Finite Group Databasebeta

← C59×#18380 #28430489

Subgroups — C59×#18380 #28430489

Order 15104.

1336 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3496C211yesmaximal, characteristic, complemented
2C59×#3186C212yesmaximal, complemented
3C59×#2543C211yesmaximal, characteristic, complemented
4C59×#1610C212yesmaximal, complemented
5C59×#1607C212yesmaximal, complemented
6C59×#1598C212yesmaximal, complemented
7C59×#1595C212yesmaximal, complemented
8C2×C59×#327C212yesmaximal, complemented
9C59×#1459C211yesmaximal, 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×C59—21no—
20C2×#121×C59C2212yes—
21C2×#121×C59C2211yescharacteristic, complemented
22C2×C4×D4×C59C2211yescharacteristic
23C59×#480—22no—
24C59×#480C2214yescomplemented
25C59×#466—24no—
26C59×#447—22no—
27C59×#447C2214yescomplemented
28C2×C22⋊C8×C59C2212yescomplemented
29C59×#398C2211yescharacteristic, complemented
30C59×#397C2211yescharacteristic, complemented
31C59×#394C2211yescharacteristic, complemented
32C59×#392C2211yescharacteristic, complemented
33C59×#388—21no—
34C59×#386—21no—
35C59×#386—21no—
36C2×#95×C59C2211yescharacteristic, 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×C59—21no—
55C23×C4×C59D411yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C2×D8×C59—24no—
58C22×C8×C59—22no—
59C4⋊Q8×C59—21no—
60#127×C59—21no—
61#124×C59—22no—
62#123×C59—24no—
63#123×C59—24no—
64C22⋊Q8×C59—24no—
65#121×C59—22no—
66#121×C59—22no—
67#121×C59—24no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—22no—
71#121×C59—22no—
72#121×C59—22no—
73#121×C59—22no—
74#121×C59—44no—
75#121×C59D414yescomplemented
76#121×C59C2314yes—
77#120×C59—24no—
78#120×C59—24no—
79C4×D4×C59—22no—
80C4×D4×C59—22no—
81C4×D4×C59—22no—
82C4×D4×C59—22no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59D412yescomplemented
86C2×C4⋊C4×C59C2311yescharacteristic
87C2×C4⋊C4×C59C2311yescharacteristic
88C2×C22⋊C4×C59—22no—
89C2×C22⋊C4×C59—22no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59D411yescharacteristic
96C2×C22⋊C4×C59C2311yescharacteristic
97C2×C42×C59D411yescharacteristic
98#107×C59—24no—
99#102×C59—24no—
100#102×C59—24no—
101C22⋊C8×C59—22no—
102C22⋊C8×C59C2314yes—
103#95×C59—21no—
104#95×C59—21no—
105#95×C59—21no—
106#95×C59C2311yescharacteristic
107#95×C59C2311yescharacteristic
108C24×C59—22no—
109C24×C59—22no—
110C24×C59—21no—
111C24×C59C2×D411yescharacteristic
112C24×C59C2×D411yescharacteristic
113#41×C59—44no—
114#41×C59—416no—
115#41×C59—44no—
116C2×Q8×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—22no—
119C2×D4×C59—24no—
120C2×D4×C59—22no—
121C2×D4×C59—24no—
122C2×D4×C59—24no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—42no—
128C2×D4×C59—416no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—44no—
132C2×D4×C59—44no—
133C2×D4×C59—42no—
134C2×D4×C59—42no—
135C2×D4×C59—42no—
136C2×D4×C59—42no—
137C2×D4×C59—81no—
138C2×D4×C59—82no—
139C2×D4×C59—82no—
140C2×D4×C59—81no—
141C2×D4×C59D814yescomplemented
142C2×D4×C59C2×D414yes—
143C22×C4×C59—22no—
144C22×C4×C59—24no—
145C22×C4×C59—22no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—42no—
152C22×C4×C59—42no—
153C22×C4×C59—81no—
154C22×C4×C59—81no—
155C22×C4×C59—81no—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D412yes—
161C22×C4×C59C2×D412yes—
162C22×C4×C59C2411yescharacteristic
163D8×C59—84no—
164C2×C8×C59—24no—
165C2×C8×C59—82no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59C2×D414yes—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—24no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—41no—
182C22⋊C4×C59—41no—
183C22⋊C4×C59—42no—
184C22⋊C4×C59—81no—
185C42×C59—21no—
186C42×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—24no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—44no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—44no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—82no—
204C23×C59—82no—
205C23×C59—82no—
206C23×C59—82no—
207C23×C59#12012yes—
208C23×C59C2×D812yes—
209C23×C59C2×D812yes—
210C23×C59AGL(1,ℤ/8)12yes—
211C23×C59C22×D411yescharacteristic
212Q8×C59—44no—
213D4×C59—28no—
214D4×C59—44no—
215D4×C59—44no—
216D4×C59—44no—
217D4×C59—44no—
218D4×C59—48no—
219D4×C59—44no—
220D4×C59—44no—
221D4×C59—48no—
222D4×C59—82no—
223D4×C59—82no—
224D4×C59—84no—
225D4×C59—84no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—24no—
232C2×C4×C59—42no—
233C2×C4×C59—41no—
234C2×C4×C59—42no—
235C2×C4×C59—41no—
236C2×C4×C59—42no—
237C2×C4×C59—48no—
238C2×C4×C59—41no—
239C2×C4×C59—44no—
240C2×C4×C59—41no—
241C2×C4×C59—44no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—81no—
251C2×C4×C59—84no—
252C2×C4×C59#12011yescharacteristic
253C2×C4×C59#12011yescharacteristic
254C2×C4×C59C2×D812yes—
255C2×C4×C59C22×D412yes—
256C8×C59—44no—
257#18380C5911yesmaximal, characteristic, direct product factor, Sylow
258C22×C59—21no—
259C22×C59—22no—
260C22×C59—21no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—44no—
269C22×C59—44no—
270C22×C59—44no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59—81no—
279C22×C59—164no—
280C22×C59—164no—
281C22×C59#44714yes—
282C22×C59C2×#12011yescharacteristic
283C22×C59C22×D811yescharacteristic
284C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—21no—
288C4×C59—21no—
289C4×C59—42no—
290C4×C59—44no—
291C4×C59—41no—
292C4×C59—41no—
293C4×C59—81no—
294C4×C59—81no—
295#3496C2×C5911yescharacteristic, complemented
296#3186C2×C5912yescomplemented
297#2543C2×C5911yescharacteristic, complemented
298#1610C2×C5912yescomplemented
299#1607C2×C5912yescomplemented
300#1598C2×C5912yescomplemented
301#1595C2×C5912yescomplemented
302C2×#327C2×C5912yescomplemented
303#1459C2×C5911yescharacteristic, complemented
304C2×C59—21no—
305C2×C59—21no—
306C2×C59—24no—
307C2×C59—42no—
308C2×C59—84no—
309C2×C59—161no—
310C2×C59C2×#44711yescharacteristic
311C2×C59#301811yescharacteristic
312C2×C59#302611yescharacteristic
313C2×H(5,𝔽2)—21no—
314#547—22no—
315#546—24no—
316D42—22no—
317#537C22×C5912yes—
318#534—21no—
319#534C22×C5912yes—
320#532—21no—
321C2×#121—21no—
322C2×#121—21no—
323C2×#121C22×C5912yes—
324C2×#121C22×C5911yescharacteristic, complemented
325C2×C4×D4C22×C5911yescharacteristic
326#480—22no—
327#480C22×C5914yescomplemented
328#466—24no—
329#447—22no—
330#447C22×C5914yescomplemented
331C2×C22⋊C8C22×C5912yescomplemented
332#398C22×C5911yescharacteristic, complemented
333#397C22×C5911yescharacteristic, complemented
334#394C22×C5911yescharacteristic, complemented
335#392C22×C5911yescharacteristic, complemented
336#388—21no—
337#386—21no—
338#386—21no—
339C2×#95C22×C5911yescharacteristic, complemented
340#327C22×C5914yescomplemented
341#327C22×C5914yescomplemented
342#324C22×C5912yescomplemented
343#323C22×C5912yescomplemented
344C59#1838011yescharacteristic, direct product factor, Sylow
345H(5,𝔽2)—44no—
346H(5,𝔽2)—44no—
347C2×#41—24no—
348C2×#41—22no—
349C22×D4—24no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4D4×C5911yescharacteristic
356C22×D4D4×C5912yescomplemented
357C22×D4C23×C5911yescharacteristic
358C23×C4—21no—
359C23×C4D4×C5911yescharacteristic
360C23×C4C23×C5911yescharacteristic
361C2×D8—24no—
362C22×C8—22no—
363C4⋊Q8—21no—
364#127—21no—
365#124—22no—
366#123—24no—
367#123—24no—
368C22⋊Q8—24no—
369#121—22no—
370#121—22no—
371#121—24no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121—22no—
377#121—22no—
378#121—44no—
379#121D4×C5914yescomplemented
380#121C23×C5914yes—
381#120—24no—
382#120—24no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4D4×C5912yescomplemented
390C2×C4⋊C4C23×C5911yescharacteristic
391C2×C4⋊C4C23×C5911yescharacteristic
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4D4×C5911yescharacteristic
400C2×C22⋊C4C23×C5911yescharacteristic
401C2×C42D4×C5911yescharacteristic
402#107—24no—
403#102—24no—
404#102—24no—
405C22⋊C8—22no—
406C22⋊C8C23×C5914yes—
407#95—21no—
408#95—21no—
409#95—21no—
410#95C23×C5911yescharacteristic
411#95C23×C5911yescharacteristic
412C24—22no—
413C24—22no—
414C24—21no—
415C24C2×D4×C5911yescharacteristic
416C24C2×D4×C5911yescharacteristic
417#41—44no—
418#41—416no—
419#41—44no—
420C2×Q8—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—416no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—81no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—81no—
445C2×D4D8×C5914yescomplemented
446C2×D4C2×D4×C5914yes—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4C2×D4×C5911yescharacteristic
461C22×C4C2×D4×C5911yescharacteristic
462C22×C4C2×D4×C5911yescharacteristic
463C22×C4C2×D4×C5911yescharacteristic
464C22×C4C2×D4×C5912yes—
465C22×C4C2×D4×C5912yes—
466C22×C4C24×C5911yescharacteristic
467D8—84no—
468C2×C8—24no—
469C2×C8—82no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4C2×D4×C5914yes—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C42—21no—
490C42—21no—
491C23—21no—
492C23—21no—
493C23—24no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—44no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—44no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23#120×C5912yes—
512C23C2×D8×C5912yes—
513C23C2×D8×C5912yes—
514C23AGL(1,ℤ/8)×C5912yes—
515C23C22×D4×C5911yescharacteristic
516Q8—44no—
517D4—28no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—48no—
523D4—48no—
524D4—44no—
525D4—44no—
526D4—82no—
527D4—82no—
528D4—84no—
529D4—84no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—81no—
555C2×C4—84no—
556C2×C4#120×C5911yescharacteristic
557C2×C4#120×C5911yescharacteristic
558C2×C4C2×D8×C5912yes—
559C2×C4C22×D4×C5912yes—
560C8—44no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—164no—
583C22—164no—
584C22C59×#44714yes—
585C22C2×#120×C5911yescharacteristic
586C22C22×D8×C5911yescharacteristic
587C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
588C4—22no—
589C4—22no—
590C4—21no—
591C4—21no—
592C4—42no—
593C4—44no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C2—21no—
599C2—21no—
600C2—24no—
601C2—42no—
602C2—84no—
603C2—161no—
604C2C2×C59×#44711yescharacteristic
605C2C59×#301811yescharacteristic
606C2C59×#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×#3186C2semidirect
3. C59×#2543C2semidirect
4. C59×#1610C2semidirect
5. C59×#1607C2semidirect
6. C59×#1598C2semidirect
7. C59×#1595C2semidirect
8. C2×C59×#327C2semidirect
9. C59×#1459C2semidirect
21. C2×#121×C59C22semidirect
24. C59×#480C22semidirect
27. C59×#447C22semidirect
28. C2×C22⋊C8×C59C22semidirect
29. C59×#398C22semidirect
30. C59×#397C22semidirect
31. C59×#394C22semidirect
32. C59×#392C22semidirect
36. C2×#95×C59C22semidirect
37. C59×#327C22semidirect
38. C59×#327C22semidirect
39. C59×#324C22semidirect
40. C59×#323C22semidirect
52. C22×D4×C59D4semidirect
75. #121×C59D4semidirect
85. C2×C4⋊C4×C59D4semidirect
141. C2×D4×C59D8semidirect
257. #18380C59direct (N × H)
295. #3496C2×C59semidirect
296. #3186C2×C59semidirect
297. #2543C2×C59semidirect
298. #1610C2×C59semidirect
299. #1607C2×C59semidirect
300. #1598C2×C59semidirect
301. #1595C2×C59semidirect
302. C2×#327C2×C59semidirect
303. #1459C2×C59semidirect
324. C2×#121C22×C59semidirect
327. #480C22×C59semidirect
330. #447C22×C59semidirect
331. C2×C22⋊C8C22×C59semidirect
332. #398C22×C59semidirect
333. #397C22×C59semidirect
334. #394C22×C59semidirect
335. #392C22×C59semidirect
339. C2×#95C22×C59semidirect
340. #327C22×C59semidirect
341. #327C22×C59semidirect
342. #324C22×C59semidirect
343. #323C22×C59semidirect
344. C59#18380direct (N × H)
356. C22×D4D4×C59semidirect
379. #121D4×C59semidirect
389. C2×C4⋊C4D4×C59semidirect
445. C2×D4D8×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34962
C59×#31862
C59×#25432
C59×#16102
C59×#16072
C59×#15982
C59×#15952
C2×C59×#3272
C59×#14592
#1838059