Finite Group Databasebeta

← C59×#24107 #28436216

Subgroups — C59×#24107 #28436216

Order 15104.

1082 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#556C211yesmaximal, characteristic
2C2×C59×#484C211yesmaximal, characteristic, complemented
3C2×C59×#428C211yesmaximal, characteristic, complemented
4C59×#2486C211yesmaximal, characteristic
5C59×#2057C212yesmaximal, complemented
6C59×#2036C212yesmaximal, complemented
7C59×#1894C211yesmaximal, characteristic, complemented
8C59×#1870C211yesmaximal, characteristic
9C59×#1865C212yesmaximal, complemented
10C59×#1854C212yesmaximal, complemented
11C59×#1825C211yesmaximal, characteristic, complemented
12C22×M16×C59C2211yescharacteristic
13C59×#556—24no—
14C59×#556—24no—
15C2×C4⋊Q8×C59C2211yescharacteristic
16C2×#126×C59—21no—
17C2×#126×C59—21no—
18C2×#124×C59—21no—
19C2×#123×C59—21no—
20C2×C22⋊Q8×C59C2211yescharacteristic
21C2×C22⋊Q8×C59C2211yescharacteristic
22C2×C4×Q8×C59—21no—
23C2×C4×Q8×C59C2211yescharacteristic
24C2×C4×D4×C59C2211yescharacteristic
25C2×#117×C59C2211yescharacteristic
26C2×#117×C59C2211yescharacteristic
27C22×C4⋊C4×C59C2211yescharacteristic
28C59×#484—24no—
29C59×#428—24no—
30C2×#107×C59C2212yes—
31C2×#107×C59C2212yes—
32C2×#106×C59C2212yes—
33C2×#103×C59C2212yes—
34C2×#103×C59C2212yes—
35C2×C22⋊C8×C59C2212yes—
36C2×C22⋊C8×C59C2211yescharacteristic
37C59×#398—21no—
38C59×#397—21no—
39C59×#389C2212yes—
40C59×#381—21no—
41C59×#380C2212yes—
42C59×#380C2211yescharacteristic
43C59×#340C2212yes—
44C59×#340C2212yes—
45C59×#340C2212yes—
46C59×#336C2212yes—
47C22×Q8×C59—21no—
48C22×Q8×C59C2311yescharacteristic
49C22×D4×C59D411yescharacteristic
50C23×C4×C59—21no—
51C23×C4×C59C2311yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C2×M16×C59—22no—
54C2×M16×C59—24no—
55C22×C8×C59C2312yes—
56C22×C8×C59C2312yes—
57C4⋊Q8×C59—22no—
58C4⋊Q8×C59—22no—
59#126×C59—28no—
60#126×C59—28no—
61#124×C59—28no—
62#123×C59—44no—
63C22⋊Q8×C59—22no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—22no—
66C22⋊Q8×C59D414yes—
67C4×Q8×C59—28no—
68C4×Q8×C59—22no—
69C4×Q8×C59—22no—
70C4×D4×C59—22no—
71C4×D4×C59—22no—
72#117×C59—22no—
73#117×C59—22no—
74#117×C59—22no—
75#117×C59D414yes—
76C2×C4⋊C4×C59—21no—
77C2×C4⋊C4×C59—21no—
78C2×C4⋊C4×C59—21no—
79C2×C4⋊C4×C59—21no—
80C2×C4⋊C4×C59—22no—
81C2×C4⋊C4×C59—22no—
82C2×C4⋊C4×C59—22no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59D411yescharacteristic
85C2×C4⋊C4×C59C2312yes—
86C2×C4⋊C4×C59C2312yes—
87C2×C4⋊C4×C59C2311yescharacteristic
88C2×C4⋊C4×C59C2311yescharacteristic
89C2×C4⋊C4×C59C2311yescharacteristic
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59—22no—
96C2×C22⋊C4×C59D411yescharacteristic
97C2×C22⋊C4×C59D411yescharacteristic
98C2×C42×C59—21no—
99C2×C42×C59—21no—
100C2×C42×C59—21no—
101C2×C42×C59C2311yescharacteristic
102#107×C59—24no—
103#107×C59—24no—
104#106×C59—24no—
105#103×C59—24no—
106#103×C59—24no—
107C22⋊C8×C59—22no—
108C22⋊C8×C59—24no—
109#95×C59—22no—
110#95×C59—21no—
111#95×C59—22no—
112C24×C59C2×D411yescharacteristic
113C24×C59C2×D411yescharacteristic
114C2×Q8×C59—24no—
115C2×Q8×C59—21no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—42no—
119C2×Q8×C59—42no—
120C2×Q8×C59C2×D412yes—
121C2×D4×C59—22no—
122C2×D4×C59—22no—
123C2×D4×C59—41no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D412yes—
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2411yescharacteristic
146M16×C59—28no—
147C2×C8×C59—22no—
148C2×C8×C59—22no—
149C2×C8×C59—22no—
150C2×C8×C59—22no—
151C2×C8×C59—22no—
152C2×C8×C59#4114yes—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—44no—
167C4⋊C4×C59C2×D412yes—
168C4⋊C4×C59C2×D412yes—
169C4⋊C4×C59C2×D412yes—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—44no—
178C42×C59—24no—
179C42×C59—24no—
180C42×C59—24no—
181C42×C59—21no—
182C42×C59—21no—
183C23×C59—22no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59—22no—
187C23×C59—21no—
188C23×C59—22no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—22no—
192C23×C59#12012yes—
193C23×C59C22×D411yescharacteristic
194Q8×C59—24no—
195Q8×C59—48no—
196Q8×C59—41no—
197Q8×C59—41no—
198D4×C59—44no—
199C2×C4×C59—21no—
200C2×C4×C59—22no—
201C2×C4×C59—21no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—24no—
205C2×C4×C59—22no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—42no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—42no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—84no—
229C2×C4×C59#12012yes—
230C2×C4×C59AGL(1,ℤ/8)12yes—
231C2×C4×C59C22×D411yescharacteristic
232C2×C4×C59C22×D411yescharacteristic
233C2×C4×C59C2×#4112yes—
234C2×C4×C592−1+412yes—
235C8×C59—24no—
236C8×C59—42no—
237#24107C5911yesmaximal, characteristic, direct product factor, Sylow
238C22×C59—22no—
239C22×C59—22no—
240C22×C59—22no—
241C22×C59—42no—
242C22×C59—42no—
243C22×C59—42no—
244C22×C59—42no—
245C22×C59—41no—
246C22×C59—41no—
247C22×C59—41no—
248C22×C59C2×#12011yescharacteristic
249C22×C59#54712yes—
250C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C59#57511yescharacteristic
252C22×C59#57812yes—
253C4×C59—22no—
254C4×C59—22no—
255C4×C59—41no—
256C4×C59—42no—
257C4×C59—42no—
258C4×C59—42no—
259C4×C59—41no—
260C4×C59—42no—
261C4×C59—82no—
262C4×C59—82no—
263C4×C59#52511yescharacteristic
264C4×C59#53611yescharacteristic
265C4×C59#54712yes—
266C2×#556C2×C5911yescharacteristic
267C2×#484C2×C5911yescharacteristic, complemented
268C2×#428C2×C5911yescharacteristic, complemented
269#2486C2×C5911yescharacteristic
270#2057C2×C5912yescomplemented
271#2036C2×C5912yescomplemented
272#1894C2×C5911yescharacteristic, complemented
273#1870C2×C5911yescharacteristic
274#1865C2×C5912yescomplemented
275#1854C2×C5912yescomplemented
276#1825C2×C5911yescharacteristic, complemented
277C2×C59—42no—
278C2×C59—81no—
279C2×C59#240611yescharacteristic
280C2×C59#300611yescharacteristic
281C2×C59#301111yescharacteristic
282C2×C59#334912yes—
283C2×C59#335312yes—
284C22×M16C22×C5911yescharacteristic
285#556—24no—
286#556—24no—
287C2×C4⋊Q8C22×C5911yescharacteristic
288C2×#126—21no—
289C2×#126—21no—
290C2×#124—21no—
291C2×#123—21no—
292C2×C22⋊Q8C22×C5911yescharacteristic
293C2×C22⋊Q8C22×C5911yescharacteristic
294C2×C4×Q8—21no—
295C2×C4×Q8C22×C5911yescharacteristic
296C2×C4×D4C22×C5911yescharacteristic
297C2×#117C22×C5911yescharacteristic
298C2×#117C22×C5911yescharacteristic
299C22×C4⋊C4C22×C5911yescharacteristic
300#484—24no—
301#428—24no—
302C2×#107C22×C5912yes—
303C2×#107C22×C5912yes—
304C2×#106C22×C5912yes—
305C2×#103C22×C5912yes—
306C2×#103C22×C5912yes—
307C2×C22⋊C8C22×C5911yescharacteristic
308C2×C22⋊C8C22×C5912yes—
309#398—21no—
310#397—21no—
311#389C22×C5912yes—
312#381—21no—
313#380C22×C5911yescharacteristic
314#380C22×C5912yes—
315#340C22×C5912yes—
316#340C22×C5912yes—
317#340C22×C5912yes—
318#336C22×C5912yes—
319C59#2410711yescharacteristic, direct product factor, Sylow
320C22×Q8—21no—
321C22×Q8C23×C5911yescharacteristic
322C22×D4D4×C5911yescharacteristic
323C23×C4—21no—
324C23×C4C23×C5911yescharacteristic
325C23×C4C23×C5911yescharacteristic
326C2×M16—22no—
327C2×M16—24no—
328C22×C8C23×C5912yes—
329C22×C8C23×C5912yes—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332#126—28no—
333#126—28no—
334#124—28no—
335#123—44no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8D4×C5914yes—
340C4×Q8—28no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×D4—22no—
344C4×D4—22no—
345#117—22no—
346#117—22no—
347#117—22no—
348#117D4×C5914yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4D4×C5911yescharacteristic
358C2×C4⋊C4C23×C5912yes—
359C2×C4⋊C4C23×C5912yes—
360C2×C4⋊C4C23×C5911yescharacteristic
361C2×C4⋊C4C23×C5911yescharacteristic
362C2×C4⋊C4C23×C5911yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4D4×C5911yescharacteristic
370C2×C22⋊C4D4×C5911yescharacteristic
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42C23×C5911yescharacteristic
375#107—24no—
376#107—24no—
377#106—24no—
378#103—24no—
379#103—24no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382#95—22no—
383#95—21no—
384#95—22no—
385C24C2×D4×C5911yescharacteristic
386C24C2×D4×C5911yescharacteristic
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×D4×C5912yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—41no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C2×D4×C5911yescharacteristic
413C22×C4C2×D4×C5911yescharacteristic
414C22×C4C2×D4×C5911yescharacteristic
415C22×C4C2×D4×C5912yes—
416C22×C4C2×D4×C5912yes—
417C22×C4C2×D4×C5911yescharacteristic
418C22×C4C24×C5911yescharacteristic
419M16—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#41×C5914yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4C2×D4×C5912yes—
441C4⋊C4C2×D4×C5912yes—
442C4⋊C4C2×D4×C5912yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—21no—
455C42—21no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23#120×C5912yes—
466C23C22×D4×C5911yescharacteristic
467Q8—24no—
468Q8—48no—
469Q8—41no—
470Q8—41no—
471D4—44no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4#120×C5912yes—
503C2×C4AGL(1,ℤ/8)×C5912yes—
504C2×C4C22×D4×C5911yescharacteristic
505C2×C4C22×D4×C5911yescharacteristic
506C2×C4C2×#41×C5912yes—
507C2×C42−1+4×C5912yes—
508C8—24no—
509C8—42no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22C2×#120×C5911yescharacteristic
521C22C59×#54712yes—
522C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
523C22C59×#57511yescharacteristic
524C22C59×#57812yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—82no—
534C4—82no—
535C4C59×#52511yescharacteristic
536C4C59×#53611yescharacteristic
537C4C59×#54712yes—
538C2—42no—
539C2—81no—
540C2C59×#240611yescharacteristic
541C2C59×#300611yescharacteristic
542C2C59×#301111yescharacteristic
543C2C59×#334912yes—
544C2C59×#335312yes—
subgroupG/H
center, socleC23×C59C22×D4
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC22×D4×C59D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C2×C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#484C2semidirect
3. C2×C59×#428C2semidirect
5. C59×#2057C2semidirect
6. C59×#2036C2semidirect
7. C59×#1894C2semidirect
9. C59×#1865C2semidirect
10. C59×#1854C2semidirect
11. C59×#1825C2semidirect
237. #24107C59direct (N × H)
267. C2×#484C2×C59semidirect
268. C2×#428C2×C59semidirect
270. #2057C2×C59semidirect
271. #2036C2×C59semidirect
272. #1894C2×C59semidirect
274. #1865C2×C59semidirect
275. #1854C2×C59semidirect
276. #1825C2×C59semidirect
319. C59#24107direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#5562
C2×C59×#4842
C2×C59×#4282
C59×#24862
C59×#20572
C59×#20362
C59×#18942
C59×#18702
C59×#18652
C59×#18542
C59×#18252
#2410759