Finite Group Databasebeta

← C59×#25584 #28437693

Subgroups — C59×#25584 #28437693

Order 15104.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#552C211yesmaximal, characteristic
2C2×C59×#491C211yesmaximal, characteristic, complemented
3C2×C59×#462C211yesmaximal, characteristic, complemented
4C59×#2542C211yesmaximal, characteristic
5C59×#2001C212yesmaximal, complemented
6C59×#2001C212yesmaximal, complemented
7C59×#1989C211yesmaximal, characteristic, complemented
8C59×#1955C211yesmaximal, characteristic
9C59×#1873C212yesmaximal, complemented
10C59×#1872C212yesmaximal, complemented
11C59×#1763C211yesmaximal, characteristic, complemented
12C22×Q16×C59C2212yes—
13C59×#552—24no—
14C59×#552—24no—
15C2×C4⋊Q8×C59C2211yescharacteristic
16C2×#126×C59—21no—
17C2×#125×C59—21no—
18C2×#125×C59C2211yescharacteristic
19C2×#123×C59—21no—
20C2×C22⋊Q8×C59C2211yescharacteristic
21C2×C22⋊Q8×C59C2211yescharacteristic
22C2×C22⋊Q8×C59C2212yes—
23C2×C22⋊Q8×C59C2212yes—
24C2×C4×Q8×C59C2211yescharacteristic
25C2×C4×D4×C59C2211yescharacteristic
26C2×#117×C59—21no—
27C2×#117×C59C2211yescharacteristic
28C22×C4⋊C4×C59C2211yescharacteristic
29C59×#491—24no—
30C59×#462—24no—
31C2×C4⋊C8×C59C2211yescharacteristic
32C2×#103×C59C2212yes—
33C2×#103×C59C2212yes—
34C2×#103×C59C2212yes—
35C2×#103×C59C2212yes—
36C2×C22⋊C8×C59C2212yes—
37C2×C22⋊C8×C59C2212yes—
38C2×#97×C59C2211yescharacteristic
39C59×#399—21no—
40C59×#399—21no—
41C59×#390—21no—
42C59×#384—21no—
43C4×C4⋊C4×C59C2211yescharacteristic
44C59×#340C2212yes—
45C59×#340C2212yes—
46C59×#340C2212yes—
47C22×Q8×C59C2312yes—
48C22×Q8×C59C2311yescharacteristic
49C22×D4×C59D411yescharacteristic
50C23×C4×C59—21no—
51C23×C4×C59C2311yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C2×Q16×C59—24no—
54C2×Q16×C59—42no—
55C2×Q16×C59—42no—
56C22×C8×C59C2312yes—
57C22×C8×C59C2312yes—
58C4⋊Q8×C59—22no—
59C4⋊Q8×C59—22no—
60#126×C59—28no—
61#125×C59—28no—
62#125×C59—22no—
63#125×C59D414yes—
64#123×C59—28no—
65C22⋊Q8×C59—22no—
66C22⋊Q8×C59—22no—
67C22⋊Q8×C59—22no—
68C22⋊Q8×C59—22no—
69C22⋊Q8×C59—24no—
70C22⋊Q8×C59—24no—
71C22⋊Q8×C59—42no—
72C22⋊Q8×C59—42no—
73C4×Q8×C59—22no—
74C4×Q8×C59D414yes—
75C4×D4×C59—22no—
76C4×D4×C59—22no—
77#117×C59—28no—
78#117×C59—22no—
79#117×C59—22no—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59—21no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59—22no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—22no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59D411yescharacteristic
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C4⋊C4×C59C2312yes—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59D411yescharacteristic
103C2×C22⋊C4×C59D411yescharacteristic
104C2×C42×C59—21no—
105C2×C42×C59—21no—
106C2×C42×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C4⋊C8×C59—22no—
109#103×C59—24no—
110#103×C59—24no—
111#103×C59—24no—
112#103×C59—24no—
113C22⋊C8×C59—42no—
114C22⋊C8×C59—42no—
115#97×C59—22no—
116#95×C59—21no—
117C24×C59C2×D411yescharacteristic
118C24×C59C2×D411yescharacteristic
119C2×Q8×C59—21no—
120C2×Q8×C59—22no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59—22no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59C2×D412yes—
129C2×D4×C59—22no—
130C2×D4×C59—22no—
131C2×D4×C59—41no—
132C22×C4×C59—24no—
133C22×C4×C59—22no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2411yescharacteristic
154Q16×C59—48no—
155C2×C8×C59—22no—
156C2×C8×C59—22no—
157C2×C8×C59—22no—
158C2×C8×C59—22no—
159C2×C8×C59—22no—
160C2×C8×C59—22no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—44no—
173C4⋊C4×C59—44no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59C2×D412yes—
177C4⋊C4×C59C2×D412yes—
178C4⋊C4×C59C2×D412yes—
179C4⋊C4×C59#4114yes—
180C22⋊C4×C59—24no—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—24no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—44no—
186C22⋊C4×C59—44no—
187C42×C59—24no—
188C42×C59—24no—
189C42×C59—42no—
190C42×C59C2×D412yes—
191C42×C59C2×D412yes—
192C23×C59—21no—
193C23×C59—22no—
194C23×C59—21no—
195C23×C59—22no—
196C23×C59—21no—
197C23×C59—22no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—22no—
201C23×C59#13712yes—
202C23×C59C22×D411yescharacteristic
203Q8×C59—22no—
204Q8×C59—22no—
205Q8×C59—42no—
206Q8×C59—42no—
207Q8×C59—42no—
208Q8×C59—42no—
209Q8×C59—42no—
210D4×C59—44no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—24no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—48no—
224C2×C4×C59—42no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—42no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59#12012yes—
242C2×C4×C59#12012yes—
243C2×C4×C59C22×D411yescharacteristic
244C2×C4×C59C22×D411yescharacteristic
245C2×C4×C59C2×#4112yes—
246C2×C4×C59H(5,𝔽2)12yes—
247C8×C59—42no—
248C8×C59—42no—
249#25584C5911yesmaximal, characteristic, direct product factor, Sylow
250C22×C59—22no—
251C22×C59—24no—
252C22×C59—41no—
253C22×C59—41no—
254C22×C59—42no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59C2×#12011yescharacteristic
261C22×C59#54612yes—
262C22×C59C2×#13711yescharacteristic
263C22×C59#57511yescharacteristic
264C22×C59#57812yes—
265C4×C59—22no—
266C4×C59—22no—
267C4×C59—41no—
268C4×C59—41no—
269C4×C59—42no—
270C4×C59—42no—
271C4×C59—42no—
272C4×C59—42no—
273C4×C59—44no—
274C4×C59—82no—
275C4×C59#53211yescharacteristic
276C4×C59#53511yescharacteristic
277C4×C59#55012yes—
278C2×#552C2×C5911yescharacteristic
279C2×#491C2×C5911yescharacteristic, complemented
280C2×#462C2×C5911yescharacteristic, complemented
281#2542C2×C5911yescharacteristic
282#2001C2×C5912yescomplemented
283#2001C2×C5912yescomplemented
284#1989C2×C5911yescharacteristic, complemented
285#1955C2×C5911yescharacteristic
286#1873C2×C5912yescomplemented
287#1872C2×C5912yescomplemented
288#1763C2×C5911yescharacteristic, complemented
289C2×C59—42no—
290C2×C59—81no—
291C2×C59#243111yescharacteristic
292C2×C59#300711yescharacteristic
293C2×C59#301111yescharacteristic
294C2×C59#331212yes—
295C2×C59#331512yes—
296C22×Q16C22×C5912yes—
297#552—24no—
298#552—24no—
299C2×C4⋊Q8C22×C5911yescharacteristic
300C2×#126—21no—
301C2×#125—21no—
302C2×#125C22×C5911yescharacteristic
303C2×#123—21no—
304C2×C22⋊Q8C22×C5911yescharacteristic
305C2×C22⋊Q8C22×C5911yescharacteristic
306C2×C22⋊Q8C22×C5912yes—
307C2×C22⋊Q8C22×C5912yes—
308C2×C4×Q8C22×C5911yescharacteristic
309C2×C4×D4C22×C5911yescharacteristic
310C2×#117—21no—
311C2×#117C22×C5911yescharacteristic
312C22×C4⋊C4C22×C5911yescharacteristic
313#491—24no—
314#462—24no—
315C2×C4⋊C8C22×C5911yescharacteristic
316C2×#103C22×C5912yes—
317C2×#103C22×C5912yes—
318C2×#103C22×C5912yes—
319C2×#103C22×C5912yes—
320C2×C22⋊C8C22×C5912yes—
321C2×C22⋊C8C22×C5912yes—
322C2×#97C22×C5911yescharacteristic
323#399—21no—
324#399—21no—
325#390—21no—
326#384—21no—
327C4×C4⋊C4C22×C5911yescharacteristic
328#340C22×C5912yes—
329#340C22×C5912yes—
330#340C22×C5912yes—
331C59#2558411yescharacteristic, direct product factor, Sylow
332C22×Q8C23×C5912yes—
333C22×Q8C23×C5911yescharacteristic
334C22×D4D4×C5911yescharacteristic
335C23×C4—21no—
336C23×C4C23×C5911yescharacteristic
337C23×C4C23×C5911yescharacteristic
338C2×Q16—24no—
339C2×Q16—42no—
340C2×Q16—42no—
341C22×C8C23×C5912yes—
342C22×C8C23×C5912yes—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345#126—28no—
346#125—28no—
347#125—22no—
348#125D4×C5914yes—
349#123—28no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—24no—
355C22⋊Q8—24no—
356C22⋊Q8—42no—
357C22⋊Q8—42no—
358C4×Q8—22no—
359C4×Q8D4×C5914yes—
360C4×D4—22no—
361C4×D4—22no—
362#117—28no—
363#117—22no—
364#117—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4D4×C5911yescharacteristic
378C2×C4⋊C4C23×C5911yescharacteristic
379C2×C4⋊C4C23×C5911yescharacteristic
380C2×C4⋊C4C23×C5911yescharacteristic
381C2×C4⋊C4C23×C5912yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4D4×C5911yescharacteristic
388C2×C22⋊C4D4×C5911yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42C23×C5911yescharacteristic
393C4⋊C8—22no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398C22⋊C8—42no—
399C22⋊C8—42no—
400#97—22no—
401#95—21no—
402C24C2×D4×C5911yescharacteristic
403C24C2×D4×C5911yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8C2×D4×C5912yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
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—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C2×D4×C5911yescharacteristic
435C22×C4C2×D4×C5911yescharacteristic
436C22×C4C2×D4×C5911yescharacteristic
437C22×C4C2×D4×C5911yescharacteristic
438C22×C4C24×C5911yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4C2×D4×C5912yes—
462C4⋊C4C2×D4×C5912yes—
463C4⋊C4C2×D4×C5912yes—
464C4⋊C4#41×C5914yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C42—24no—
473C42—24no—
474C42—42no—
475C42C2×D4×C5912yes—
476C42C2×D4×C5912yes—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—22no—
481C23—21no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23#137×C5912yes—
487C23C22×D4×C5911yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495D4—44no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4#120×C5912yes—
527C2×C4#120×C5912yes—
528C2×C4C22×D4×C5911yescharacteristic
529C2×C4C22×D4×C5911yescharacteristic
530C2×C4C2×#41×C5912yes—
531C2×C4H(5,𝔽2)×C5912yes—
532C8—42no—
533C8—42no—
534C22—22no—
535C22—24no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22C2×#120×C5911yescharacteristic
545C22C59×#54612yes—
546C22C2×#137×C5911yescharacteristic
547C22C59×#57511yescharacteristic
548C22C59×#57812yes—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—82no—
559C4C59×#53211yescharacteristic
560C4C59×#53511yescharacteristic
561C4C59×#55012yes—
562C2—42no—
563C2—81no—
564C2C59×#243111yescharacteristic
565C2C59×#300711yescharacteristic
566C2C59×#301111yescharacteristic
567C2C59×#331212yes—
568C2C59×#331512yes—
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×#491C2semidirect
3. C2×C59×#462C2semidirect
5. C59×#2001C2semidirect
6. C59×#2001C2semidirect
7. C59×#1989C2semidirect
9. C59×#1873C2semidirect
10. C59×#1872C2semidirect
11. C59×#1763C2semidirect
249. #25584C59direct (N × H)
279. C2×#491C2×C59semidirect
280. C2×#462C2×C59semidirect
282. #2001C2×C59semidirect
283. #2001C2×C59semidirect
284. #1989C2×C59semidirect
286. #1873C2×C59semidirect
287. #1872C2×C59semidirect
288. #1763C2×C59semidirect
331. C59#25584direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#5522
C2×C59×#4912
C2×C59×#4622
C59×#25422
C59×#20012
C59×#19892
C59×#19552
C59×#18732
C59×#18722
C59×#17632
#2558459