Finite Group Databasebeta

← C59×#9117 #28421226

Subgroups — C59×#9117 #28421226

Order 15104.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#407C211yesmaximal, characteristic, complemented
2C59×#2847C211yesmaximal, characteristic
3C2×C59×#327C212yesmaximal, complemented
4C59×C22⋊#95C211yesmaximal, characteristic, complemented
5C59×#1294C212yesmaximal, complemented
6C24×C4×C59C2211yescharacteristic
7C22×M16×C59—21no—
8C2×#121×C59C411yescharacteristic, complemented
9C2×#121×C59C411yescharacteristic, complemented
10C2×#121×C59C2211yescharacteristic
11C2×#120×C59—21no—
12C22×C22⋊C4×C59—21no—
13C59×#407—24no—
14C2×C22⋊C8×C59—21no—
15C2×C22⋊C8×C59C2212yes—
16C59×#386—21no—
17C59×#386—21no—
18C59×#386—21no—
19C59×#380C411yescharacteristic, complemented
20C59×#380C411yescharacteristic, complemented
21C59×#380C2211yescharacteristic
22C2×#95×C59—21no—
23C2×#95×C59C2212yes—
24C59×#327—24no—
25C25×C59D411yescharacteristic
26C22×D4×C59—21no—
27C22×D4×C59—21no—
28C22×D4×C59—21no—
29C22×D4×C59C2×C411yescharacteristic
30C23×C4×C59—22no—
31C23×C4×C59—22no—
32C23×C4×C59—22no—
33C23×C4×C59—42no—
34C23×C4×C59—42no—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C2×M16×C59—24no—
42C2×M16×C59—42no—
43C2×M16×C59—42no—
44C22×C8×C59—22no—
45#121×C59—22no—
46#121×C59—22no—
47#121×C59—42no—
48#121×C59—42no—
49#120×C59—44no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59—24no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59—42no—
61C2×C22⋊C4×C59—42no—
62C22⋊C8×C59—24no—
63C22⋊C8×C59—22no—
64C22⋊C8×C59D414yescomplemented
65#95×C59—21no—
66#95×C59—21no—
67#95×C59—21no—
68#95×C59—22no—
69#95×C59—22no—
70#95×C59—42no—
71C24×C59—22no—
72C24×C59—22no—
73C24×C59—21no—
74C24×C59—21no—
75C24×C59—22no—
76C24×C59—22no—
77C24×C59—21no—
78C24×C59—41no—
79C24×C59—41no—
80C24×C59C22⋊C411yescharacteristic
81C24×C59C22⋊C411yescharacteristic
82C24×C59C2×D411yescharacteristic
83C2×D4×C59—22no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—42no—
87C2×D4×C59—42no—
88C2×D4×C59—42no—
89C2×D4×C59—42no—
90C2×D4×C59—42no—
91C2×D4×C59—42no—
92C2×D4×C59—42no—
93C2×D4×C59—42no—
94C2×D4×C59—41no—
95C2×D4×C59—41no—
96C22×C4×C59—22no—
97C22×C4×C59—22no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—21no—
107C22×C4×C59—21no—
108C22×C4×C59—24no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—41no—
113C22×C4×C59—41no—
114C22×C4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59—41no—
117C22×C4×C59—41no—
118C22×C4×C59—41no—
119C22×C4×C59—48no—
120C22×C4×C59—48no—
121C22×C4×C59—42no—
122C22×C4×C59—42no—
123C22×C4×C59—42no—
124C22×C4×C59—42no—
125C22×C4×C59—42no—
126C22×C4×C59—42no—
127C22×C4×C59C22⋊C411yescharacteristic
128C22×C4×C59C22⋊C411yescharacteristic
129C22×C4×C59C22×C411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D412yes—
132C22×C4×C59#4112yes—
133M16×C59—48no—
134C2×C8×C59—24no—
135C2×C8×C59—42no—
136C2×C8×C59—42no—
137C4⋊C4×C59—22no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C22⋊C4×C59—42no—
141C22⋊C4×C59—48no—
142C22⋊C4×C59—42no—
143C22⋊C4×C59—42no—
144C22⋊C4×C59—42no—
145C22⋊C4×C59—42no—
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—24no—
153C23×C59—24no—
154C23×C59—21no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—21no—
159C23×C59—21no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—44no—
167C23×C59—44no—
168C23×C59—44no—
169C23×C59—44no—
170C23×C59—42no—
171C23×C59—42no—
172C23×C59—42no—
173C23×C59—42no—
174C23×C59—41no—
175C23×C59—41no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59#9912yes—
182C23×C59#10412yes—
183C23×C59C2×C22⋊C411yescharacteristic
184C23×C59#12112yes—
185C23×C59AGL(1,ℤ/8)11yescharacteristic
186C23×C59AGL(1,ℤ/8)12yes—
187C23×C59#13711yescharacteristic
188D4×C59—82no—
189D4×C59—84no—
190D4×C59—84no—
191D4×C59—82no—
192D4×C59—82no—
193C2×C4×C59—24no—
194C2×C4×C59—24no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—42no—
205C2×C4×C59—42no—
206C2×C4×C59—44no—
207C2×C4×C59—44no—
208C2×C4×C59—44no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—44no—
212C2×C4×C59—44no—
213C2×C4×C59—44no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—82no—
224C2×C4×C59—82no—
225C2×C4×C59—82no—
226C2×C4×C59—82no—
227C2×C4×C59#9912yes—
228C2×C4×C59C4×D412yes—
229C2×C4×C59#12011yescharacteristic
230C2×C4×C59#12311yescharacteristic
231C8×C59—44no—
232#9117C5911yesmaximal, characteristic, direct product factor, Sylow
233C22×C59—22no—
234C22×C59—22no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—22no—
242C22×C59—22no—
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—22no—
247C22×C59—48no—
248C22×C59—48no—
249C22×C59—41no—
250C22×C59—41no—
251C22×C59—41no—
252C22×C59—41no—
253C22×C59—41no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59#38611yescharacteristic
265C22×C59C2×#9911yescharacteristic
266C22×C59C2×#9911yescharacteristic
267C22×C59#41711yescharacteristic
268C22×C59#41811yescharacteristic
269C22×C59C2×#10411yescharacteristic
270C22×C59#42111yescharacteristic
271C22×C59#46812yes—
272C22×C59#46912yes—
273C4×C59—21no—
274C4×C59—22no—
275C4×C59—21no—
276C4×C59—22no—
277C4×C59—41no—
278C4×C59—84no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C2×#407C2×C5911yescharacteristic, complemented
283#2847C2×C5911yescharacteristic
284C2×#327C2×C5912yescomplemented
285C22⋊#95C2×C5911yescharacteristic, complemented
286#1294C2×C5912yescomplemented
287C2×C59—22no—
288C2×C59—22no—
289C2×C59—42no—
290C2×C59—42no—
291C2×C59—161no—
292C2×C59#150411yescharacteristic
293C2×C59#150711yescharacteristic
294C2×C59#151011yescharacteristic
295C2×C59#151311yescharacteristic
296C2×C59#187111yescharacteristic
297C2×C59#189511yescharacteristic
298C2×C59#189811yescharacteristic
299C24×C4C22×C5911yescharacteristic
300C22×M16—21no—
301C2×#121C4×C5911yescharacteristic, complemented
302C2×#121C4×C5911yescharacteristic, complemented
303C2×#121C22×C5911yescharacteristic
304C2×#120—21no—
305C22×C22⋊C4—21no—
306#407—24no—
307C2×C22⋊C8—21no—
308C2×C22⋊C8C22×C5912yes—
309#386—21no—
310#386—21no—
311#386—21no—
312#380C4×C5911yescharacteristic, complemented
313#380C4×C5911yescharacteristic, complemented
314#380C22×C5911yescharacteristic
315C2×#95—21no—
316C2×#95C22×C5912yes—
317#327—24no—
318C59#911711yescharacteristic, direct product factor, Sylow
319C25D4×C5911yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C2×C4×C5911yescharacteristic
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C2×C4×C5911yescharacteristic
330C23×C4C2×C4×C5911yescharacteristic
331C23×C4D4×C5911yescharacteristic
332C23×C4D4×C5911yescharacteristic
333C23×C4D4×C5911yescharacteristic
334C23×C4C23×C5911yescharacteristic
335C2×M16—24no—
336C2×M16—42no—
337C2×M16—42no—
338C22×C8—22no—
339#121—22no—
340#121—22no—
341#121—42no—
342#121—42no—
343#120—44no—
344C2×C4⋊C4C2×C4×C5911yescharacteristic
345C2×C4⋊C4C2×C4×C5911yescharacteristic
346C2×C4⋊C4C2×C4×C5911yescharacteristic
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8D4×C5914yescomplemented
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—42no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—41no—
373C24—41no—
374C24C22⋊C4×C5911yescharacteristic
375C24C22⋊C4×C5911yescharacteristic
376C24C2×D4×C5911yescharacteristic
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C5911yescharacteristic
422C22×C4C22⋊C4×C5911yescharacteristic
423C22×C4C22×C4×C5911yescharacteristic
424C22×C4C2×D4×C5911yescharacteristic
425C22×C4C2×D4×C5912yes—
426C22×C4#41×C5912yes—
427M16—48no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—48no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#99×C5912yes—
476C23#104×C5912yes—
477C23C2×C22⋊C4×C5911yescharacteristic
478C23#121×C5912yes—
479C23AGL(1,ℤ/8)×C5911yescharacteristic
480C23AGL(1,ℤ/8)×C5912yes—
481C23#137×C5911yescharacteristic
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#99×C5912yes—
522C2×C4C4×D4×C5912yes—
523C2×C4#120×C5911yescharacteristic
524C2×C4#123×C5911yescharacteristic
525C8—44no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C59×#38611yescharacteristic
558C22C2×#99×C5911yescharacteristic
559C22C2×#99×C5911yescharacteristic
560C22C59×#41711yescharacteristic
561C22C59×#41811yescharacteristic
562C22C2×#104×C5911yescharacteristic
563C22C59×#42111yescharacteristic
564C22C59×#46812yes—
565C22C59×#46912yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2C59×#150411yescharacteristic
581C2C59×#150711yescharacteristic
582C2C59×#151011yescharacteristic
583C2C59×#151311yescharacteristic
584C2C59×#187111yescharacteristic
585C2C59×#189511yescharacteristic
586C2C59×#189811yescharacteristic
subgroupG/H
center, socleC23×C59C2×C22⋊C4
commutatorC22×C4C22×C4×C59
Frattini, cosocleC23×C4C23×C59
prime radicalC59×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ 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×#407C2semidirect
3. C2×C59×#327C2semidirect
4. C59×C22⋊#95C2semidirect
5. C59×#1294C2semidirect
8. C2×#121×C59C4semidirect
9. C2×#121×C59C4semidirect
19. C59×#380C4semidirect
20. C59×#380C4semidirect
64. C22⋊C8×C59D4semidirect
232. #9117C59direct (N × H)
282. C2×#407C2×C59semidirect
284. C2×#327C2×C59semidirect
285. C22⋊#95C2×C59semidirect
286. #1294C2×C59semidirect
301. C2×#121C4×C59semidirect
302. C2×#121C4×C59semidirect
312. #380C4×C59semidirect
313. #380C4×C59semidirect
318. C59#9117direct (N × H)
358. C22⋊C8D4×C59semidirect

Maximal subgroups

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

subgroupindex
C2×C59×#4072
C59×#28472
C2×C59×#3272
C59×C22⋊#952
C59×#12942
#911759