Finite Group Databasebeta

← C59×#18336 #28430445

Subgroups — C59×#18336 #28430445

Order 15104.

1002 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×#3459C211yesmaximal, characteristic
2C59×#3218C212yesmaximal, complemented
3C59×#2563C211yesmaximal, characteristic, complemented
4C59×#1606C214yesmaximal, complemented
5C59×#1602C212yesmaximal, complemented
6C59×#1601C212yesmaximal, complemented
7C59×#1509C212yesmaximal, complemented
8C59×#1466C211yesmaximal, characteristic, complemented
9C59×#543C2211yescharacteristic
10C59×#539—22no—
11C59×#539—21no—
12C59×#538—22no—
13C59×#538C2212yes—
14C59×#534—21no—
15C59×#534C2211yescharacteristic
16C2×C4⋊Q8×C59C2211yescharacteristic, complemented
17C59×#529—22no—
18C59×#529C2212yes—
19C59×#525—21no—
20C59×#525—22no—
21C2×C4×D4×C59—21no—
22C2×C4×D4×C59C2211yescharacteristic
23C59×#497—22no—
24C59×#491—22no—
25C59×#462—22no—
26C59×#461—22no—
27C59×#451C2214yescomplemented
28C59×#448C2214yes—
29C59×#432C2212yescomplemented
30C59×#393C2212yes—
31C59×#393C2212yescomplemented
32C59×#390—21no—
33C59×#382—21no—
34C59×#380—21no—
35C59×#380—21no—
36C4×C4⋊C4×C59C2211yescharacteristic, complemented
37C59×#328C2214yescomplemented
38C59×#327C2214yes—
39C59×#323C2212yes—
40C59×#323C2212yescomplemented
41C2×#41×C59—21no—
42C22×Q8×C59C2312yes—
43C22×D4×C59—21no—
44C22×D4×C59D411yescharacteristic
45C23×C4×C59—21no—
46C23×C4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C2×Q16×C59—24no—
49C2×SD16×C59—24no—
50C4⋊Q8×C59—22no—
51C4⋊Q8×C59—22no—
52#126×C59—22no—
53#126×C59—22no—
54#126×C59—22no—
55#125×C59—22no—
56#125×C59D412yes—
57#124×C59—22no—
58#124×C59D412yes—
59#123×C59—22no—
60#123×C59—22no—
61#123×C59—22no—
62#123×C59—22no—
63#123×C59—22no—
64#123×C59—22no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—22no—
67C22⋊Q8×C59—22no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—22no—
70C22⋊Q8×C59C2312yes—
71#121×C59—22no—
72#121×C59—21no—
73#121×C59—22no—
74#121×C59—22no—
75#121×C59—21no—
76#121×C59C2312yes—
77#120×C59—22no—
78#120×C59—22no—
79#120×C59—22no—
80C4×Q8×C59D412yes—
81C4×D4×C59—22no—
82C4×D4×C59—22no—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—22no—
87C4×D4×C59—22no—
88C4×D4×C59D412yes—
89#117×C59—21no—
90#117×C59—22no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59D411yescharacteristic
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59—22no—
100C2×C22⋊C4×C59C2311yescharacteristic
101C2×C22⋊C4×C59C2311yescharacteristic
102C2×C42×C59—21no—
103C2×C42×C59—21no—
104C2×C42×C59C2311yescharacteristic
105C4⋊C8×C59—22no—
106#103×C59—24no—
107#103×C59—24no—
108C22⋊C8×C59C2314yes—
109#97×C59—22no—
110#95×C59—22no—
111#95×C59—22no—
112#95×C59C2312yes—
113C24×C59—21no—
114C24×C59C2×D411yescharacteristic
115C24×C59C2×D411yescharacteristic
116#41×C59—82no—
117C2×Q8×C59—21no—
118C2×Q8×C59—24no—
119C2×Q8×C59—82no—
120C2×Q8×C59C2×D412yes—
121C2×D4×C59—22no—
122C2×D4×C59—22no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—21no—
126C2×D4×C59—22no—
127C2×D4×C59—22no—
128C2×D4×C59—41no—
129C2×D4×C59—42no—
130C2×D4×C59—42no—
131C2×D4×C59C2×D412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—22no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—22no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—82no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2411yescharacteristic
151Q16×C59—84no—
152SD16×C59—84no—
153C2×C8×C59—24no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
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—42no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—41no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59C2×D412yes—
168C4⋊C4×C59C2×D412yes—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—82no—
182C22⋊C4×C59—82no—
183C42×C59—22no—
184C42×C59—22no—
185C42×C59—42no—
186C42×C59C2×D412yes—
187C42×C59C2×D412yes—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—22no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—42no—
199C23×C59—81no—
200C23×C59—81no—
201C23×C59#13712yes—
202C23×C59C22×D411yescharacteristic
203Q8×C59—42no—
204Q8×C59—44no—
205Q8×C59—44no—
206Q8×C59—81no—
207D4×C59—44no—
208D4×C59—42no—
209D4×C59—44no—
210D4×C59—42no—
211D4×C59—42no—
212D4×C59—82no—
213D4×C59—81no—
214D4×C59—82no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—41no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—44no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—81no—
237C2×C4×C59—81no—
238C2×C4×C59—81no—
239C2×C4×C59—82no—
240C2×C4×C59—81no—
241C2×C4×C59#12012yes—
242C2×C4×C59#12011yescharacteristic
243C2×C4×C59#12011yescharacteristic
244C2×C4×C59C22×D412yes—
245C8×C59—44no—
246#18336C5911yesmaximal, characteristic, direct product factor, Sylow
247C22×C59—21no—
248C22×C59—21no—
249C22×C59—22no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—41no—
253C22×C59—41no—
254C22×C59—41no—
255C22×C59—42no—
256C22×C59—41no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—41no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—84no—
263C22×C59C2×#12011yescharacteristic
264C22×C59C2×#13711yescharacteristic
265C22×C59#57511yescharacteristic
266C4×C59—22no—
267C4×C59—21no—
268C4×C59—22no—
269C4×C59—42no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—42no—
273C4×C59—42no—
274C4×C59—82no—
275C4×C59—82no—
276C4×C59#45512yes—
277#3459C2×C5911yescharacteristic
278#3218C2×C5912yescomplemented
279#2563C2×C5911yescharacteristic, complemented
280#1606C2×C5914yescomplemented
281#1602C2×C5912yescomplemented
282#1601C2×C5912yescomplemented
283#1509C2×C5912yescomplemented
284#1466C2×C5911yescharacteristic, complemented
285C2×C59—21no—
286C2×C59—21no—
287C2×C59—42no—
288C2×C59—81no—
289C2×C59—82no—
290C2×C59#300711yescharacteristic
291C2×C59C2×#45511yescharacteristic
292C2×C59#302711yescharacteristic
293#543C22×C5911yescharacteristic
294#539—22no—
295#539—21no—
296#538—22no—
297#538C22×C5912yes—
298#534—21no—
299#534C22×C5911yescharacteristic
300C2×C4⋊Q8C22×C5911yescharacteristic, complemented
301#529—22no—
302#529C22×C5912yes—
303#525—21no—
304#525—22no—
305C2×C4×D4—21no—
306C2×C4×D4C22×C5911yescharacteristic
307#497—22no—
308#491—22no—
309#462—22no—
310#461—22no—
311#451C22×C5914yescomplemented
312#448C22×C5914yes—
313#432C22×C5912yescomplemented
314#393C22×C5912yes—
315#393C22×C5912yescomplemented
316#390—21no—
317#382—21no—
318#380—21no—
319#380—21no—
320C4×C4⋊C4C22×C5911yescharacteristic, complemented
321#328C22×C5914yescomplemented
322#327C22×C5914yes—
323#323C22×C5912yes—
324#323C22×C5912yescomplemented
325C59#1833611yescharacteristic, direct product factor, Sylow
326C2×#41—21no—
327C22×Q8C23×C5912yes—
328C22×D4—21no—
329C22×D4D4×C5911yescharacteristic
330C23×C4—21no—
331C23×C4D4×C5911yescharacteristic
332C23×C4D4×C5911yescharacteristic
333C2×Q16—24no—
334C2×SD16—24no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337#126—22no—
338#126—22no—
339#126—22no—
340#125—22no—
341#125D4×C5912yes—
342#124—22no—
343#124D4×C5912yes—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8C23×C5912yes—
356#121—22no—
357#121—21no—
358#121—22no—
359#121—22no—
360#121—21no—
361#121C23×C5912yes—
362#120—22no—
363#120—22no—
364#120—22no—
365C4×Q8D4×C5912yes—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4D4×C5912yes—
374#117—21no—
375#117—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4D4×C5911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4C23×C5911yescharacteristic
386C2×C22⋊C4C23×C5911yescharacteristic
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C23×C5911yescharacteristic
390C4⋊C8—22no—
391#103—24no—
392#103—24no—
393C22⋊C8C23×C5914yes—
394#97—22no—
395#95—22no—
396#95—22no—
397#95C23×C5912yes—
398C24—21no—
399C24C2×D4×C5911yescharacteristic
400C24C2×D4×C5911yescharacteristic
401#41—82no—
402C2×Q8—21no—
403C2×Q8—24no—
404C2×Q8—82no—
405C2×Q8C2×D4×C5912yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—41no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C2×D4×C5912yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4C2×D4×C5911yescharacteristic
432C22×C4C2×D4×C5911yescharacteristic
433C22×C4C2×D4×C5911yescharacteristic
434C22×C4C2×D4×C5911yescharacteristic
435C22×C4C24×C5911yescharacteristic
436Q16—84no—
437SD16—84no—
438C2×C8—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4C2×D4×C5912yes—
453C4⋊C4C2×D4×C5912yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42C2×D4×C5912yes—
472C42C2×D4×C5912yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23#137×C5912yes—
487C23C22×D4×C5911yescharacteristic
488Q8—42no—
489Q8—44no—
490Q8—44no—
491Q8—81no—
492D4—44no—
493D4—42no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—82no—
498D4—81no—
499D4—82no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4#120×C5912yes—
527C2×C4#120×C5911yescharacteristic
528C2×C4#120×C5911yescharacteristic
529C2×C4C22×D4×C5912yes—
530C8—44no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—84no—
547C22C2×#120×C5911yescharacteristic
548C22C2×#137×C5911yescharacteristic
549C22C59×#57511yescharacteristic
550C4—22no—
551C4—21no—
552C4—22no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4C59×#45512yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—81no—
565C2—82no—
566C2C59×#300711yescharacteristic
567C2C2×C59×#45511yescharacteristic
568C2C59×#302711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3459C2
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
2. C59×#3218C2semidirect
3. C59×#2563C2semidirect
4. C59×#1606C2semidirect
5. C59×#1602C2semidirect
6. C59×#1601C2semidirect
7. C59×#1509C2semidirect
8. C59×#1466C2semidirect
16. C2×C4⋊Q8×C59C22semidirect
27. C59×#451C22semidirect
29. C59×#432C22semidirect
31. C59×#393C22semidirect
36. C4×C4⋊C4×C59C22semidirect
37. C59×#328C22semidirect
40. C59×#323C22semidirect
246. #18336C59direct (N × H)
278. #3218C2×C59semidirect
279. #2563C2×C59semidirect
280. #1606C2×C59semidirect
281. #1602C2×C59semidirect
282. #1601C2×C59semidirect
283. #1509C2×C59semidirect
284. #1466C2×C59semidirect
300. C2×C4⋊Q8C22×C59semidirect
311. #451C22×C59semidirect
313. #432C22×C59semidirect
315. #393C22×C59semidirect
320. C4×C4⋊C4C22×C59semidirect
321. #328C22×C59semidirect
324. #323C22×C59semidirect
325. C59#18336direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34592
C59×#32182
C59×#25632
C59×#16062
C59×#16022
C59×#16012
C59×#15092
C59×#14662
#1833659