Finite Group Databasebeta

← C59×#24088 #28436197

Subgroups — C59×#24088 #28436197

Order 15104.

1354 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#547C211yesmaximal, characteristic
2C2×C59×#480C211yesmaximal, characteristic, complemented
3C22×#107×C59C211yesmaximal, characteristic, complemented
4C59×#2486C211yesmaximal, characteristic, complemented
5C59×#2057C212yesmaximal, complemented
6C59×#2033C212yesmaximal, complemented
7C59×#1893C211yesmaximal, characteristic, complemented
8C59×#1869C211yesmaximal, characteristic, complemented
9C59×#1862C212yesmaximal, complemented
10C59×#1852C212yesmaximal, complemented
11C59×#1823C211yesmaximal, characteristic, complemented
12C22×#41×C59—21no—
13C23×C8×C59C2211yescharacteristic
14C59×#547—24no—
15C59×#547—24no—
16C2×C4⋊Q8×C59C2211yescharacteristic
17C2×#123×C59—21no—
18C2×#123×C59—21no—
19C2×#123×C59—21no—
20C2×C22⋊Q8×C59—21no—
21C2×#121×C59C2211yescharacteristic
22C2×#121×C59C2211yescharacteristic
23C2×C4×D4×C59C2211yescharacteristic
24C2×C4×D4×C59C2211yescharacteristic
25C22×C4⋊C4×C59C2211yescharacteristic
26C22×C4⋊C4×C59C2211yescharacteristic
27C22×C4⋊C4×C59C2211yescharacteristic
28C59×#480—24no—
29C2×#107×C59—24no—
30C2×#107×C59C2212yes—
31C2×#107×C59C2212yes—
32C2×#107×C59C2212yes—
33C2×#102×C59C2212yes—
34C2×#102×C59C2212yes—
35C2×C22⋊C8×C59C2212yes—
36C2×C22⋊C8×C59C2211yescharacteristic
37C59×#398—21no—
38C59×#397—21no—
39C59×#389C2212yes—
40C59×#381—21no—
41C59×#380C2212yes—
42C59×#380C2211yescharacteristic
43C59×#340C2212yes—
44C59×#340C2212yes—
45C59×#340C2212yes—
46C59×#336C2212yes—
47C2×#41×C59—28no—
48C2×#41×C59—44no—
49C2×#41×C59—44no—
50C22×Q8×C59—21no—
51C22×D4×C59—21no—
52C22×D4×C59—21no—
53C22×D4×C59D411yescharacteristic
54C22×D4×C59C2311yescharacteristic
55C23×C4×C59—21no—
56C23×C4×C59—21no—
57C23×C4×C59—21no—
58C23×C4×C59—21no—
59C23×C4×C59C2311yescharacteristic
60C23×C4×C59C2311yescharacteristic
61C22×C8×C59—22no—
62C22×C8×C59—24no—
63C22×C8×C59C2312yes—
64C22×C8×C59C2312yes—
65C4⋊Q8×C59—22no—
66C4⋊Q8×C59—22no—
67#123×C59—28no—
68#123×C59—28no—
69#123×C59—44no—
70C22⋊Q8×C59—28no—
71#121×C59—22no—
72#121×C59—22no—
73#121×C59—22no—
74#121×C59D414yes—
75C4×D4×C59—22no—
76C4×D4×C59—22no—
77C4×D4×C59—22no—
78C4×D4×C59—22no—
79C2×C4⋊C4×C59—21no—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59—22no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59—22no—
84C2×C4⋊C4×C59—22no—
85C2×C4⋊C4×C59—22no—
86C2×C4⋊C4×C59—22no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59D411yescharacteristic
90C2×C4⋊C4×C59D414yes—
91C2×C4⋊C4×C59C2312yes—
92C2×C4⋊C4×C59C2312yes—
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59D411yescharacteristic
102C2×C22⋊C4×C59D411yescharacteristic
103C2×C42×C59C2311yescharacteristic
104#107×C59—24no—
105#107×C59—24no—
106#107×C59—24no—
107#102×C59—24no—
108#102×C59—24no—
109C22⋊C8×C59—22no—
110C22⋊C8×C59—24no—
111#95×C59—22no—
112#95×C59—21no—
113#95×C59—22no—
114C24×C59—21no—
115C24×C59C2×D411yescharacteristic
116C24×C59C2×D411yescharacteristic
117#41×C59—432no—
118C2×Q8×C59—24no—
119C2×Q8×C59—42no—
120C2×Q8×C59—42no—
121C2×D4×C59—22no—
122C2×D4×C59—22no—
123C2×D4×C59—24no—
124C2×D4×C59—24no—
125C2×D4×C59—21no—
126C2×D4×C59—22no—
127C2×D4×C59—22no—
128C2×D4×C59—42no—
129C2×D4×C59—42no—
130C2×D4×C59—42no—
131C2×D4×C59—42no—
132C2×D4×C59D814yes—
133C2×D4×C59C2×D412yes—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—24no—
137C22×C4×C59—24no—
138C22×C4×C59—24no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—42no—
153C22×C4×C59—42no—
154C22×C4×C59—42no—
155C22×C4×C59—42no—
156C22×C4×C59—42no—
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D412yes—
161C22×C4×C59C2×D412yes—
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59C2411yescharacteristic
164C2×C8×C59—28no—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59#4114yescomplemented
171C4⋊C4×C59—24no—
172C4⋊C4×C59—24no—
173C4⋊C4×C59—24no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—44no—
184C4⋊C4×C59C2×D412yes—
185C4⋊C4×C59C2×D412yes—
186C4⋊C4×C59C2×D412yes—
187C22⋊C4×C59—24no—
188C22⋊C4×C59—24no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—24no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—42no—
193C22⋊C4×C59—44no—
194C42×C59—21no—
195C42×C59—21no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—21no—
199C23×C59—22no—
200C23×C59—21no—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—42no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—42no—
207C23×C59#12012yes—
208C23×C59C2×D812yes—
209C23×C59C2×D812yes—
210C23×C59AGL(1,ℤ/8)12yes—
211C23×C59C22×D411yescharacteristic
212Q8×C59—48no—
213D4×C59—28no—
214D4×C59—24no—
215D4×C59—48no—
216D4×C59—48no—
217D4×C59—41no—
218D4×C59—41no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—24no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—48no—
233C2×C4×C59—48no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—48no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—84no—
251C2×C4×C59#12012yes—
252C2×C4×C59C2×D812yes—
253C2×C4×C59C22×D411yescharacteristic
254C2×C4×C59C22×D411yescharacteristic
255C2×C4×C59C2×#4112yes—
256C2×C4×C592−1+412yes—
257C8×C59—24no—
258C8×C59—42no—
259#24088C5911yesmaximal, characteristic, direct product factor, Sylow
260C22×C59—24no—
261C22×C59—24no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—24no—
265C22×C59—22no—
266C22×C59—42no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—82no—
273C22×C59—82no—
274C22×C59#44714yes—
275C22×C59C2×#12011yescharacteristic
276C22×C59#54712yes—
277C22×C59C22×D811yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59#57812yes—
280C4×C59—22no—
281C4×C59—22no—
282C4×C59—41no—
283C4×C59—42no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—82no—
288C4×C59—82no—
289C4×C59#52511yescharacteristic
290C4×C59#53611yescharacteristic
291C4×C59#54712yes—
292C2×#547C2×C5911yescharacteristic
293C2×#480C2×C5911yescharacteristic, complemented
294C22×#107C2×C5911yescharacteristic, complemented
295#2486C2×C5911yescharacteristic, complemented
296#2057C2×C5912yescomplemented
297#2033C2×C5912yescomplemented
298#1893C2×C5911yescharacteristic, complemented
299#1869C2×C5911yescharacteristic, complemented
300#1862C2×C5912yescomplemented
301#1852C2×C5912yescomplemented
302#1823C2×C5911yescharacteristic, complemented
303C2×C59—24no—
304C2×C59—42no—
305C2×C59—82no—
306C2×C59#240611yescharacteristic
307C2×C59C2×#44711yescharacteristic
308C2×C59#301111yescharacteristic
309C2×C59#333412yes—
310C2×C59#334312yes—
311C22×#41—21no—
312C23×C8C22×C5911yescharacteristic
313#547—24no—
314#547—24no—
315C2×C4⋊Q8C22×C5911yescharacteristic
316C2×#123—21no—
317C2×#123—21no—
318C2×#123—21no—
319C2×C22⋊Q8—21no—
320C2×#121C22×C5911yescharacteristic
321C2×#121C22×C5911yescharacteristic
322C2×C4×D4C22×C5911yescharacteristic
323C2×C4×D4C22×C5911yescharacteristic
324C22×C4⋊C4C22×C5911yescharacteristic
325C22×C4⋊C4C22×C5911yescharacteristic
326C22×C4⋊C4C22×C5911yescharacteristic
327#480—24no—
328C2×#107—24no—
329C2×#107C22×C5912yes—
330C2×#107C22×C5912yes—
331C2×#107C22×C5912yes—
332C2×#102C22×C5912yes—
333C2×#102C22×C5912yes—
334C2×C22⋊C8C22×C5911yescharacteristic
335C2×C22⋊C8C22×C5912yes—
336#398—21no—
337#397—21no—
338#389C22×C5912yes—
339#381—21no—
340#380C22×C5911yescharacteristic
341#380C22×C5912yes—
342#340C22×C5912yes—
343#340C22×C5912yes—
344#340C22×C5912yes—
345#336C22×C5912yes—
346C59#2408811yescharacteristic, direct product factor, Sylow
347C2×#41—28no—
348C2×#41—44no—
349C2×#41—44no—
350C22×Q8—21no—
351C22×D4—21no—
352C22×D4—21no—
353C22×D4D4×C5911yescharacteristic
354C22×D4C23×C5911yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—21no—
359C23×C4C23×C5911yescharacteristic
360C23×C4C23×C5911yescharacteristic
361C22×C8—22no—
362C22×C8—24no—
363C22×C8C23×C5912yes—
364C22×C8C23×C5912yes—
365C4⋊Q8—22no—
366C4⋊Q8—22no—
367#123—28no—
368#123—28no—
369#123—44no—
370C22⋊Q8—28no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121D4×C5914yes—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4D4×C5911yescharacteristic
390C2×C4⋊C4D4×C5914yes—
391C2×C4⋊C4C23×C5912yes—
392C2×C4⋊C4C23×C5912yes—
393C2×C4⋊C4C23×C5911yescharacteristic
394C2×C4⋊C4C23×C5911yescharacteristic
395C2×C4⋊C4C23×C5911yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4D4×C5911yescharacteristic
402C2×C22⋊C4D4×C5911yescharacteristic
403C2×C42C23×C5911yescharacteristic
404#107—24no—
405#107—24no—
406#107—24no—
407#102—24no—
408#102—24no—
409C22⋊C8—22no—
410C22⋊C8—24no—
411#95—22no—
412#95—21no—
413#95—22no—
414C24—21no—
415C24C2×D4×C5911yescharacteristic
416C24C2×D4×C5911yescharacteristic
417#41—432no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4D8×C5914yes—
433C2×D4C2×D4×C5912yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×D4×C5911yescharacteristic
458C22×C4C2×D4×C5911yescharacteristic
459C22×C4C2×D4×C5911yescharacteristic
460C22×C4C2×D4×C5912yes—
461C22×C4C2×D4×C5912yes—
462C22×C4C2×D4×C5911yescharacteristic
463C22×C4C24×C5911yescharacteristic
464C2×C8—28no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8#41×C5914yescomplemented
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4C2×D4×C5912yes—
485C4⋊C4C2×D4×C5912yes—
486C4⋊C4C2×D4×C5912yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C42—21no—
495C42—21no—
496C23—22no—
497C23—22no—
498C23—21no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23#120×C5912yes—
508C23C2×D8×C5912yes—
509C23C2×D8×C5912yes—
510C23AGL(1,ℤ/8)×C5912yes—
511C23C22×D4×C5911yescharacteristic
512Q8—48no—
513D4—28no—
514D4—24no—
515D4—48no—
516D4—48no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4#120×C5912yes—
552C2×C4C2×D8×C5912yes—
553C2×C4C22×D4×C5911yescharacteristic
554C2×C4C22×D4×C5911yescharacteristic
555C2×C4C2×#41×C5912yes—
556C2×C42−1+4×C5912yes—
557C8—24no—
558C8—42no—
559C22—22no—
560C22—24no—
561C22—24no—
562C22—22no—
563C22—22no—
564C22—24no—
565C22—41no—
566C22—42no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22C59×#44714yes—
574C22C2×#120×C5911yescharacteristic
575C22C59×#54712yes—
576C22C22×D8×C5911yescharacteristic
577C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
578C22C59×#57812yes—
579C4—22no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—82no—
587C4—82no—
588C4C59×#52511yescharacteristic
589C4C59×#53611yescharacteristic
590C4C59×#54712yes—
591C2—24no—
592C2—42no—
593C2—82no—
594C2C59×#240611yescharacteristic
595C2C2×C59×#44711yescharacteristic
596C2C59×#301111yescharacteristic
597C2C59×#333412yes—
598C2C59×#334312yes—
subgroupG/H
center, socleC23×C59C22×D4
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC2×C59×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C2×C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#480C2semidirect
3. C22×#107×C59C2semidirect
4. C59×#2486C2semidirect
5. C59×#2057C2semidirect
6. C59×#2033C2semidirect
7. C59×#1893C2semidirect
8. C59×#1869C2semidirect
9. C59×#1862C2semidirect
10. C59×#1852C2semidirect
11. C59×#1823C2semidirect
170. C2×C8×C59#41semidirect
259. #24088C59direct (N × H)
293. C2×#480C2×C59semidirect
294. C22×#107C2×C59semidirect
295. #2486C2×C59semidirect
296. #2057C2×C59semidirect
297. #2033C2×C59semidirect
298. #1893C2×C59semidirect
299. #1869C2×C59semidirect
300. #1862C2×C59semidirect
301. #1852C2×C59semidirect
302. #1823C2×C59semidirect
346. C59#24088direct (N × H)
470. C2×C8#41×C59semidirect

Maximal subgroups

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

subgroupindex
C2×C59×#5472
C2×C59×#4802
C22×#107×C592
C59×#24862
C59×#20572
C59×#20332
C59×#18932
C59×#18692
C59×#18622
C59×#18522
C59×#18232
#2408859