Finite Group Databasebeta

← C59×#18351 #28430460

Subgroups — C59×#18351 #28430460

Order 15104.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3496C211yesmaximal, characteristic
2C59×#3193C212yesmaximal, complemented
3C59×#2638C211yesmaximal, characteristic, complemented
4C59×#1610C212yesmaximal, complemented
5C59×#1607C212yesmaximal, complemented
6C59×#1604C212yesmaximal, complemented
7C59×#1601C212yesmaximal, complemented
8C59×#1510C212yesmaximal, complemented
9C59×#1465C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C59—21no—
11C59×#547—22no—
12C59×#546—24no—
13D42×C59—22no—
14C59×#537C2212yes—
15C59×#534—21no—
16C59×#534C2212yes—
17C59×#532—21no—
18C2×C22⋊Q8×C59C2211yescharacteristic, complemented
19C2×#121×C59—21no—
20C2×#121×C59C2212yes—
21C2×C4×D4×C59C2211yescharacteristic
22C59×#484—22no—
23C59×#480C2214yescomplemented
24C59×#468—24no—
25C59×#450—22no—
26C59×#448C2214yescomplemented
27C59×#407C2212yescomplemented
28C59×#398C2211yescharacteristic, complemented
29C59×#397—21no—
30C59×#397C2211yescharacteristic, complemented
31C59×#395C2211yescharacteristic, complemented
32C59×#394—21no—
33C59×#393C2211yescharacteristic, complemented
34C59×#387—21no—
35C59×#385—21no—
36C59×#381C2211yescharacteristic, complemented
37C59×#327C2214yescomplemented
38C59×#327C2214yescomplemented
39C59×#324C2212yescomplemented
40C59×#323C2212yescomplemented
41H(5,𝔽2)×C59—44no—
42H(5,𝔽2)×C59—44no—
43C2×#41×C59—24no—
44C2×#41×C59—22no—
45C22×Q8×C59C2311yescharacteristic
46C22×D4×C59—24no—
47C22×D4×C59—22no—
48C22×D4×C59—22no—
49C22×D4×C59—21no—
50C22×D4×C59D411yescharacteristic
51C22×D4×C59D412yes—
52C23×C4×C59D411yescharacteristic
53C23×C4×C59C2311yescharacteristic
54C2×SD16×C59—24no—
55C2×M16×C59—22no—
56C4⋊Q8×C59—21no—
57#127×C59—21no—
58#124×C59—22no—
59#123×C59—24no—
60#123×C59—24no—
61C22⋊Q8×C59—24no—
62C22⋊Q8×C59—22no—
63C22⋊Q8×C59—22no—
64#121×C59—22no—
65#121×C59—22no—
66#121×C59—24no—
67#121×C59—22no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—22no—
71#121×C59D414yes—
72#121×C59C2314yes—
73#120×C59—24no—
74#120×C59—24no—
75C4×D4×C59—22no—
76C4×D4×C59—22no—
77C4×D4×C59—22no—
78C4×D4×C59—22no—
79C2×C4⋊C4×C59—21no—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59—21no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59D412yes—
84C2×C4⋊C4×C59C2311yescharacteristic
85C2×C4⋊C4×C59C2311yescharacteristic
86C2×C22⋊C4×C59—22no—
87C2×C22⋊C4×C59—22no—
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59D411yescharacteristic
93C2×C22⋊C4×C59C2311yescharacteristic
94C2×C42×C59D411yescharacteristic
95#107×C59—24no—
96#103×C59—24no—
97#102×C59—24no—
98C22⋊C8×C59—22no—
99C22⋊C8×C59C2314yes—
100#95×C59—21no—
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—22no—
109C24×C59C2×D411yescharacteristic
110C24×C59C2×D411yescharacteristic
111#41×C59—44no—
112#41×C59—416no—
113#41×C59—44no—
114C2×Q8×C59—22no—
115C2×Q8×C59—22no—
116C2×Q8×C59—81no—
117C2×D4×C59—22no—
118C2×D4×C59—22no—
119C2×D4×C59—24no—
120C2×D4×C59—22no—
121C2×D4×C59—24no—
122C2×D4×C59—24no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—416no—
128C2×D4×C59—44no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—44no—
132C2×D4×C59—82no—
133C2×D4×C59—82no—
134C2×D4×C59C2×D414yes—
135C22×C4×C59—22no—
136C22×C4×C59—24no—
137C22×C4×C59—22no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—81no—
145C22×C4×C59—81no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D412yes—
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2411yescharacteristic
153SD16×C59—84no—
154M16×C59—82no—
155C2×C8×C59—24no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—41no—
162C4⋊C4×C59—41no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C2×D414yes—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—81no—
175C22⋊C4×C59—81no—
176C42×C59—21no—
177C42×C59—21no—
178C23×C59—21no—
179C23×C59—21no—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—24no—
183C23×C59—41no—
184C23×C59—41no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59—44no—
188C23×C59—44no—
189C23×C59—82no—
190C23×C59—82no—
191C23×C59—82no—
192C23×C59—82no—
193C23×C59#12012yes—
194C23×C59AGL(1,ℤ/8)12yes—
195C23×C59C22×D411yescharacteristic
196Q8×C59—44no—
197Q8×C59—44no—
198D4×C59—28no—
199D4×C59—44no—
200D4×C59—44no—
201D4×C59—44no—
202D4×C59—48no—
203D4×C59—48no—
204D4×C59—44no—
205D4×C59—44no—
206D4×C59—82no—
207D4×C59—82no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—24no—
213C2×C4×C59—42no—
214C2×C4×C59—41no—
215C2×C4×C59—42no—
216C2×C4×C59—41no—
217C2×C4×C59—42no—
218C2×C4×C59—48no—
219C2×C4×C59—44no—
220C2×C4×C59—44no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
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—81no—
236C2×C4×C59#12011yescharacteristic
237C2×C4×C59#12011yescharacteristic
238C2×C4×C59AGL(1,ℤ/8)12yes—
239C2×C4×C59C22×D412yes—
240C8×C59—44no—
241#18351C5911yesmaximal, characteristic, direct product factor, Sylow
242C22×C59—21no—
243C22×C59—21no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—22no—
247C22×C59—22no—
248C22×C59—42no—
249C22×C59—41no—
250C22×C59—41no—
251C22×C59—44no—
252C22×C59—44no—
253C22×C59—44no—
254C22×C59—81no—
255C22×C59—81no—
256C22×C59—81no—
257C22×C59—164no—
258C22×C59—164no—
259C22×C59C2×#12011yescharacteristic
260C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
262C4×C59—22no—
263C4×C59—22no—
264C4×C59—21no—
265C4×C59—21no—
266C4×C59—44no—
267C4×C59—41no—
268C4×C59—41no—
269C4×C59—81no—
270C4×C59—81no—
271C4×C59—81no—
272C4×C59—81no—
273#3496C2×C5911yescharacteristic
274#3193C2×C5912yescomplemented
275#2638C2×C5911yescharacteristic, complemented
276#1610C2×C5912yescomplemented
277#1607C2×C5912yescomplemented
278#1604C2×C5912yescomplemented
279#1601C2×C5912yescomplemented
280#1510C2×C5912yescomplemented
281#1465C2×C5911yescharacteristic, complemented
282C2×C59—21no—
283C2×C59—21no—
284C2×C59—42no—
285C2×C59—42no—
286C2×C59—84no—
287C2×C59#300311yescharacteristic
288C2×C59#302211yescharacteristic
289C2×C59#302611yescharacteristic
290C2×H(5,𝔽2)—21no—
291#547—22no—
292#546—24no—
293D42—22no—
294#537C22×C5912yes—
295#534—21no—
296#534C22×C5912yes—
297#532—21no—
298C2×C22⋊Q8C22×C5911yescharacteristic, complemented
299C2×#121—21no—
300C2×#121C22×C5912yes—
301C2×C4×D4C22×C5911yescharacteristic
302#484—22no—
303#480C22×C5914yescomplemented
304#468—24no—
305#450—22no—
306#448C22×C5914yescomplemented
307#407C22×C5912yescomplemented
308#398C22×C5911yescharacteristic, complemented
309#397—21no—
310#397C22×C5911yescharacteristic, complemented
311#395C22×C5911yescharacteristic, complemented
312#394—21no—
313#393C22×C5911yescharacteristic, complemented
314#387—21no—
315#385—21no—
316#381C22×C5911yescharacteristic, complemented
317#327C22×C5914yescomplemented
318#327C22×C5914yescomplemented
319#324C22×C5912yescomplemented
320#323C22×C5912yescomplemented
321C59#1835111yescharacteristic, direct product factor, Sylow
322H(5,𝔽2)—44no—
323H(5,𝔽2)—44no—
324C2×#41—24no—
325C2×#41—22no—
326C22×Q8C23×C5911yescharacteristic
327C22×D4—24no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—21no—
331C22×D4D4×C5911yescharacteristic
332C22×D4D4×C5912yes—
333C23×C4D4×C5911yescharacteristic
334C23×C4C23×C5911yescharacteristic
335C2×SD16—24no—
336C2×M16—22no—
337C4⋊Q8—21no—
338#127—21no—
339#124—22no—
340#123—24no—
341#123—24no—
342C22⋊Q8—24no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345#121—22no—
346#121—22no—
347#121—24no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121—22no—
352#121D4×C5914yes—
353#121C23×C5914yes—
354#120—24no—
355#120—24no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4D4×C5912yes—
365C2×C4⋊C4C23×C5911yescharacteristic
366C2×C4⋊C4C23×C5911yescharacteristic
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4D4×C5911yescharacteristic
374C2×C22⋊C4C23×C5911yescharacteristic
375C2×C42D4×C5911yescharacteristic
376#107—24no—
377#103—24no—
378#102—24no—
379C22⋊C8—22no—
380C22⋊C8C23×C5914yes—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95C23×C5911yescharacteristic
387#95C23×C5911yescharacteristic
388C24—22no—
389C24—22no—
390C24C2×D4×C5911yescharacteristic
391C24C2×D4×C5911yescharacteristic
392#41—44no—
393#41—416no—
394#41—44no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—81no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—416no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C2×D4×C5914yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4C2×D4×C5911yescharacteristic
428C22×C4C2×D4×C5911yescharacteristic
429C22×C4C2×D4×C5911yescharacteristic
430C22×C4C2×D4×C5911yescharacteristic
431C22×C4C2×D4×C5912yes—
432C22×C4C2×D4×C5912yes—
433C22×C4C24×C5911yescharacteristic
434SD16—84no—
435M16—82no—
436C2×C8—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4C2×D4×C5914yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23#120×C5912yes—
475C23AGL(1,ℤ/8)×C5912yes—
476C23C22×D4×C5911yescharacteristic
477Q8—44no—
478Q8—44no—
479D4—28no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—48no—
484D4—48no—
485D4—44no—
486D4—44no—
487D4—82no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4#120×C5911yescharacteristic
518C2×C4#120×C5911yescharacteristic
519C2×C4AGL(1,ℤ/8)×C5912yes—
520C2×C4C22×D4×C5912yes—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—164no—
538C22—164no—
539C22C2×#120×C5911yescharacteristic
540C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
541C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C2—21no—
554C2—21no—
555C2—42no—
556C2—42no—
557C2—84no—
558C2C59×#300311yescharacteristic
559C2C59×#302211yescharacteristic
560C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3193C2semidirect
3. C59×#2638C2semidirect
4. C59×#1610C2semidirect
5. C59×#1607C2semidirect
6. C59×#1604C2semidirect
7. C59×#1601C2semidirect
8. C59×#1510C2semidirect
9. C59×#1465C2semidirect
18. C2×C22⋊Q8×C59C22semidirect
23. C59×#480C22semidirect
26. C59×#448C22semidirect
27. C59×#407C22semidirect
28. C59×#398C22semidirect
30. C59×#397C22semidirect
31. C59×#395C22semidirect
33. C59×#393C22semidirect
36. C59×#381C22semidirect
37. C59×#327C22semidirect
38. C59×#327C22semidirect
39. C59×#324C22semidirect
40. C59×#323C22semidirect
241. #18351C59direct (N × H)
274. #3193C2×C59semidirect
275. #2638C2×C59semidirect
276. #1610C2×C59semidirect
277. #1607C2×C59semidirect
278. #1604C2×C59semidirect
279. #1601C2×C59semidirect
280. #1510C2×C59semidirect
281. #1465C2×C59semidirect
298. C2×C22⋊Q8C22×C59semidirect
303. #480C22×C59semidirect
306. #448C22×C59semidirect
307. #407C22×C59semidirect
308. #398C22×C59semidirect
310. #397C22×C59semidirect
311. #395C22×C59semidirect
313. #393C22×C59semidirect
316. #381C22×C59semidirect
317. #327C22×C59semidirect
318. #327C22×C59semidirect
319. #324C22×C59semidirect
320. #323C22×C59semidirect
321. C59#18351direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34962
C59×#31932
C59×#26382
C59×#16102
C59×#16072
C59×#16042
C59×#16012
C59×#15102
C59×#14652
#1835159