Finite Group Databasebeta

← C59×#18350 #28430459

Subgroups — C59×#18350 #28430459

Order 15104.

1276 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
1C59×#3496C211yesmaximal, characteristic, complemented
2C59×#3192C212yesmaximal, complemented
3C59×#2635C211yesmaximal, characteristic, complemented
4C59×#1616C212yesmaximal, complemented
5C59×#1613C212yesmaximal, complemented
6C59×#1598C212yesmaximal, complemented
7C59×#1595C212yesmaximal, complemented
8C59×#1510C212yesmaximal, complemented
9C59×#1465C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C59—21no—
11C59×#547—22no—
12C59×#546—24no—
13D42×C59—22no—
14C59×#537C2212yes—
15C59×#534—21no—
16C59×#534C2212yes—
17C59×#532—21no—
18C2×#121×C59—21no—
19C2×#121×C59C2212yes—
20C2×#121×C59C2211yescharacteristic, complemented
21C2×C4×D4×C59C2211yescharacteristic
22C59×#481—22no—
23C59×#481C2214yescomplemented
24C59×#469—24no—
25C59×#447—22no—
26C59×#447C2214yescomplemented
27C59×#407C2212yescomplemented
28C59×#400C2211yescharacteristic, complemented
29C59×#399C2211yescharacteristic, complemented
30C59×#394—21no—
31C59×#394C2211yescharacteristic, complemented
32C59×#392—21no—
33C59×#392C2211yescharacteristic, complemented
34C59×#388—21no—
35C59×#386—21no—
36C59×#381C2211yescharacteristic, complemented
37C59×#327C2214yescomplemented
38C59×#327C2214yescomplemented
39C59×#324C2212yescomplemented
40C59×#323C2212yescomplemented
41H(5,𝔽2)×C59—44no—
42H(5,𝔽2)×C59—44no—
43C2×#41×C59—24no—
44C2×#41×C59—22no—
45C22×D4×C59—24no—
46C22×D4×C59—22no—
47C22×D4×C59—22no—
48C22×D4×C59—21no—
49C22×D4×C59—21no—
50C22×D4×C59—21no—
51C22×D4×C59D411yescharacteristic
52C22×D4×C59D412yescomplemented
53C22×D4×C59C2311yescharacteristic
54C23×C4×C59D411yescharacteristic
55C23×C4×C59C2311yescharacteristic
56C2×D8×C59—24no—
57C2×M16×C59—22no—
58C4⋊Q8×C59—21no—
59#127×C59—21no—
60#124×C59—22no—
61#123×C59—24no—
62#123×C59—24no—
63C22⋊Q8×C59—24no—
64#121×C59—22no—
65#121×C59—22no—
66#121×C59—24no—
67#121×C59—22no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—22no—
71#121×C59—22no—
72#121×C59—22no—
73#121×C59D414yescomplemented
74#121×C59C2314yes—
75#120×C59—24no—
76#120×C59—24no—
77C4×D4×C59—22no—
78C4×D4×C59—22no—
79C4×D4×C59—22no—
80C4×D4×C59—22no—
81C2×C4⋊C4×C59—21no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59D412yescomplemented
84C2×C4⋊C4×C59C2311yescharacteristic
85C2×C4⋊C4×C59C2311yescharacteristic
86C2×C22⋊C4×C59—22no—
87C2×C22⋊C4×C59—22no—
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59D411yescharacteristic
95C2×C22⋊C4×C59C2311yescharacteristic
96C2×C42×C59D411yescharacteristic
97#106×C59—24no—
98#102×C59—24no—
99#102×C59—24no—
100C22⋊C8×C59—22no—
101C22⋊C8×C59C2314yes—
102#95×C59—21no—
103#95×C59—21no—
104#95×C59—21no—
105#95×C59C2311yescharacteristic
106#95×C59C2311yescharacteristic
107C24×C59—22no—
108C24×C59—22no—
109C24×C59—21no—
110C24×C59C2×D411yescharacteristic
111C24×C59C2×D411yescharacteristic
112#41×C59—44no—
113#41×C59—416no—
114#41×C59—44no—
115C2×Q8×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—24no—
119C2×D4×C59—22no—
120C2×D4×C59—24no—
121C2×D4×C59—24no—
122C2×D4×C59—22no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—416no—
128C2×D4×C59—44no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—44no—
132C2×D4×C59—42no—
133C2×D4×C59—42no—
134C2×D4×C59—81no—
135C2×D4×C59—82no—
136C2×D4×C59—82no—
137C2×D4×C59—81no—
138C2×D4×C59—81no—
139C2×D4×C59—81no—
140C2×D4×C59—81no—
141C2×D4×C59C2×D414yes—
142C22×C4×C59—22no—
143C22×C4×C59—24no—
144C22×C4×C59—22no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—81no—
151C22×C4×C59—81no—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D412yes—
157C22×C4×C59C2×D412yes—
158C22×C4×C59C2411yescharacteristic
159D8×C59—84no—
160M16×C59—82no—
161C2×C8×C59—24no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—81no—
168C4⋊C4×C59C2×D414yes—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—41no—
176C22⋊C4×C59—41no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—81no—
179C22⋊C4×C59—81no—
180C22⋊C4×C59—81no—
181C22⋊C4×C59—81no—
182C42×C59—21no—
183C42×C59—21no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—24no—
189C23×C59—41no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—44no—
194C23×C59—44no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—82no—
203C23×C59—82no—
204C23×C59—82no—
205C23×C59—82no—
206C23×C59#12012yes—
207C23×C59AGL(1,ℤ/8)12yes—
208C23×C59C22×D411yescharacteristic
209Q8×C59—44no—
210D4×C59—28no—
211D4×C59—44no—
212D4×C59—44no—
213D4×C59—44no—
214D4×C59—44no—
215D4×C59—48no—
216D4×C59—48no—
217D4×C59—44no—
218D4×C59—44no—
219D4×C59—82no—
220D4×C59—82no—
221D4×C59—84no—
222D4×C59—84no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—24no—
228C2×C4×C59—42no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—41no—
232C2×C4×C59—42no—
233C2×C4×C59—48no—
234C2×C4×C59—44no—
235C2×C4×C59—44no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—81no—
248C2×C4×C59#12011yescharacteristic
249C2×C4×C59#12011yescharacteristic
250C2×C4×C59AGL(1,ℤ/8)12yes—
251C2×C4×C59C22×D412yes—
252C8×C59—44no—
253#18350C5911yesmaximal, characteristic, direct product factor, Sylow
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—42no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—44no—
264C22×C59—44no—
265C22×C59—44no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—164no—
277C22×C59—164no—
278C22×C59C2×#12011yescharacteristic
279C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—21no—
284C4×C59—21no—
285C4×C59—44no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59—81no—
291#3496C2×C5911yescharacteristic, complemented
292#3192C2×C5912yescomplemented
293#2635C2×C5911yescharacteristic, complemented
294#1616C2×C5912yescomplemented
295#1613C2×C5912yescomplemented
296#1598C2×C5912yescomplemented
297#1595C2×C5912yescomplemented
298#1510C2×C5912yescomplemented
299#1465C2×C5911yescharacteristic, complemented
300C2×C59—21no—
301C2×C59—21no—
302C2×C59—42no—
303C2×C59—42no—
304C2×C59—84no—
305C2×C59—161no—
306C2×C59#300311yescharacteristic
307C2×C59#302111yescharacteristic
308C2×C59#302611yescharacteristic
309C2×H(5,𝔽2)—21no—
310#547—22no—
311#546—24no—
312D42—22no—
313#537C22×C5912yes—
314#534—21no—
315#534C22×C5912yes—
316#532—21no—
317C2×#121—21no—
318C2×#121C22×C5912yes—
319C2×#121C22×C5911yescharacteristic, complemented
320C2×C4×D4C22×C5911yescharacteristic
321#481—22no—
322#481C22×C5914yescomplemented
323#469—24no—
324#447—22no—
325#447C22×C5914yescomplemented
326#407C22×C5912yescomplemented
327#400C22×C5911yescharacteristic, complemented
328#399C22×C5911yescharacteristic, complemented
329#394—21no—
330#394C22×C5911yescharacteristic, complemented
331#392—21no—
332#392C22×C5911yescharacteristic, complemented
333#388—21no—
334#386—21no—
335#381C22×C5911yescharacteristic, complemented
336#327C22×C5914yescomplemented
337#327C22×C5914yescomplemented
338#324C22×C5912yescomplemented
339#323C22×C5912yescomplemented
340C59#1835011yescharacteristic, direct product factor, Sylow
341H(5,𝔽2)—44no—
342H(5,𝔽2)—44no—
343C2×#41—24no—
344C2×#41—22no—
345C22×D4—24no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4—21no—
351C22×D4D4×C5911yescharacteristic
352C22×D4D4×C5912yescomplemented
353C22×D4C23×C5911yescharacteristic
354C23×C4D4×C5911yescharacteristic
355C23×C4C23×C5911yescharacteristic
356C2×D8—24no—
357C2×M16—22no—
358C4⋊Q8—21no—
359#127—21no—
360#124—22no—
361#123—24no—
362#123—24no—
363C22⋊Q8—24no—
364#121—22no—
365#121—22no—
366#121—24no—
367#121—22no—
368#121—22no—
369#121—22no—
370#121—22no—
371#121—22no—
372#121—22no—
373#121D4×C5914yescomplemented
374#121C23×C5914yes—
375#120—24no—
376#120—24no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4D4×C5912yescomplemented
384C2×C4⋊C4C23×C5911yescharacteristic
385C2×C4⋊C4C23×C5911yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4D4×C5911yescharacteristic
395C2×C22⋊C4C23×C5911yescharacteristic
396C2×C42D4×C5911yescharacteristic
397#106—24no—
398#102—24no—
399#102—24no—
400C22⋊C8—22no—
401C22⋊C8C23×C5914yes—
402#95—21no—
403#95—21no—
404#95—21no—
405#95C23×C5911yescharacteristic
406#95C23×C5911yescharacteristic
407C24—22no—
408C24—22no—
409C24—21no—
410C24C2×D4×C5911yescharacteristic
411C24C2×D4×C5911yescharacteristic
412#41—44no—
413#41—416no—
414#41—44no—
415C2×Q8—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—416no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—81no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4C2×D4×C5914yes—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4C2×D4×C5911yescharacteristic
453C22×C4C2×D4×C5911yescharacteristic
454C22×C4C2×D4×C5911yescharacteristic
455C22×C4C2×D4×C5911yescharacteristic
456C22×C4C2×D4×C5912yes—
457C22×C4C2×D4×C5912yes—
458C22×C4C24×C5911yescharacteristic
459D8—84no—
460M16—82no—
461C2×C8—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—81no—
468C4⋊C4C2×D4×C5914yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C42—21no—
483C42—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—44no—
494C23—44no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23#120×C5912yes—
507C23AGL(1,ℤ/8)×C5912yes—
508C23C22×D4×C5911yescharacteristic
509Q8—44no—
510D4—28no—
511D4—44no—
512D4—44no—
513D4—44no—
514D4—44no—
515D4—48no—
516D4—48no—
517D4—44no—
518D4—44no—
519D4—82no—
520D4—82no—
521D4—84no—
522D4—84no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—81no—
548C2×C4#120×C5911yescharacteristic
549C2×C4#120×C5911yescharacteristic
550C2×C4AGL(1,ℤ/8)×C5912yes—
551C2×C4C22×D4×C5912yes—
552C8—44no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—164no—
576C22—164no—
577C22C2×#120×C5911yescharacteristic
578C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
579C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
580C4—22no—
581C4—22no—
582C4—21no—
583C4—21no—
584C4—44no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C2—21no—
591C2—21no—
592C2—42no—
593C2—42no—
594C2—84no—
595C2—161no—
596C2C59×#300311yescharacteristic
597C2C59×#302111yescharacteristic
598C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3496C2semidirect
2. C59×#3192C2semidirect
3. C59×#2635C2semidirect
4. C59×#1616C2semidirect
5. C59×#1613C2semidirect
6. C59×#1598C2semidirect
7. C59×#1595C2semidirect
8. C59×#1510C2semidirect
9. C59×#1465C2semidirect
20. C2×#121×C59C22semidirect
23. C59×#481C22semidirect
26. C59×#447C22semidirect
27. C59×#407C22semidirect
28. C59×#400C22semidirect
29. C59×#399C22semidirect
31. C59×#394C22semidirect
33. C59×#392C22semidirect
36. C59×#381C22semidirect
37. C59×#327C22semidirect
38. C59×#327C22semidirect
39. C59×#324C22semidirect
40. C59×#323C22semidirect
52. C22×D4×C59D4semidirect
73. #121×C59D4semidirect
83. C2×C4⋊C4×C59D4semidirect
253. #18350C59direct (N × H)
291. #3496C2×C59semidirect
292. #3192C2×C59semidirect
293. #2635C2×C59semidirect
294. #1616C2×C59semidirect
295. #1613C2×C59semidirect
296. #1598C2×C59semidirect
297. #1595C2×C59semidirect
298. #1510C2×C59semidirect
299. #1465C2×C59semidirect
319. C2×#121C22×C59semidirect
322. #481C22×C59semidirect
325. #447C22×C59semidirect
326. #407C22×C59semidirect
327. #400C22×C59semidirect
328. #399C22×C59semidirect
330. #394C22×C59semidirect
332. #392C22×C59semidirect
335. #381C22×C59semidirect
336. #327C22×C59semidirect
337. #327C22×C59semidirect
338. #324C22×C59semidirect
339. #323C22×C59semidirect
340. C59#18350direct (N × H)
352. C22×D4D4×C59semidirect
373. #121D4×C59semidirect
383. C2×C4⋊C4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34962
C59×#31922
C59×#26352
C59×#16162
C59×#16132
C59×#15982
C59×#15952
C59×#15102
C59×#14652
#1835059