Finite Group Databasebeta

← C59×#18152 #28430261

Subgroups — C59×#18152 #28430261

Order 15104.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3449C211yesmaximal, characteristic, complemented
2C59×#3109C212yesmaximal, complemented
3C59×#2598C211yesmaximal, characteristic, complemented
4C59×#1616C212yesmaximal, complemented
5C59×#1613C212yesmaximal, complemented
6C59×#1598C212yesmaximal, complemented
7C59×#1595C212yesmaximal, complemented
8C59×#1514C212yesmaximal, complemented
9C59×#1464C211yesmaximal, characteristic, complemented
10C22×#41×C59—21no—
11C59×#537—21no—
12C59×#537C2212yes—
13C59×#536—22no—
14C59×#535—22no—
15C59×#534—21no—
16C59×#534C2212yes—
17C59×#532—21no—
18C59×#532C2212yes—
19C59×#525—24no—
20C2×C4×D4×C59C2211yescharacteristic, complemented
21C2×#117×C59C2211yescharacteristic
22C59×#481C2214yescomplemented
23C59×#476—22no—
24C59×#459—22no—
25C59×#447C2214yescomplemented
26C59×#442—24no—
27C59×#432C2212yescomplemented
28C59×#400C2211yescharacteristic, complemented
29C59×#399C2211yescharacteristic
30C59×#394C2211yescharacteristic, complemented
31C59×#392C2211yescharacteristic
32C59×#388—21no—
33C59×#386—21no—
34C59×#386—21no—
35C59×#385—21no—
36C59×#383C2211yescharacteristic, complemented
37C59×#327C2214yescomplemented
38C59×#327C2214yescomplemented
39C59×#324C2212yescomplemented
40C59×#323C2212yes—
41C2×#41×C59—22no—
42C2×#41×C59—22no—
43C2×#41×C59—22no—
44C2×#41×C59—22no—
45C2×#41×C59—44no—
46C2×#41×C59—44no—
47C22×Q8×C59—21no—
48C22×D4×C59—21no—
49C22×D4×C59—22no—
50C22×D4×C59—21no—
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59—22no—
53C23×C4×C59—21no—
54C23×C4×C59D411yescharacteristic
55C2×D8×C59—24no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59D412yescomplemented
58#127×C59—21no—
59#127×C59D412yescomplemented
60#124×C59—22no—
61#124×C59—22no—
62#123×C59—24no—
63#123×C59—24no—
64C22⋊Q8×C59—24no—
65C22⋊Q8×C59—24no—
66#121×C59—24no—
67#121×C59—24no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—22no—
71#121×C59C2314yes—
72#120×C59—24no—
73#120×C59—24no—
74C4×D4×C59—22no—
75C4×D4×C59—22no—
76C4×D4×C59—22no—
77C4×D4×C59—22no—
78C4×D4×C59—22no—
79C4×D4×C59D414yescomplemented
80#117×C59—22no—
81#117×C59—22no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59D411yescharacteristic
84C2×C4⋊C4×C59C2311yescharacteristic
85C2×C4⋊C4×C59C2311yescharacteristic
86C2×C22⋊C4×C59—21no—
87C2×C22⋊C4×C59—21no—
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59D411yescharacteristic
92C2×C22⋊C4×C59C2311yescharacteristic
93C2×C42×C59D411yescharacteristic
94C2×C42×C59C2311yescharacteristic
95#106×C59—24no—
96C4⋊C8×C59—22no—
97#102×C59—24no—
98#102×C59—24no—
99C22⋊C8×C59C2314yes—
100#97×C59—22no—
101#95×C59—21no—
102#95×C59—21no—
103#95×C59—21no—
104#95×C59—21no—
105#95×C59C2311yescharacteristic
106#95×C59C2311yescharacteristic
107C24×C59—22no—
108C24×C59—21no—
109C24×C59C2×D411yescharacteristic
110#41×C59—48no—
111#41×C59—48no—
112#41×C59—44no—
113#41×C59—44no—
114#41×C59—44no—
115#41×C59—44no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—44no—
119C2×D4×C59—24no—
120C2×D4×C59—22no—
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—44no—
128C2×D4×C59—42no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—42no—
132C2×D4×C59—42no—
133C2×D4×C59—81no—
134C2×D4×C59C2×D414yes—
135C22×C4×C59—22no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—22no—
139C22×C4×C59—24no—
140C22×C4×C59—22no—
141C22×C4×C59—24no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—48no—
146C22×C4×C59—42no—
147C22×C4×C59—81no—
148C22×C4×C59—81no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155D8×C59—48no—
156C2×C8×C59—24no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59C2×D414yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—22no—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—41no—
173C22⋊C4×C59—41no—
174C22⋊C4×C59—42no—
175C22⋊C4×C59—81no—
176C22⋊C4×C59—81no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59C2×D412yes—
180C42×C59C2×D412yes—
181C23×C59—21no—
182C23×C59—21no—
183C23×C59—22no—
184C23×C59—24no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—44no—
189C23×C59—41no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—82no—
195C23×C59—82no—
196C23×C59AGL(1,ℤ/8)12yes—
197C23×C59C22×D411yescharacteristic
198Q8×C59—44no—
199Q8×C59—44no—
200D4×C59—24no—
201D4×C59—44no—
202D4×C59—42no—
203D4×C59—44no—
204D4×C59—44no—
205D4×C59—48no—
206D4×C59—44no—
207D4×C59—44no—
208D4×C59—82no—
209D4×C59—82no—
210D4×C59—82no—
211D4×C59—84no—
212C2×C4×C59—24no—
213C2×C4×C59—24no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—24no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—48no—
224C2×C4×C59—48no—
225C2×C4×C59—44no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—84no—
238C2×C4×C59#12011yescharacteristic
239C2×C4×C59#12011yescharacteristic
240C2×C4×C59#12012yes—
241C2×C4×C59AGL(1,ℤ/8)12yes—
242C2×C4×C59C22×D412yes—
243C8×C59—44no—
244#18152C5911yesmaximal, characteristic, direct product factor, Sylow
245C22×C59—21no—
246C22×C59—21no—
247C22×C59—22no—
248C22×C59—42no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—41no—
252C22×C59—41no—
253C22×C59—44no—
254C22×C59—44no—
255C22×C59—42no—
256C22×C59—44no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—88no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59C2×#12011yescharacteristic
264C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
266C4×C59—21no—
267C4×C59—22no—
268C4×C59—21no—
269C4×C59—22no—
270C4×C59—22no—
271C4×C59—44no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—81no—
275C4×C59—81no—
276C4×C59—81no—
277#3449C2×C5911yescharacteristic, complemented
278#3109C2×C5912yescomplemented
279#2598C2×C5911yescharacteristic, complemented
280#1616C2×C5912yescomplemented
281#1613C2×C5912yescomplemented
282#1598C2×C5912yescomplemented
283#1595C2×C5912yescomplemented
284#1514C2×C5912yescomplemented
285#1464C2×C5911yescharacteristic, complemented
286C2×C59—21no—
287C2×C59—21no—
288C2×C59—42no—
289C2×C59—82no—
290C2×C59—84no—
291C2×C59#300311yescharacteristic
292C2×C59#302111yescharacteristic
293C2×C59#302511yescharacteristic
294C22×#41—21no—
295#537—21no—
296#537C22×C5912yes—
297#536—22no—
298#535—22no—
299#534—21no—
300#534C22×C5912yes—
301#532—21no—
302#532C22×C5912yes—
303#525—24no—
304C2×C4×D4C22×C5911yescharacteristic, complemented
305C2×#117C22×C5911yescharacteristic
306#481C22×C5914yescomplemented
307#476—22no—
308#459—22no—
309#447C22×C5914yescomplemented
310#442—24no—
311#432C22×C5912yescomplemented
312#400C22×C5911yescharacteristic, complemented
313#399C22×C5911yescharacteristic
314#394C22×C5911yescharacteristic, complemented
315#392C22×C5911yescharacteristic
316#388—21no—
317#386—21no—
318#386—21no—
319#385—21no—
320#383C22×C5911yescharacteristic, complemented
321#327C22×C5914yescomplemented
322#327C22×C5914yescomplemented
323#324C22×C5912yescomplemented
324#323C22×C5912yes—
325C59#1815211yescharacteristic, direct product factor, Sylow
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—44no—
331C2×#41—44no—
332C22×Q8—21no—
333C22×D4—21no—
334C22×D4—22no—
335C22×D4—21no—
336C22×D4C23×C5911yescharacteristic
337C23×C4—22no—
338C23×C4—21no—
339C23×C4D4×C5911yescharacteristic
340C2×D8—24no—
341C4⋊Q8—21no—
342C4⋊Q8D4×C5912yescomplemented
343#127—21no—
344#127D4×C5912yescomplemented
345#124—22no—
346#124—22no—
347#123—24no—
348#123—24no—
349C22⋊Q8—24no—
350C22⋊Q8—24no—
351#121—24no—
352#121—24no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121C23×C5914yes—
357#120—24no—
358#120—24no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4D4×C5914yescomplemented
365#117—22no—
366#117—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4D4×C5911yescharacteristic
369C2×C4⋊C4C23×C5911yescharacteristic
370C2×C4⋊C4C23×C5911yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4D4×C5911yescharacteristic
377C2×C22⋊C4C23×C5911yescharacteristic
378C2×C42D4×C5911yescharacteristic
379C2×C42C23×C5911yescharacteristic
380#106—24no—
381C4⋊C8—22no—
382#102—24no—
383#102—24no—
384C22⋊C8C23×C5914yes—
385#97—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95C23×C5911yescharacteristic
391#95C23×C5911yescharacteristic
392C24—22no—
393C24—21no—
394C24C2×D4×C5911yescharacteristic
395#41—48no—
396#41—48no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4C2×D4×C5914yes—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C2×D4×C5911yescharacteristic
435C22×C4C2×D4×C5911yescharacteristic
436C22×C4C2×D4×C5911yescharacteristic
437C22×C4C2×D4×C5911yescharacteristic
438C22×C4C2×D4×C5911yescharacteristic
439C22×C4C24×C5911yescharacteristic
440D8—48no—
441C2×C8—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—81no—
449C4⋊C4C2×D4×C5914yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—42no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C42—21no—
463C42—21no—
464C42C2×D4×C5912yes—
465C42C2×D4×C5912yes—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—24no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—44no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—82no—
480C23—82no—
481C23AGL(1,ℤ/8)×C5912yes—
482C23C22×D4×C5911yescharacteristic
483Q8—44no—
484Q8—44no—
485D4—24no—
486D4—44no—
487D4—42no—
488D4—44no—
489D4—44no—
490D4—48no—
491D4—44no—
492D4—44no—
493D4—82no—
494D4—82no—
495D4—82no—
496D4—84no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4#120×C5911yescharacteristic
524C2×C4#120×C5911yescharacteristic
525C2×C4#120×C5912yes—
526C2×C4AGL(1,ℤ/8)×C5912yes—
527C2×C4C22×D4×C5912yes—
528C8—44no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—88no—
545C22—81no—
546C22—81no—
547C22C2×#120×C5911yescharacteristic
548C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
549C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
550C4—21no—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—82no—
565C2—84no—
566C2C59×#300311yescharacteristic
567C2C59×#302111yescharacteristic
568C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3449C2semidirect
2. C59×#3109C2semidirect
3. C59×#2598C2semidirect
4. C59×#1616C2semidirect
5. C59×#1613C2semidirect
6. C59×#1598C2semidirect
7. C59×#1595C2semidirect
8. C59×#1514C2semidirect
9. C59×#1464C2semidirect
20. C2×C4×D4×C59C22semidirect
22. C59×#481C22semidirect
25. C59×#447C22semidirect
27. C59×#432C22semidirect
28. C59×#400C22semidirect
30. C59×#394C22semidirect
36. C59×#383C22semidirect
37. C59×#327C22semidirect
38. C59×#327C22semidirect
39. C59×#324C22semidirect
57. C4⋊Q8×C59D4semidirect
59. #127×C59D4semidirect
79. C4×D4×C59D4semidirect
244. #18152C59direct (N × H)
277. #3449C2×C59semidirect
278. #3109C2×C59semidirect
279. #2598C2×C59semidirect
280. #1616C2×C59semidirect
281. #1613C2×C59semidirect
282. #1598C2×C59semidirect
283. #1595C2×C59semidirect
284. #1514C2×C59semidirect
285. #1464C2×C59semidirect
304. C2×C4×D4C22×C59semidirect
306. #481C22×C59semidirect
309. #447C22×C59semidirect
311. #432C22×C59semidirect
312. #400C22×C59semidirect
314. #394C22×C59semidirect
320. #383C22×C59semidirect
321. #327C22×C59semidirect
322. #327C22×C59semidirect
323. #324C22×C59semidirect
325. C59#18152direct (N × H)
342. C4⋊Q8D4×C59semidirect
344. #127D4×C59semidirect
364. C4×D4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34492
C59×#31092
C59×#25982
C59×#16162
C59×#16132
C59×#15982
C59×#15952
C59×#15142
C59×#14642
#1815259