Finite Group Databasebeta

← C59×#18169 #28430278

Subgroups — C59×#18169 #28430278

Order 15104.

1144 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3490C211yesmaximal, characteristic
2C59×#3092C212yesmaximal, complemented
3C2×C59×#397C211yesmaximal, characteristic, complemented
4C59×#1612C214yesmaximal, complemented
5C59×#1608C212yesmaximal, complemented
6C59×#1607C212yesmaximal, complemented
7C59×#1504C212yesmaximal, complemented
8C59×#1462C211yesmaximal, characteristic, complemented
9C59×#551—22no—
10C59×#551—22no—
11C59×#546—22no—
12C59×#546—22no—
13C59×#543C2211yescharacteristic
14C59×#538—21no—
15C59×#538C2212yes—
16C59×#534C2211yescharacteristic
17C59×#525—21no—
18C59×#525C2212yes—
19C2×#120×C59—21no—
20C59×#518—21no—
21C22×C4⋊C4×C59C2211yescharacteristic, complemented
22C22×C22⋊C4×C59—21no—
23C22×C22⋊C4×C59C2211yescharacteristic
24C59×#484—22no—
25C59×#483C2214yescomplemented
26C59×#481—22no—
27C59×#480C2214yescomplemented
28C59×#428—22no—
29C59×#417—22no—
30C59×#407C2212yescomplemented
31C59×#397—21no—
32C59×#397—21no—
33C59×#397C2212yes—
34C59×#397C2212yescomplemented
35C2×#95×C59—21no—
36C2×#95×C59C2211yescharacteristic, complemented
37C59×#328C2214yescomplemented
38C59×#327C2214yescomplemented
39C59×#323C2212yes—
40C59×#323C2212yescomplemented
41C25×C59D411yescharacteristic
42C2×#41×C59—21no—
43C2×#41×C59D412yes—
44C22×D4×C59—21no—
45C22×D4×C59—21no—
46C22×D4×C59—21no—
47C23×C4×C59—21no—
48C23×C4×C59—21no—
49C23×C4×C59D411yescharacteristic
50C23×C4×C59C2311yescharacteristic
51C2×M16×C59—22no—
52#126×C59—22no—
53#126×C59—22no—
54#124×C59—22no—
55#124×C59—22no—
56#123×C59—22no—
57#123×C59—22no—
58#123×C59—22no—
59#123×C59—22no—
60#123×C59—22no—
61C22⋊Q8×C59—21no—
62C22⋊Q8×C59—22no—
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59D412yes—
66C22⋊Q8×C59C2312yes—
67#121×C59—21no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—21no—
71#121×C59D412yes—
72#121×C59C2312yes—
73#120×C59—22no—
74#120×C59—22no—
75#120×C59—22no—
76#120×C59—22no—
77#120×C59—22no—
78C4×D4×C59—22no—
79C4×D4×C59—22no—
80C4×D4×C59—22no—
81C4×D4×C59—22no—
82C4×D4×C59—22no—
83C4×D4×C59—22no—
84#117×C59—22no—
85#117×C59—21no—
86#117×C59D412yes—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59—22no—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59C2312yes—
91C2×C22⋊C4×C59—22no—
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59—22no—
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—22no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59—44no—
100C2×C22⋊C4×C59D411yescharacteristic
101C2×C22⋊C4×C59D411yescharacteristic
102C2×C22⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59C2311yescharacteristic
104#107×C59—24no—
105#106×C59—24no—
106#103×C59—24no—
107#102×C59—24no—
108C22⋊C8×C59—22no—
109C22⋊C8×C59C2314yes—
110#95×C59—22no—
111#95×C59—22no—
112#95×C59—21no—
113#95×C59C2312yes—
114C24×C59—21no—
115C24×C59—22no—
116C24×C59—21no—
117C24×C59—21no—
118C24×C59—41no—
119C24×C59—42no—
120C24×C59—42no—
121C24×C59C2×D411yescharacteristic
122C24×C59C2×D411yescharacteristic
123C24×C59C2×D411yescharacteristic
124#41×C59—44no—
125#41×C59—82no—
126C2×Q8×C59—21no—
127C2×Q8×C59C2×D412yes—
128C2×D4×C59—22no—
129C2×D4×C59—22no—
130C2×D4×C59—22no—
131C2×D4×C59—22no—
132C2×D4×C59—21no—
133C2×D4×C59—22no—
134C2×D4×C59—22no—
135C2×D4×C59—42no—
136C2×D4×C59—42no—
137C2×D4×C59—82no—
138C2×D4×C59—82no—
139C2×D4×C59C2×D412yes—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—22no—
145C22×C4×C59—21no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
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—41no—
157C22×C4×C59—41no—
158C22×C4×C59C2×D412yes—
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D412yes—
163C22×C4×C59C2411yescharacteristic
164M16×C59—44no—
165C2×C8×C59—24no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—24no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59C2×D412yes—
175C4⋊C4×C59C2×D412yes—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—22no—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—22no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—44no—
191C22⋊C4×C59—44no—
192C42×C59—22no—
193C42×C59—22no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—22no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—42no—
205C23×C59—42no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—48no—
211C23×C59—42no—
212C23×C59—42no—
213C23×C59—42no—
214C23×C59—81no—
215C23×C59—81no—
216C23×C59—84no—
217C23×C59—81no—
218C23×C59—81no—
219C23×C59#12011yescharacteristic
220C23×C59#12011yescharacteristic
221C23×C59AGL(1,ℤ/8)12yes—
222C23×C59C22×D411yescharacteristic
223Q8×C59—42no—
224Q8×C59—81no—
225D4×C59—42no—
226D4×C59—44no—
227D4×C59—42no—
228D4×C59—42no—
229D4×C59—44no—
230D4×C59—44no—
231D4×C59—82no—
232D4×C59—82no—
233D4×C59—81no—
234C2×C4×C59—24no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—21no—
240C2×C4×C59—42no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
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—82no—
254C2×C4×C59—81no—
255C2×C4×C59—81no—
256C2×C4×C59—81no—
257C2×C4×C59—82no—
258C2×C4×C59—84no—
259C2×C4×C59—82no—
260C2×C4×C59—82no—
261C2×C4×C59#12012yes—
262C2×C4×C59C22×D412yes—
263C8×C59—44no—
264#18169C5911yesmaximal, characteristic, direct product factor, Sylow
265C22×C59—24no—
266C22×C59—24no—
267C22×C59—21no—
268C22×C59—21no—
269C22×C59—21no—
270C22×C59—21no—
271C22×C59—21no—
272C22×C59—21no—
273C22×C59—22no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—42no—
279C22×C59—41no—
280C22×C59—42no—
281C22×C59—41no—
282C22×C59—42no—
283C22×C59—41no—
284C22×C59—81no—
285C22×C59—84no—
286C22×C59—84no—
287C22×C59—81no—
288C22×C59—162no—
289C22×C59#45512yes—
290C22×C59#45812yes—
291C22×C59C2×#12011yescharacteristic
292C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C59#57511yescharacteristic
294C4×C59—22no—
295C4×C59—22no—
296C4×C59—42no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—42no—
300C4×C59—42no—
301C4×C59—82no—
302C4×C59—82no—
303#3490C2×C5911yescharacteristic
304#3092C2×C5912yescomplemented
305C2×#397C2×C5911yescharacteristic, complemented
306#1612C2×C5914yescomplemented
307#1608C2×C5912yescomplemented
308#1607C2×C5912yescomplemented
309#1504C2×C5912yescomplemented
310#1462C2×C5911yescharacteristic, complemented
311C2×C59—21no—
312C2×C59—21no—
313C2×C59—22no—
314C2×C59—22no—
315C2×C59—42no—
316C2×C59—81no—
317C2×C59—82no—
318C2×C59#300611yescharacteristic
319C2×C59C2×#45511yescharacteristic
320C2×C59C2×#45811yescharacteristic
321#551—22no—
322#551—22no—
323#546—22no—
324#546—22no—
325#543C22×C5911yescharacteristic
326#538—21no—
327#538C22×C5912yes—
328#534C22×C5911yescharacteristic
329#525—21no—
330#525C22×C5912yes—
331C2×#120—21no—
332#518—21no—
333C22×C4⋊C4C22×C5911yescharacteristic, complemented
334C22×C22⋊C4—21no—
335C22×C22⋊C4C22×C5911yescharacteristic
336#484—22no—
337#483C22×C5914yescomplemented
338#481—22no—
339#480C22×C5914yescomplemented
340#428—22no—
341#417—22no—
342#407C22×C5912yescomplemented
343#397—21no—
344#397—21no—
345#397C22×C5912yes—
346#397C22×C5912yescomplemented
347C2×#95—21no—
348C2×#95C22×C5911yescharacteristic, complemented
349#328C22×C5914yescomplemented
350#327C22×C5914yescomplemented
351#323C22×C5912yes—
352#323C22×C5912yescomplemented
353C59#1816911yescharacteristic, direct product factor, Sylow
354C25D4×C5911yescharacteristic
355C2×#41—21no—
356C2×#41D4×C5912yes—
357C22×D4—21no—
358C22×D4—21no—
359C22×D4—21no—
360C23×C4—21no—
361C23×C4—21no—
362C23×C4D4×C5911yescharacteristic
363C23×C4C23×C5911yescharacteristic
364C2×M16—22no—
365#126—22no—
366#126—22no—
367#124—22no—
368#124—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8D4×C5912yes—
379C22⋊Q8C23×C5912yes—
380#121—21no—
381#121—22no—
382#121—22no—
383#121—21no—
384#121D4×C5912yes—
385#121C23×C5912yes—
386#120—22no—
387#120—22no—
388#120—22no—
389#120—22no—
390#120—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397#117—22no—
398#117—21no—
399#117D4×C5912yes—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4C23×C5912yes—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—44no—
413C2×C22⋊C4D4×C5911yescharacteristic
414C2×C22⋊C4D4×C5911yescharacteristic
415C2×C22⋊C4C23×C5911yescharacteristic
416C2×C22⋊C4C23×C5911yescharacteristic
417#107—24no—
418#106—24no—
419#103—24no—
420#102—24no—
421C22⋊C8—22no—
422C22⋊C8C23×C5914yes—
423#95—22no—
424#95—22no—
425#95—21no—
426#95C23×C5912yes—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—41no—
432C24—42no—
433C24—42no—
434C24C2×D4×C5911yescharacteristic
435C24C2×D4×C5911yescharacteristic
436C24C2×D4×C5911yescharacteristic
437#41—44no—
438#41—82no—
439C2×Q8—21no—
440C2×Q8C2×D4×C5912yes—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4C2×D4×C5912yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4C2×D4×C5912yes—
472C22×C4C2×D4×C5911yescharacteristic
473C22×C4C2×D4×C5911yescharacteristic
474C22×C4C2×D4×C5911yescharacteristic
475C22×C4C2×D4×C5912yes—
476C22×C4C24×C5911yescharacteristic
477M16—44no—
478C2×C8—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4C2×D4×C5912yes—
488C4⋊C4C2×D4×C5912yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C42—22no—
506C42—22no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—81no—
528C23—81no—
529C23—84no—
530C23—81no—
531C23—81no—
532C23#120×C5911yescharacteristic
533C23#120×C5911yescharacteristic
534C23AGL(1,ℤ/8)×C5912yes—
535C23C22×D4×C5911yescharacteristic
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—82no—
545D4—82no—
546D4—81no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4#120×C5912yes—
575C2×C4C22×D4×C5912yes—
576C8—44no—
577C22—24no—
578C22—24no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—81no—
597C22—84no—
598C22—84no—
599C22—81no—
600C22—162no—
601C22C59×#45512yes—
602C22C59×#45812yes—
603C22C2×#120×C5911yescharacteristic
604C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
605C22C59×#57511yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—82no—
614C4—82no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—21no—
619C2—42no—
620C2—81no—
621C2—82no—
622C2C59×#300611yescharacteristic
623C2C2×C59×#45511yescharacteristic
624C2C2×C59×#45811yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3490C2
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
2. C59×#3092C2semidirect
3. C2×C59×#397C2semidirect
4. C59×#1612C2semidirect
5. C59×#1608C2semidirect
6. C59×#1607C2semidirect
7. C59×#1504C2semidirect
8. C59×#1462C2semidirect
21. C22×C4⋊C4×C59C22semidirect
25. C59×#483C22semidirect
27. C59×#480C22semidirect
30. C59×#407C22semidirect
34. C59×#397C22semidirect
36. C2×#95×C59C22semidirect
37. C59×#328C22semidirect
38. C59×#327C22semidirect
40. C59×#323C22semidirect
264. #18169C59direct (N × H)
304. #3092C2×C59semidirect
305. C2×#397C2×C59semidirect
306. #1612C2×C59semidirect
307. #1608C2×C59semidirect
308. #1607C2×C59semidirect
309. #1504C2×C59semidirect
310. #1462C2×C59semidirect
333. C22×C4⋊C4C22×C59semidirect
337. #483C22×C59semidirect
339. #480C22×C59semidirect
342. #407C22×C59semidirect
346. #397C22×C59semidirect
348. C2×#95C22×C59semidirect
349. #328C22×C59semidirect
350. #327C22×C59semidirect
352. #323C22×C59semidirect
353. C59#18169direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34902
C59×#30922
C2×C59×#3972
C59×#16122
C59×#16082
C59×#16072
C59×#15042
C59×#14622
#1816959