Finite Group Databasebeta

← C59×#24006 #28436115

Subgroups — C59×#24006 #28436115

Order 15104.

1500 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C59C211yesmaximal, characteristic
2C2×C59×#484C212yesmaximal, complemented
3C2×C59×#380C211yesmaximal, characteristic
4C59×#2036C214yesmaximal, complemented
5C59×#1872C212yesmaximal, complemented
6C59×#1852C212yesmaximal, complemented
7C59×#1787C212yesmaximal, complemented
8C59×#1779C211yesmaximal, characteristic
9C23×Q8×C59C2211yescharacteristic
10C24×C4×C59C2211yescharacteristic
11C2×C22⋊Q8×C59—21no—
12C2×C22⋊Q8×C59—21no—
13C2×C22⋊Q8×C59—28no—
14C2×C22⋊Q8×C59C2212yes—
15C2×C22⋊Q8×C59C2212yes—
16C22×C4⋊C4×C59—21no—
17C22×C4⋊C4×C59C2211yescharacteristic
18C22×C4⋊C4×C59C2211yescharacteristic
19C22×C4⋊C4×C59C2211yescharacteristic
20C22×C22⋊C4×C59—21no—
21C22×C22⋊C4×C59—21no—
22C59×#484—28no—
23C2×#107×C59C2214yescomplemented
24C2×#103×C59C2214yescomplemented
25C2×#103×C59C2214yescomplemented
26C2×C22⋊C8×C59C2212yes—
27C2×C22⋊C8×C59C2212yes—
28C2×C22⋊C8×C59C2212yes—
29C59×#380—21no—
30C59×#380—21no—
31C59×#380C2212yes—
32C59×#380C2212yes—
33C2×#95×C59—21no—
34C59×#340C2214yescomplemented
35C25×C59D411yescharacteristic
36C22×Q8×C59—24no—
37C22×Q8×C59—21no—
38C22×Q8×C59—24no—
39C22×Q8×C59—42no—
40C22×Q8×C59C2312yes—
41C23×C4×C59—21no—
42C23×C4×C59—21no—
43C23×C4×C59—24no—
44C23×C4×C59—24no—
45C23×C4×C59—21no—
46C23×C4×C59—21no—
47C23×C4×C59—42no—
48C23×C4×C59D411yescharacteristic
49C23×C4×C59D411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C23×C4×C59C2311yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C23×C4×C59C2311yescharacteristic
54C22×C8×C59C2314yes—
55C22⋊Q8×C59—28no—
56C22⋊Q8×C59—28no—
57C22⋊Q8×C59—24no—
58C22⋊Q8×C59—24no—
59C22⋊Q8×C59—24no—
60C22⋊Q8×C59D418yes—
61C2×C4⋊C4×C59—28no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59—22no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—24no—
66C2×C4⋊C4×C59—24no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59—42no—
70C2×C4⋊C4×C59C2312yes—
71C2×C4⋊C4×C59C2312yes—
72C2×C4⋊C4×C59C2312yes—
73C2×C22⋊C4×C59—28no—
74C2×C22⋊C4×C59—22no—
75C2×C22⋊C4×C59—22no—
76C2×C22⋊C4×C59—22no—
77C2×C22⋊C4×C59—22no—
78C2×C22⋊C4×C59—44no—
79#107×C59—28no—
80#103×C59—28no—
81#103×C59—44no—
82C22⋊C8×C59—24no—
83C22⋊C8×C59—24no—
84C22⋊C8×C59—42no—
85#95×C59—22no—
86#95×C59—22no—
87C24×C59—22no—
88C24×C59—22no—
89C24×C59—22no—
90C24×C59—22no—
91C24×C59—41no—
92C24×C59—41no—
93C24×C59Q1614yes—
94C24×C59C2×D411yescharacteristic
95C24×C59C2×D411yescharacteristic
96C24×C59C2×D411yescharacteristic
97C2×Q8×C59—24no—
98C2×Q8×C59—22no—
99C2×Q8×C59—24no—
100C2×Q8×C59—24no—
101C2×Q8×C59—416no—
102C2×Q8×C59—42no—
103C2×Q8×C59—42no—
104C2×Q8×C59C2×D414yes—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—24no—
110C22×C4×C59—24no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—41no—
125C22×C4×C59—41no—
126C22×C4×C59—41no—
127C22×C4×C59—416no—
128C22×C4×C59—42no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59—42no—
132C22×C4×C59—42no—
133C22×C4×C59—42no—
134C22×C4×C59—42no—
135C22×C4×C59—42no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D414yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59#4114yes—
143C22×C4×C59C2411yescharacteristic
144C2×C8×C59—24no—
145C2×C8×C59—24no—
146C2×C8×C59—24no—
147C4⋊C4×C59—24no—
148C4⋊C4×C59—28no—
149C4⋊C4×C59—28no—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—24no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—42no—
154C4⋊C4×C59—42no—
155C4⋊C4×C59C2×D414yes—
156C22⋊C4×C59—28no—
157C22⋊C4×C59—28no—
158C22⋊C4×C59—44no—
159C22⋊C4×C59—44no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—21no—
167C23×C59—21no—
168C23×C59—21no—
169C23×C59—21no—
170C23×C59—21no—
171C23×C59—21no—
172C23×C59—21no—
173C23×C59—21no—
174C23×C59—28no—
175C23×C59—28no—
176C23×C59—44no—
177C23×C59—44no—
178C23×C59—44no—
179C23×C59—44no—
180C23×C59—44no—
181C23×C59#12012yes—
182C23×C59#12312yes—
183C23×C59C2×Q1612yes—
184C23×C59C2×Q1612yes—
185C23×C59C2×Q1612yes—
186C23×C59AGL(1,ℤ/8)12yes—
187C23×C59#13712yes—
188C23×C59C22×D411yescharacteristic
189Q8×C59—28no—
190Q8×C59—48no—
191Q8×C59—44no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—28no—
197C2×C4×C59—28no—
198C2×C4×C59—24no—
199C2×C4×C59—24no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—44no—
204C2×C4×C59—44no—
205C2×C4×C59—44no—
206C2×C4×C59—48no—
207C2×C4×C59—48no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—42no—
212C2×C4×C59—44no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—84no—
220C2×C4×C59—84no—
221C2×C4×C59#12012yes—
222C2×C4×C59#12312yes—
223C2×C4×C59C22×D412yes—
224C2×C4×C59C2×#4112yes—
225C2×C4×C59H(5,𝔽2)12yes—
226C8×C59—44no—
227#24006C5911yesmaximal, characteristic, direct product factor, Sylow
228C22×C59—22no—
229C22×C59—22no—
230C22×C59—22no—
231C22×C59—22no—
232C22×C59—22no—
233C22×C59—22no—
234C22×C59—22no—
235C22×C59—22no—
236C22×C59—22no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—22no—
240C22×C59—22no—
241C22×C59—22no—
242C22×C59—22no—
243C22×C59—416no—
244C22×C59—41no—
245C22×C59—41no—
246C22×C59—41no—
247C22×C59—41no—
248C22×C59#45114yes—
249C22×C59#48414yescomplemented
250C22×C59C2×#12011yescharacteristic
251C22×C59C2×#12311yescharacteristic
252C22×C59#53411yescharacteristic
253C22×C59C22×Q1611yescharacteristic
254C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C59C2×#13711yescharacteristic
256C22×C59#57511yescharacteristic
257C4×C59—22no—
258C4×C59—22no—
259C4×C59—41no—
260C4×C59—44no—
261C4×C59—44no—
262C4×C59—82no—
263C4×C59—82no—
264C4×C59#54614yes—
265C22×C22⋊Q8C2×C5911yescharacteristic
266C2×#484C2×C5912yescomplemented
267C2×#380C2×C5911yescharacteristic
268#2036C2×C5914yescomplemented
269#1872C2×C5912yescomplemented
270#1852C2×C5912yescomplemented
271#1787C2×C5912yescomplemented
272#1779C2×C5911yescharacteristic
273C2×C59—24no—
274C2×C59—24no—
275C2×C59—42no—
276C2×C59#240511yescharacteristic
277C2×C59C2×#45111yescharacteristic
278C2×C59#300611yescharacteristic
279C2×C59C2×#48411yescharacteristic
280C2×C59#309511yescharacteristic
281C2×C59#318911yescharacteristic
282C2×C59#319311yescharacteristic
283C23×Q8C22×C5911yescharacteristic
284C24×C4C22×C5911yescharacteristic
285C2×C22⋊Q8—21no—
286C2×C22⋊Q8—21no—
287C2×C22⋊Q8—28no—
288C2×C22⋊Q8C22×C5912yes—
289C2×C22⋊Q8C22×C5912yes—
290C22×C4⋊C4—21no—
291C22×C4⋊C4C22×C5911yescharacteristic
292C22×C4⋊C4C22×C5911yescharacteristic
293C22×C4⋊C4C22×C5911yescharacteristic
294C22×C22⋊C4—21no—
295C22×C22⋊C4—21no—
296#484—28no—
297C2×#107C22×C5914yescomplemented
298C2×#103C22×C5914yescomplemented
299C2×#103C22×C5914yescomplemented
300C2×C22⋊C8C22×C5912yes—
301C2×C22⋊C8C22×C5912yes—
302C2×C22⋊C8C22×C5912yes—
303#380—21no—
304#380—21no—
305#380C22×C5912yes—
306#380C22×C5912yes—
307C2×#95—21no—
308#340C22×C5914yescomplemented
309C59#2400611yescharacteristic, direct product factor, Sylow
310C25D4×C5911yescharacteristic
311C22×Q8—24no—
312C22×Q8—21no—
313C22×Q8—24no—
314C22×Q8—42no—
315C22×Q8C23×C5912yes—
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—24no—
319C23×C4—24no—
320C23×C4—21no—
321C23×C4—21no—
322C23×C4—42no—
323C23×C4D4×C5911yescharacteristic
324C23×C4D4×C5911yescharacteristic
325C23×C4D4×C5911yescharacteristic
326C23×C4C23×C5911yescharacteristic
327C23×C4C23×C5911yescharacteristic
328C23×C4C23×C5911yescharacteristic
329C22×C8C23×C5914yes—
330C22⋊Q8—28no—
331C22⋊Q8—28no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8D4×C5918yes—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4C23×C5912yes—
346C2×C4⋊C4C23×C5912yes—
347C2×C4⋊C4C23×C5912yes—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—44no—
354#107—28no—
355#103—28no—
356#103—44no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—42no—
360#95—22no—
361#95—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24Q16×C5914yes—
369C24C2×D4×C5911yescharacteristic
370C24C2×D4×C5911yescharacteristic
371C24C2×D4×C5911yescharacteristic
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—416no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8C2×D4×C5914yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—416no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4C2×D4×C5911yescharacteristic
414C22×C4C2×D4×C5911yescharacteristic
415C22×C4C2×D4×C5914yes—
416C22×C4C2×D4×C5911yescharacteristic
417C22×C4#41×C5914yes—
418C22×C4C24×C5911yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4C2×D4×C5914yes—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—28no—
444C23—28no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#120×C5912yes—
457C23#123×C5912yes—
458C23C2×Q16×C5912yes—
459C23C2×Q16×C5912yes—
460C23C2×Q16×C5912yes—
461C23AGL(1,ℤ/8)×C5912yes—
462C23#137×C5912yes—
463C23C22×D4×C5911yescharacteristic
464Q8—28no—
465Q8—48no—
466Q8—44no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4#120×C5912yes—
497C2×C4#123×C5912yes—
498C2×C4C22×D4×C5912yes—
499C2×C4C2×#41×C5912yes—
500C2×C4H(5,𝔽2)×C5912yes—
501C8—44no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22C59×#45114yes—
523C22C59×#48414yescomplemented
524C22C2×#120×C5911yescharacteristic
525C22C2×#123×C5911yescharacteristic
526C22C59×#53411yescharacteristic
527C22C22×Q16×C5911yescharacteristic
528C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
529C22C2×#137×C5911yescharacteristic
530C22C59×#57511yescharacteristic
531C4—22no—
532C4—22no—
533C4—44no—
534C4—41no—
535C4—44no—
536C4—82no—
537C4—82no—
538C4C59×#54614yes—
539C2—24no—
540C2—24no—
541C2—42no—
542C2C59×#240511yescharacteristic
543C2C2×C59×#45111yescharacteristic
544C2C59×#300611yescharacteristic
545C2C2×C59×#48411yescharacteristic
546C2C59×#309511yescharacteristic
547C2C59×#318911yescharacteristic
548C2C59×#319311yescharacteristic
subgroupG/H
center, socleC23×C59C22×D4
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC25×C59D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C24×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#484C2semidirect
4. C59×#2036C2semidirect
5. C59×#1872C2semidirect
6. C59×#1852C2semidirect
7. C59×#1787C2semidirect
23. C2×#107×C59C22semidirect
24. C2×#103×C59C22semidirect
25. C2×#103×C59C22semidirect
34. C59×#340C22semidirect
227. #24006C59direct (N × H)
249. C22×C59#484semidirect
266. C2×#484C2×C59semidirect
268. #2036C2×C59semidirect
269. #1872C2×C59semidirect
270. #1852C2×C59semidirect
271. #1787C2×C59semidirect
297. C2×#107C22×C59semidirect
298. C2×#103C22×C59semidirect
299. C2×#103C22×C59semidirect
308. #340C22×C59semidirect
309. C59#24006direct (N × H)
523. C22C59×#484semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C592
C2×C59×#4842
C2×C59×#3802
C59×#20362
C59×#18722
C59×#18522
C59×#17872
C59×#17792
#2400659