
400#95—42no—
401#95—42no—
402#95—42no—
403#95—42no—
404#95—42no—
405#95—42no—
406#95C2×C4×C5918yes—
407#95C2×C4×C5914yes—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—21no—
412C24—22no—
413C24—22no—
414C24—21no—
415C24—22no—
416C24—22no—
417C24—21no—
418C24C22×C4×C5912yes—
419C24C24×C5911yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4C22×C4×C5912yes—
471C22×C4C22×C4×C5914yes—
472C22×C4C22×C4×C5914yes—
473C22×C4C22×C4×C5912yes—
474C22×C4C2×D4×C5912yes—
475C22×C4C2×D4×C5912yes—
476C22×C4C2×D4×C5912yes—
477C22×C4#41×C5912yes—
478C4⋊C4—28no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C22⋊C4—28no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C42—44no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23C23×C4×C5911yescharacteristic
513C23C22×D4×C5911yescharacteristic
514C23C2×#41×C5911yescharacteristic
515C23H(5,𝔽2)×C5911yescharacteristic
516C23H(5,𝔽2)×C5911yescharacteristic
517C23H(5,𝔽2)×C5912yes—
518C23H(5,𝔽2)×C5911yescharacteristic
519C23H(5,𝔽2)×C5912yes—
520C232−1+4×C5912yes—
521C232−1+4×C5912yes—
522C232−1+4×C5911yescharacteristic
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4C4×D4×C5914yes—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22C2×C4×D4×C5911yescharacteristic
593C22C59×#51811yescharacteristic
594C22C59×#52012yes—
595C22C59×#52011yescharacteristic
596C22C59×#52012yes—
597C22C59×#53411yescharacteristic
598C22C59×#53511yescharacteristic
599C22C59×#53511yescharacteristic
600C22C59×#53611yescharacteristic
601C22C59×#53611yescharacteristic
602C22C59×#53711yescharacteristic
603C22C59×#53912yes—
604C22C59×#54011yescharacteristic
605C22C59×#54211yescharacteristic
606C22C59×#54212yes—
607C22C59×#56012yes—
608C22C59×#56012yes—
609C22C59×#56212yes—
610C22C59×#56212yes—
611C22C59×#56312yes—
612C22C59×#56312yes—
613C22C59×#56312yes—
614C22C59×#56312yes—
615C4—24no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—42no—
620C4—44no—
621C4—81no—
622C4—82no—
623C4—81no—
624C4—82no—
625C2—82no—
626C2C59×#232211yescharacteristic
627C2C59×#232411yescharacteristic
628C2C59×#232611yescharacteristic
629C2C59×#237612yes—
630C2C59×#238312yes—
631C2C59×#281411yescharacteristic
632C2C59×#281511yescharacteristic
633C2C59×#282011yescharacteristic
634C2C59×#282111yescharacteristic
635C2C59×#283712yes—
636C2C59×#283812yes—
subgroupG/H
center, socleC24×C59C24
commutatorC23C23×C4×C59
Frattini, cosocleC24C24×C59
prime radicalC25×C59C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#389C2semidirect
4. C59×#1445C2semidirect
5. C59×#1444C2semidirect
6. C59×#1444C2semidirect
7. C59×#1443C2semidirect
11. C59×#1434C2semidirect
23. C59×#397C4semidirect
262. #14630C59direct (N × H)
308. C2×#389C2×C59semidirect
310. #1445C2×C59semidirect
311. #1444C2×C59semidirect
312. #1444C2×C59semidirect
313. #1443C2×C59semidirect
317. #1434C2×C59semidirect
341. #397C4×C59semidirect
356. C59#14630direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#3972
C2×C59×#3892
C2×C59×#3882
C59×#14452
C59×#14442
C59×#14432
C59×#14392
C59×#14342
#1463059