Finite Group Databasebeta

← C59×#8412 #28420521

Subgroups — C59×#8412 #28420521

Order 15104.

768 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3514C211yesmaximal, characteristic
2C59×#1759C211yesmaximal, characteristic, complemented
3C59×#1529C211yesmaximal, characteristic, complemented
4C59×#1505C211yesmaximal, characteristic, complemented
5C59×#1500C211yesmaximal, characteristic, complemented
6C59×#1488C211yesmaximal, characteristic, complemented
7C59×#1484C211yesmaximal, characteristic, complemented
8C59×#559C411yescharacteristic, complemented
9C59×#556C411yescharacteristic, complemented
10C59×#555—21no—
11C59×#551—22no—
12D4×Q8×C59C411yescharacteristic, complemented
13C59×#548—21no—
14C59×#546—21no—
15C59×#546—21no—
16D42×C59C411yescharacteristic, complemented
17C59×#542—21no—
18C59×#538—22no—
19C59×#532C2211yescharacteristic
20C59×#529—21no—
21C59×#529C2211yescharacteristic
22C2×#124×C59—21no—
23C59×#518—21no—
24C4×#41×C59C2211yescharacteristic
25C59×#432C2211yescharacteristic, complemented
26C59×#431C2211yescharacteristic, complemented
27C2×#104×C59—21no—
28C4×M16×C59C2211yescharacteristic, complemented
29C59×#376C2211yescharacteristic, complemented
30C59×#339—21no—
31C59×#330—21no—
32C59×#329—21no—
33C59×#328—21no—
34C59×#327—21no—
35C59×#327—22no—
36C59×Q8⋊C8—21no—
37C59×D4⋊C8—22no—
38C59×D4⋊C8—21no—
39C2×#41×C59—21no—
40C2×#41×C59D411yescharacteristic
41C22×Q8×C59—21no—
42C22×D4×C59—21no—
43C22×D4×C59—21no—
44C2×M16×C59—21no—
45C4⋊Q8×C59—21no—
46C4⋊Q8×C59C2×C411yescharacteristic
47#127×C59C2×C411yescharacteristic
48#126×C59—22no—
49#126×C59—22no—
50#126×C59—21no—
51#125×C59D411yescharacteristic
52#124×C59—22no—
53#124×C59—22no—
54#124×C59—22no—
55#124×C59—21no—
56#124×C59—21no—
57#124×C59—21no—
58#124×C59—21no—
59#124×C59D411yescharacteristic
60#123×C59—22no—
61#123×C59—22no—
62#123×C59—22no—
63#123×C59—21no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C22⋊Q8×C59—22no—
68C22⋊Q8×C59D411yescharacteristic
69#121×C59—22no—
70#121×C59—22no—
71#121×C59—21no—
72#121×C59—21no—
73#121×C59—21no—
74#121×C59D411yescharacteristic
75#121×C59D412yes—
76#120×C59—22no—
77#120×C59—22no—
78C4×Q8×C59—21no—
79C4×Q8×C59C2×C411yescharacteristic
80C4×Q8×C59C2×C411yescharacteristic
81C4×Q8×C59D411yescharacteristic
82C4×D4×C59—22no—
83C4×D4×C59—22no—
84C4×D4×C59—21no—
85C4×D4×C59—21no—
86C4×D4×C59—21no—
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
94C4×D4×C59D412yes—
95#117×C59—22no—
96#117×C59—21no—
97#117×C59—21no—
98#117×C59—21no—
99#117×C59D411yescharacteristic
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
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—
117C24×C59—22no—
118#41×C59—22no—
119#41×C59—41no—
120#41×C59—82no—
121C2×Q8×C59—22no—
122C2×Q8×C59—21no—
123C2×Q8×C59—82no—
124C2×Q8×C59C22⋊C411yescharacteristic
125C2×Q8×C59C22⋊C411yescharacteristic
126C2×D4×C59—22no—
127C2×D4×C59—21no—
128C2×D4×C59—22no—
129C2×D4×C59—21no—
130C2×D4×C59—22no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—42no—
134C2×D4×C59—42no—
135C2×D4×C59—82no—
136C2×D4×C59C22⋊C411yescharacteristic
137C2×D4×C59C22⋊C411yescharacteristic
138C2×D4×C59C22⋊C412yes—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—22no—
146C22×C4×C59—41no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150M16×C59—22no—
151M16×C59—41no—
152C2×C8×C59—41no—
153C2×C8×C59—41no—
154C2×C8×C59—41no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59C22⋊C411yescharacteristic
165C4⋊C4×C59C22⋊C411yescharacteristic
166C4⋊C4×C59C22⋊C411yescharacteristic
167C4⋊C4×C59C22⋊C411yescharacteristic
168C4⋊C4×C59C22⋊C412yes—
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—21no—
177C22⋊C4×C59—21no—
178C22⋊C4×C59—21no—
179C22⋊C4×C59—21no—
180C22⋊C4×C59—21no—
181C22⋊C4×C59—21no—
182C22⋊C4×C59—21no—
183C22⋊C4×C59—21no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—22no—
186C42×C59—22no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—22no—
192C42×C59—41no—
193C42×C59C22×C411yescharacteristic
194C42×C59C2×D411yescharacteristic
195C42×C59C2×D411yescharacteristic
196C42×C59C2×D411yescharacteristic
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—22no—
200C23×C59—44no—
201C23×C59—82no—
202C23×C59—82no—
203C23×C59#12011yescharacteristic
204Q8×C59—21no—
205Q8×C59—21no—
206Q8×C59—41no—
207Q8×C59—42no—
208Q8×C59—82no—
209D4×C59—21no—
210D4×C59—41no—
211D4×C59—42no—
212D4×C59—42no—
213D4×C59—42no—
214D4×C59—44no—
215D4×C59—41no—
216D4×C59—41no—
217D4×C59—81no—
218D4×C59—82no—
219D4×C59—82no—
220D4×C59#10412yes—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—42no—
237C2×C4×C59—81no—
238C2×C4×C59—82no—
239C2×C4×C59—82no—
240C2×C4×C59—82no—
241C2×C4×C59—81no—
242C2×C4×C59—81no—
243C2×C4×C59C2×C22⋊C411yescharacteristic
244C2×C4×C59C2×C22⋊C411yescharacteristic
245C2×C4×C59C2×C22⋊C411yescharacteristic
246C2×C4×C59#12011yescharacteristic
247C2×C4×C59#12011yescharacteristic
248C2×C4×C59#12011yescharacteristic
249C8×C59—42no—
250C8×C59—81no—
251C8×C59—81no—
252#8412C5911yesmaximal, characteristic, direct product factor, Sylow
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—22no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—42no—
260C22×C59—41no—
261C22×C59—42no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—42no—
265C22×C59—164no—
266C22×C59#37911yescharacteristic
267C4×C59—21no—
268C4×C59—21no—
269C4×C59—21no—
270C4×C59—21no—
271C4×C59—21no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—42no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—82no—
280C4×C59C2×#10411yescharacteristic
281C4×C59#42111yescharacteristic
282#3514C2×C5911yescharacteristic
283#1759C2×C5911yescharacteristic, complemented
284#1529C2×C5911yescharacteristic, complemented
285#1505C2×C5911yescharacteristic, complemented
286#1500C2×C5911yescharacteristic, complemented
287#1488C2×C5911yescharacteristic, complemented
288#1484C2×C5911yescharacteristic, complemented
289C2×C59—41no—
290C2×C59—42no—
291C2×C59—82no—
292C2×C59—81no—
293C2×C59#178911yescharacteristic
294C2×C59#179411yescharacteristic
295C2×C59#179511yescharacteristic
296#559C4×C5911yescharacteristic, complemented
297#556C4×C5911yescharacteristic, complemented
298#555—21no—
299#551—22no—
300D4×Q8C4×C5911yescharacteristic, complemented
301#548—21no—
302#546—21no—
303#546—21no—
304D42C4×C5911yescharacteristic, complemented
305#542—21no—
306#538—22no—
307#532C22×C5911yescharacteristic
308#529—21no—
309#529C22×C5911yescharacteristic
310C2×#124—21no—
311#518—21no—
312C4×#41C22×C5911yescharacteristic
313#432C22×C5911yescharacteristic, complemented
314#431C22×C5911yescharacteristic, complemented
315C2×#104—21no—
316C4×M16C22×C5911yescharacteristic, complemented
317#376C22×C5911yescharacteristic, complemented
318#339—21no—
319#330—21no—
320#329—21no—
321#328—21no—
322#327—21no—
323#327—22no—
324Q8⋊C8—21no—
325D4⋊C8—22no—
326D4⋊C8—21no—
327C59#841211yescharacteristic, direct product factor, Sylow
328C2×#41—21no—
329C2×#41D4×C5911yescharacteristic
330C22×Q8—21no—
331C22×D4—21no—
332C22×D4—21no—
333C2×M16—21no—
334C4⋊Q8—21no—
335C4⋊Q8C2×C4×C5911yescharacteristic
336#127C2×C4×C5911yescharacteristic
337#126—22no—
338#126—22no—
339#126—21no—
340#125D4×C5911yescharacteristic
341#124—22no—
342#124—22no—
343#124—22no—
344#124—21no—
345#124—21no—
346#124—21no—
347#124—21no—
348#124D4×C5911yescharacteristic
349#123—22no—
350#123—22no—
351#123—22no—
352#123—21no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8D4×C5911yescharacteristic
358#121—22no—
359#121—22no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121D4×C5911yescharacteristic
364#121D4×C5912yes—
365#120—22no—
366#120—22no—
367C4×Q8—21no—
368C4×Q8C2×C4×C5911yescharacteristic
369C4×Q8C2×C4×C5911yescharacteristic
370C4×Q8D4×C5911yescharacteristic
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C2×C4×C5911yescharacteristic
381C4×D4C2×C4×C5911yescharacteristic
382C4×D4D4×C5911yescharacteristic
383C4×D4D4×C5912yes—
384#117—22no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117D4×C5911yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C23×C5911yescharacteristic
395C4⋊C8—21no—
396C4⋊C8—21no—
397#104—24no—
398C22⋊C8—21no—
399C22⋊C8—21no—
400#97—21no—
401#97—21no—
402C4×C8—21no—
403C4×C8—21no—
404#95—21no—
405#95—21no—
406C24—22no—
407#41—22no—
408#41—41no—
409#41—82no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—82no—
413C2×Q8C22⋊C4×C5911yescharacteristic
414C2×Q8C22⋊C4×C5911yescharacteristic
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—82no—
425C2×D4C22⋊C4×C5911yescharacteristic
426C2×D4C22⋊C4×C5911yescharacteristic
427C2×D4C22⋊C4×C5912yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4C2×D4×C5911yescharacteristic
437C22×C4C2×D4×C5911yescharacteristic
438C22×C4C2×D4×C5911yescharacteristic
439M16—22no—
440M16—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4C22⋊C4×C5911yescharacteristic
454C4⋊C4C22⋊C4×C5911yescharacteristic
455C4⋊C4C22⋊C4×C5911yescharacteristic
456C4⋊C4C22⋊C4×C5911yescharacteristic
457C4⋊C4C22⋊C4×C5912yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—22no—
481C42—41no—
482C42C22×C4×C5911yescharacteristic
483C42C2×D4×C5911yescharacteristic
484C42C2×D4×C5911yescharacteristic
485C42C2×D4×C5911yescharacteristic
486C23—21no—
487C23—21no—
488C23—22no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23#120×C5911yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—42no—
497Q8—82no—
498D4—21no—
499D4—41no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—41no—
505D4—41no—
506D4—81no—
507D4—82no—
508D4—82no—
509D4#104×C5912yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4C2×C22⋊C4×C5911yescharacteristic
533C2×C4C2×C22⋊C4×C5911yescharacteristic
534C2×C4C2×C22⋊C4×C5911yescharacteristic
535C2×C4#120×C5911yescharacteristic
536C2×C4#120×C5911yescharacteristic
537C2×C4#120×C5911yescharacteristic
538C8—42no—
539C8—81no—
540C8—81no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—21no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—164no—
554C22C59×#37911yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—82no—
568C4C2×#104×C5911yescharacteristic
569C4C59×#42111yescharacteristic
570C2—41no—
571C2—42no—
572C2—81no—
573C2—82no—
574C2C59×#178911yescharacteristic
575C2C59×#179411yescharacteristic
576C2C59×#179511yescharacteristic
subgroupG/H
center, socleC22×C59#379
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3514C2
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×#1505C2semidirect
5. C59×#1500C2semidirect
6. C59×#1488C2semidirect
7. C59×#1484C2semidirect
8. C59×#559C4semidirect
9. C59×#556C4semidirect
12. D4×Q8×C59C4semidirect
16. D42×C59C4semidirect
25. C59×#432C22semidirect
26. C59×#431C22semidirect
28. C4×M16×C59C22semidirect
29. C59×#376C22semidirect
252. #8412C59direct (N × H)
283. #1759C2×C59semidirect
284. #1529C2×C59semidirect
285. #1505C2×C59semidirect
286. #1500C2×C59semidirect
287. #1488C2×C59semidirect
288. #1484C2×C59semidirect
296. #559C4×C59semidirect
297. #556C4×C59semidirect
300. D4×Q8C4×C59semidirect
304. D42C4×C59semidirect
313. #432C22×C59semidirect
314. #431C22×C59semidirect
316. C4×M16C22×C59semidirect
317. #376C22×C59semidirect
327. C59#8412direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35142
C59×#17592
C59×#15292
C59×#15052
C59×#15002
C59×#14882
C59×#14842
#841259