Finite Group Databasebeta

← C59×#8413 #28420522

Subgroups — C59×#8413 #28420522

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3518C211yesmaximal, characteristic
2C59×#1759C211yesmaximal, characteristic, complemented
3C59×#1529C211yesmaximal, characteristic, complemented
4C59×#1506C211yesmaximal, characteristic, complemented
5C59×#1500C211yesmaximal, characteristic, complemented
6C59×#1488C211yesmaximal, characteristic, complemented
7C59×#1485C211yesmaximal, characteristic, complemented
8Q82×C59C411yescharacteristic, complemented
9C59×#556—21no—
10C59×#556—22no—
11C59×#556—21no—
12C59×#556C411yescharacteristic, complemented
13C59×#555—21no—
14C59×#555C411yescharacteristic, complemented
15D4×Q8×C59C411yescharacteristic, complemented
16C59×#548—21no—
17C59×#542—22no—
18C59×#542—21no—
19C59×#533—21no—
20C59×#533C2211yescharacteristic
21C59×#529—21no—
22C59×#529C2211yescharacteristic
23C59×#519—21no—
24C4×#41×C59C2211yescharacteristic
25C59×#432C2211yescharacteristic
26C59×#431C2211yescharacteristic
27C2×#104×C59—21no—
28C4×M16×C59C2211yescharacteristic
29C59×#376C2211yescharacteristic
30C59×#339—21no—
31C59×#330—21no—
32C59×#329—21no—
33C59×#328—21no—
34C59×#328—22no—
35C59×#327—21no—
36C59×Q8⋊C8—22no—
37C59×Q8⋊C8—21no—
38C59×D4⋊C8—21no—
39C2×#41×C59D411yescharacteristic
40C22×Q8×C59—21no—
41C2×M16×C59—21no—
42C4⋊Q8×C59—22no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59—21no—
45C4⋊Q8×C59—21no—
46C4⋊Q8×C59C2×C411yescharacteristic
47C4⋊Q8×C59C2×C411yescharacteristic
48#126×C59—22no—
49#126×C59—22no—
50#126×C59—22no—
51#126×C59—22no—
52#126×C59—21no—
53#125×C59—21no—
54#125×C59D411yescharacteristic
55#124×C59—22no—
56#124×C59—22no—
57#124×C59—22no—
58#124×C59—22no—
59#124×C59—21no—
60#124×C59D411yescharacteristic
61#123×C59—22no—
62#123×C59—21no—
63C22⋊Q8×C59—22no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C22⋊Q8×C59—21no—
68C22⋊Q8×C59—22no—
69C22⋊Q8×C59D411yescharacteristic
70C22⋊Q8×C59D412yes—
71#121×C59—22no—
72#121×C59—21no—
73#121×C59—21no—
74#121×C59D411yescharacteristic
75C4×Q8×C59—22no—
76C4×Q8×C59—22no—
77C4×Q8×C59—22no—
78C4×Q8×C59—21no—
79C4×Q8×C59—21no—
80C4×Q8×C59—21no—
81C4×Q8×C59—21no—
82C4×Q8×C59C2×C411yescharacteristic
83C4×Q8×C59C2×C411yescharacteristic
84C4×Q8×C59D411yescharacteristic
85C4×Q8×C59D412yes—
86C4×D4×C59—22no—
87C4×D4×C59—21no—
88C4×D4×C59—21no—
89C4×D4×C59—21no—
90C4×D4×C59—21no—
91C4×D4×C59C2×C411yescharacteristic
92C4×D4×C59C2×C411yescharacteristic
93C4×D4×C59D411yescharacteristic
94#117×C59—22no—
95#117×C59—22no—
96#117×C59—21no—
97#117×C59—21no—
98#117×C59—21no—
99#117×C59—21no—
100#117×C59—21no—
101#117×C59—21no—
102#117×C59D411yescharacteristic
103C2×C42×C59—21no—
104C2×C42×C59—21no—
105C2×C42×C59C2311yescharacteristic
106C4⋊C8×C59—21no—
107C4⋊C8×C59—21no—
108#104×C59—24no—
109C22⋊C8×C59—21no—
110C22⋊C8×C59—21no—
111#97×C59—21no—
112#97×C59—21no—
113C4×C8×C59—21no—
114C4×C8×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#41×C59—22no—
118#41×C59—41no—
119C2×Q8×C59—22no—
120C2×Q8×C59—22no—
121C2×Q8×C59—21no—
122C2×Q8×C59—42no—
123C2×Q8×C59—42no—
124C2×Q8×C59C22⋊C411yescharacteristic
125C2×Q8×C59C22⋊C411yescharacteristic
126C2×Q8×C59C22⋊C412yes—
127C2×D4×C59—21no—
128C2×D4×C59—22no—
129C2×D4×C59—21no—
130C2×D4×C59C22⋊C411yescharacteristic
131C2×D4×C59C22⋊C411yescharacteristic
132C22×C4×C59—22no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—41no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141M16×C59—22no—
142M16×C59—41no—
143C2×C8×C59—41no—
144C2×C8×C59—41no—
145C2×C8×C59—41no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59—22no—
149C4⋊C4×C59—22no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59C22⋊C411yescharacteristic
162C4⋊C4×C59C22⋊C411yescharacteristic
163C4⋊C4×C59C22⋊C411yescharacteristic
164C4⋊C4×C59C22⋊C411yescharacteristic
165C4⋊C4×C59C22⋊C412yes—
166C22⋊C4×C59—22no—
167C22⋊C4×C59—22no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—21no—
173C22⋊C4×C59—21no—
174C22⋊C4×C59—21no—
175C22⋊C4×C59—21no—
176C22⋊C4×C59—21no—
177C22⋊C4×C59—21no—
178C22⋊C4×C59—21no—
179C22⋊C4×C59—21no—
180C42×C59—22no—
181C42×C59—22no—
182C42×C59—22no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—22no—
188C42×C59—22no—
189C42×C59—41no—
190C42×C59C22×C411yescharacteristic
191C42×C59C2×D411yescharacteristic
192C42×C59C2×D411yescharacteristic
193C42×C59C2×D411yescharacteristic
194C23×C59—21no—
195C23×C59—21no—
196C23×C59#12011yescharacteristic
197Q8×C59—21no—
198Q8×C59—42no—
199Q8×C59—41no—
200Q8×C59—44no—
201Q8×C59—41no—
202Q8×C59—41no—
203Q8×C59—82no—
204Q8×C59#10412yes—
205D4×C59—21no—
206D4×C59—21no—
207D4×C59—42no—
208D4×C59—41no—
209D4×C59—81no—
210D4×C59—82no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—44no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—42no—
229C2×C4×C59—81no—
230C2×C4×C59—81no—
231C2×C4×C59—81no—
232C2×C4×C59C2×C22⋊C411yescharacteristic
233C2×C4×C59C2×C22⋊C411yescharacteristic
234C2×C4×C59C2×C22⋊C411yescharacteristic
235C2×C4×C59#12011yescharacteristic
236C2×C4×C59#12011yescharacteristic
237C2×C4×C59#12011yescharacteristic
238C8×C59—42no—
239C8×C59—81no—
240C8×C59—81no—
241#8413C5911yesmaximal, characteristic, direct product factor, Sylow
242C22×C59—21no—
243C22×C59—21no—
244C22×C59—21no—
245C22×C59—41no—
246C22×C59—41no—
247C22×C59—41no—
248C22×C59—41no—
249C22×C59—41no—
250C22×C59—41no—
251C22×C59#37911yescharacteristic
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—21no—
256C4×C59—21no—
257C4×C59—22no—
258C4×C59—41no—
259C4×C59—42no—
260C4×C59—41no—
261C4×C59—41no—
262C4×C59—41no—
263C4×C59—42no—
264C4×C59—41no—
265C4×C59—82no—
266C4×C59C2×#10411yescharacteristic
267C4×C59#42111yescharacteristic
268#3518C2×C5911yescharacteristic
269#1759C2×C5911yescharacteristic, complemented
270#1529C2×C5911yescharacteristic, complemented
271#1506C2×C5911yescharacteristic, complemented
272#1500C2×C5911yescharacteristic, complemented
273#1488C2×C5911yescharacteristic, complemented
274#1485C2×C5911yescharacteristic, complemented
275C2×C59—41no—
276C2×C59—81no—
277C2×C59—81no—
278C2×C59#178911yescharacteristic
279C2×C59#179411yescharacteristic
280C2×C59#179611yescharacteristic
281Q82C4×C5911yescharacteristic, complemented
282#556—21no—
283#556—22no—
284#556—21no—
285#556C4×C5911yescharacteristic, complemented
286#555—21no—
287#555C4×C5911yescharacteristic, complemented
288D4×Q8C4×C5911yescharacteristic, complemented
289#548—21no—
290#542—22no—
291#542—21no—
292#533—21no—
293#533C22×C5911yescharacteristic
294#529—21no—
295#529C22×C5911yescharacteristic
296#519—21no—
297C4×#41C22×C5911yescharacteristic
298#432C22×C5911yescharacteristic
299#431C22×C5911yescharacteristic
300C2×#104—21no—
301C4×M16C22×C5911yescharacteristic
302#376C22×C5911yescharacteristic
303#339—21no—
304#330—21no—
305#329—21no—
306#328—21no—
307#328—22no—
308#327—21no—
309Q8⋊C8—22no—
310Q8⋊C8—21no—
311D4⋊C8—21no—
312C59#841311yescharacteristic, direct product factor, Sylow
313C2×#41D4×C5911yescharacteristic
314C22×Q8—21no—
315C2×M16—21no—
316C4⋊Q8—22no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8C2×C4×C5911yescharacteristic
321C4⋊Q8C2×C4×C5911yescharacteristic
322#126—22no—
323#126—22no—
324#126—22no—
325#126—22no—
326#126—21no—
327#125—21no—
328#125D4×C5911yescharacteristic
329#124—22no—
330#124—22no—
331#124—22no—
332#124—22no—
333#124—21no—
334#124D4×C5911yescharacteristic
335#123—22no—
336#123—21no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—22no—
343C22⋊Q8D4×C5911yescharacteristic
344C22⋊Q8D4×C5912yes—
345#121—22no—
346#121—21no—
347#121—21no—
348#121D4×C5911yescharacteristic
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8C2×C4×C5911yescharacteristic
357C4×Q8C2×C4×C5911yescharacteristic
358C4×Q8D4×C5911yescharacteristic
359C4×Q8D4×C5912yes—
360C4×D4—22no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4C2×C4×C5911yescharacteristic
366C4×D4C2×C4×C5911yescharacteristic
367C4×D4D4×C5911yescharacteristic
368#117—22no—
369#117—22no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117D4×C5911yescharacteristic
377C2×C42—21no—
378C2×C42—21no—
379C2×C42C23×C5911yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382#104—24no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385#97—21no—
386#97—21no—
387C4×C8—21no—
388C4×C8—21no—
389#95—21no—
390#95—21no—
391#41—22no—
392#41—41no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8C22⋊C4×C5911yescharacteristic
399C2×Q8C22⋊C4×C5911yescharacteristic
400C2×Q8C22⋊C4×C5912yes—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4C22⋊C4×C5911yescharacteristic
405C2×D4C22⋊C4×C5911yescharacteristic
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4C2×D4×C5911yescharacteristic
413C22×C4C2×D4×C5911yescharacteristic
414C22×C4C2×D4×C5911yescharacteristic
415M16—22no—
416M16—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4C22⋊C4×C5911yescharacteristic
436C4⋊C4C22⋊C4×C5911yescharacteristic
437C4⋊C4C22⋊C4×C5911yescharacteristic
438C4⋊C4C22⋊C4×C5911yescharacteristic
439C4⋊C4C22⋊C4×C5912yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42C22×C4×C5911yescharacteristic
465C42C2×D4×C5911yescharacteristic
466C42C2×D4×C5911yescharacteristic
467C42C2×D4×C5911yescharacteristic
468C23—21no—
469C23—21no—
470C23#120×C5911yescharacteristic
471Q8—21no—
472Q8—42no—
473Q8—41no—
474Q8—44no—
475Q8—41no—
476Q8—41no—
477Q8—82no—
478Q8#104×C5912yes—
479D4—21no—
480D4—21no—
481D4—42no—
482D4—41no—
483D4—81no—
484D4—82no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4C2×C22⋊C4×C5911yescharacteristic
507C2×C4C2×C22⋊C4×C5911yescharacteristic
508C2×C4C2×C22⋊C4×C5911yescharacteristic
509C2×C4#120×C5911yescharacteristic
510C2×C4#120×C5911yescharacteristic
511C2×C4#120×C5911yescharacteristic
512C8—42no—
513C8—81no—
514C8—81no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22C59×#37911yescharacteristic
525C4—21no—
526C4—22no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—42no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4C2×#104×C5911yescharacteristic
540C4C59×#42111yescharacteristic
541C2—41no—
542C2—81no—
543C2—81no—
544C2C59×#178911yescharacteristic
545C2C59×#179411yescharacteristic
546C2C59×#179611yescharacteristic
subgroupG/H
center, socleC22×C59#379
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3518C2
LCSG ▷ C42 ▷ 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×#1759C2semidirect
3. C59×#1529C2semidirect
4. C59×#1506C2semidirect
5. C59×#1500C2semidirect
6. C59×#1488C2semidirect
7. C59×#1485C2semidirect
8. Q82×C59C4semidirect
12. C59×#556C4semidirect
14. C59×#555C4semidirect
15. D4×Q8×C59C4semidirect
241. #8413C59direct (N × H)
269. #1759C2×C59semidirect
270. #1529C2×C59semidirect
271. #1506C2×C59semidirect
272. #1500C2×C59semidirect
273. #1488C2×C59semidirect
274. #1485C2×C59semidirect
281. Q82C4×C59semidirect
285. #556C4×C59semidirect
287. #555C4×C59semidirect
288. D4×Q8C4×C59semidirect
312. C59#8413direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35182
C59×#17592
C59×#15292
C59×#15062
C59×#15002
C59×#14882
C59×#14852
#841359