Finite Group Databasebeta

← C59×#24057 #28436166

Subgroups — C59×#24057 #28436166

Order 15104.

1392 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3485C211yesmaximal, characteristic
2C59×#2853C212yesmaximal, complemented
3C59×#2297C211yesmaximal, characteristic, complemented
4C59×#2044C214yesmaximal, complemented
5C59×#1899C212yesmaximal, complemented
6C59×#1867C212yesmaximal, complemented
7C59×#1790C212yesmaximal, complemented
8C59×#1782C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C59C2211yescharacteristic
10C22×#41×C59C2211yescharacteristic
11C59×#547—24no—
12C59×#546—24no—
13C59×#536—21no—
14C59×#534—21no—
15C59×#534C2212yes—
16C59×#525—21no—
17C2×#123×C59—21no—
18C2×#123×C59C2212yes—
19C59×#518C2211yescharacteristic
20C2×#117×C59C2211yescharacteristic, complemented
21C22×C42×C59C2211yescharacteristic, complemented
22C2×#99×C59C2212yes—
23C2×#99×C59C2214yes—
24C2×#99×C59C2212yescomplemented
25C59×#391—21no—
26C59×#390—21no—
27C59×#388—24no—
28C59×#387—24no—
29C59×#386C2212yes—
30C59×#386C2214yes—
31C59×#386C2212yescomplemented
32C59×#385C2214yescomplemented
33C59×#381C2212yescomplemented
34C4×C22⋊C4×C59—21no—
35C4×C22⋊C4×C59—21no—
36C59×#342C2214yescomplemented
37H(5,𝔽2)×C59—24no—
38H(5,𝔽2)×C59—42no—
39C2×#41×C59—24no—
40C2×#41×C59—21no—
41C2×#41×C59—24no—
42C2×#41×C59—22no—
43C2×#41×C59—42no—
44C22×Q8×C59D411yescharacteristic
45C22×D4×C59—21no—
46C22×D4×C59—22no—
47C22×D4×C59D411yescharacteristic
48C22×D4×C59C2312yes—
49C22×D4×C59C2311yescharacteristic
50C22×D4×C59C2311yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59D411yescharacteristic
53C23×C4×C59C2312yes—
54C23×C4×C59C2311yescharacteristic
55C4⋊Q8×C59—22no—
56#124×C59—22no—
57#123×C59—24no—
58#123×C59—22no—
59#123×C59—22no—
60#123×C59—22no—
61#123×C59—22no—
62#123×C59—24no—
63#123×C59—22no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—24no—
66#121×C59—22no—
67#121×C59—24no—
68#121×C59—24no—
69#120×C59—22no—
70#120×C59—22no—
71#120×C59—22no—
72C4×D4×C59—24no—
73C4×D4×C59—24no—
74#117×C59—21no—
75#117×C59—42no—
76C2×C4⋊C4×C59—22no—
77C2×C4⋊C4×C59—24no—
78C2×C4⋊C4×C59—21no—
79C2×C4⋊C4×C59D414yes—
80C2×C22⋊C4×C59—22no—
81C2×C22⋊C4×C59—21no—
82C2×C22⋊C4×C59—22no—
83C2×C22⋊C4×C59—22no—
84C2×C22⋊C4×C59—24no—
85C2×C22⋊C4×C59D414yes—
86C2×C22⋊C4×C59C2312yes—
87C2×C22⋊C4×C59C2314yes—
88C2×C22⋊C4×C59C2312yes—
89C2×C42×C59—21no—
90C2×C42×C59—21no—
91C2×C42×C59—24no—
92C2×C42×C59—21no—
93#99×C59—24no—
94#99×C59—44no—
95#99×C59—42no—
96#95×C59—22no—
97#95×C59—22no—
98#95×C59—24no—
99#95×C59—24no—
100#95×C59—24no—
101C24×C59—21no—
102C24×C59C2×D412yes—
103C24×C59C2×D411yescharacteristic
104C24×C59C2×D411yescharacteristic
105C24×C59C2411yescharacteristic
106#41×C59—416no—
107#41×C59—48no—
108#41×C59—42no—
109#41×C59—42no—
110C2×Q8×C59—21no—
111C2×Q8×C59—22no—
112C2×Q8×C59—22no—
113C2×Q8×C59—41no—
114C2×D4×C59—22no—
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—21no—
118C2×D4×C59—22no—
119C2×D4×C59—22no—
120C2×D4×C59—22no—
121C2×D4×C59—22no—
122C2×D4×C59—22no—
123C2×D4×C59—24no—
124C2×D4×C59—22no—
125C2×D4×C59—24no—
126C2×D4×C59—24no—
127C2×D4×C59—22no—
128C2×D4×C59—41no—
129C2×D4×C59—48no—
130C2×D4×C59—42no—
131C2×D4×C59—42no—
132C2×D4×C59#4114yes—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—21no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—24no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—24no—
145C22×C4×C59—22no—
146C22×C4×C59—44no—
147C22×C4×C59—41no—
148C22×C4×C59—41no—
149C22×C4×C59C2×D412yes—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D414yes—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D414yes—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—24no—
160C4⋊C4×C59—48no—
161C4⋊C4×C59—42no—
162C22⋊C4×C59—22no—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—24no—
165C22⋊C4×C59—22no—
166C22⋊C4×C59—22no—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—44no—
173C22⋊C4×C59—44no—
174C22⋊C4×C59—48no—
175C42×C59—22no—
176C42×C59—42no—
177C42×C59—42no—
178C42×C59—42no—
179C23×C59—21no—
180C23×C59—21no—
181C23×C59—22no—
182C23×C59—21no—
183C23×C59—21no—
184C23×C59—21no—
185C23×C59—24no—
186C23×C59—42no—
187C23×C59—42no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—44no—
191C23×C59—82no—
192C23×C59#12312yes—
193C23×C59C22×D411yescharacteristic
194C23×C59C22×D412yes—
195C23×C59C2×#4112yes—
196C23×C59H(5,𝔽2)12yes—
197Q8×C59—44no—
198Q8×C59—42no—
199D4×C59—28no—
200D4×C59—44no—
201D4×C59—42no—
202D4×C59—44no—
203D4×C59—42no—
204D4×C59—42no—
205D4×C59—44no—
206D4×C59—44no—
207D4×C59—84no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—28no—
215C2×C4×C59—24no—
216C2×C4×C59—28no—
217C2×C4×C59—24no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—24no—
223C2×C4×C59—44no—
224C2×C4×C59—44no—
225C2×C4×C59—44no—
226C2×C4×C59—44no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—44no—
231C2×C4×C59—44no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—44no—
235C2×C4×C59—42no—
236C2×C4×C59—44no—
237C2×C4×C59—82no—
238C2×C4×C59#12012yes—
239C2×C4×C59#12012yes—
240C2×C4×C59#12312yes—
241#24057C5911yesmaximal, characteristic, direct product factor, Sylow
242C22×C59—24no—
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—42no—
246C22×C59—42no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—42no—
250C22×C59—41no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—44no—
254C22×C59—81no—
255C22×C59—81no—
256C22×C59—84no—
257C22×C59—162no—
258C22×C59C2×#12011yescharacteristic
259C22×C59C2×#12311yescharacteristic
260C22×C59#53411yescharacteristic
261C22×C59#54614yes—
262C4×C59—22no—
263C4×C59—22no—
264C4×C59—22no—
265C4×C59—42no—
266C4×C59—41no—
267C4×C59—44no—
268C4×C59—44no—
269C4×C59—82no—
270C4×C59—84no—
271#3485C2×C5911yescharacteristic
272#2853C2×C5912yescomplemented
273#2297C2×C5911yescharacteristic, complemented
274#2044C2×C5914yescomplemented
275#1899C2×C5912yescomplemented
276#1867C2×C5912yescomplemented
277#1790C2×C5912yescomplemented
278#1782C2×C5911yescharacteristic, complemented
279C2×C59—21no—
280C2×C59—24no—
281C2×C59—21no—
282C2×C59—42no—
283C2×C59—81no—
284C2×C59—82no—
285C2×C59#240511yescharacteristic
286C2×C59#302512yes—
287C2×H(5,𝔽2)C22×C5911yescharacteristic
288C22×#41C22×C5911yescharacteristic
289#547—24no—
290#546—24no—
291#536—21no—
292#534—21no—
293#534C22×C5912yes—
294#525—21no—
295C2×#123—21no—
296C2×#123C22×C5912yes—
297#518C22×C5911yescharacteristic
298C2×#117C22×C5911yescharacteristic, complemented
299C22×C42C22×C5911yescharacteristic, complemented
300C2×#99C22×C5912yes—
301C2×#99C22×C5914yes—
302C2×#99C22×C5912yescomplemented
303#391—21no—
304#390—21no—
305#388—24no—
306#387—24no—
307#386C22×C5914yes—
308#386C22×C5912yes—
309#386C22×C5912yescomplemented
310#385C22×C5914yescomplemented
311#381C22×C5912yescomplemented
312C4×C22⋊C4—21no—
313C4×C22⋊C4—21no—
314#342C22×C5914yescomplemented
315C59#2405711yescharacteristic, direct product factor, Sylow
316H(5,𝔽2)—24no—
317H(5,𝔽2)—42no—
318C2×#41—24no—
319C2×#41—21no—
320C2×#41—24no—
321C2×#41—22no—
322C2×#41—42no—
323C22×Q8D4×C5911yescharacteristic
324C22×D4—21no—
325C22×D4—22no—
326C22×D4D4×C5911yescharacteristic
327C22×D4C23×C5912yes—
328C22×D4C23×C5911yescharacteristic
329C22×D4C23×C5911yescharacteristic
330C23×C4D4×C5911yescharacteristic
331C23×C4D4×C5911yescharacteristic
332C23×C4C23×C5911yescharacteristic
333C23×C4C23×C5912yes—
334C4⋊Q8—22no—
335#124—22no—
336#123—24no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—22no—
341#123—24no—
342#123—22no—
343C22⋊Q8—22no—
344C22⋊Q8—24no—
345#121—22no—
346#121—24no—
347#121—24no—
348#120—22no—
349#120—22no—
350#120—22no—
351C4×D4—24no—
352C4×D4—24no—
353#117—21no—
354#117—42no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4D4×C5914yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4D4×C5914yes—
365C2×C22⋊C4C23×C5912yes—
366C2×C22⋊C4C23×C5914yes—
367C2×C22⋊C4C23×C5912yes—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—24no—
371C2×C42—21no—
372#99—24no—
373#99—44no—
374#99—42no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—21no—
381C24C2×D4×C5912yes—
382C24C2×D4×C5911yescharacteristic
383C24C2×D4×C5911yescharacteristic
384C24C24×C5911yescharacteristic
385#41—416no—
386#41—48no—
387#41—42no—
388#41—42no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—41no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4#41×C5914yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C2×D4×C5912yes—
429C22×C4C2×D4×C5911yescharacteristic
430C22×C4C2×D4×C5914yes—
431C22×C4C2×D4×C5911yescharacteristic
432C22×C4C2×D4×C5911yescharacteristic
433C22×C4C2×D4×C5911yescharacteristic
434C22×C4C2×D4×C5914yes—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—48no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—42no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—82no—
471C23#123×C5912yes—
472C23C22×D4×C5911yescharacteristic
473C23C22×D4×C5912yes—
474C23C2×#41×C5912yes—
475C23H(5,𝔽2)×C5912yes—
476Q8—44no—
477Q8—42no—
478D4—28no—
479D4—44no—
480D4—42no—
481D4—44no—
482D4—42no—
483D4—42no—
484D4—44no—
485D4—44no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—82no—
517C2×C4#120×C5912yes—
518C2×C4#120×C5912yes—
519C2×C4#123×C5912yes—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—41no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—84no—
535C22—162no—
536C22C2×#120×C5911yescharacteristic
537C22C2×#123×C5911yescharacteristic
538C22C59×#53411yescharacteristic
539C22C59×#54614yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—42no—
546C4—41no—
547C4—82no—
548C4—84no—
549C2—21no—
550C2—24no—
551C2—21no—
552C2—42no—
553C2—81no—
554C2—82no—
555C2C59×#240511yescharacteristic
556C2C59×#302512yes—
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC24C24×C59
prime radicalC59×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2853C2semidirect
3. C59×#2297C2semidirect
4. C59×#2044C2semidirect
5. C59×#1899C2semidirect
6. C59×#1867C2semidirect
7. C59×#1790C2semidirect
8. C59×#1782C2semidirect
20. C2×#117×C59C22semidirect
21. C22×C42×C59C22semidirect
24. C2×#99×C59C22semidirect
31. C59×#386C22semidirect
32. C59×#385C22semidirect
33. C59×#381C22semidirect
36. C59×#342C22semidirect
241. #24057C59direct (N × H)
272. #2853C2×C59semidirect
273. #2297C2×C59semidirect
274. #2044C2×C59semidirect
275. #1899C2×C59semidirect
276. #1867C2×C59semidirect
277. #1790C2×C59semidirect
278. #1782C2×C59semidirect
298. C2×#117C22×C59semidirect
299. C22×C42C22×C59semidirect
302. C2×#99C22×C59semidirect
309. #386C22×C59semidirect
310. #385C22×C59semidirect
311. #381C22×C59semidirect
314. #342C22×C59semidirect
315. C59#24057direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34852
C59×#28532
C59×#22972
C59×#20442
C59×#18992
C59×#18672
C59×#17902
C59×#17822
#2405759