Finite Group Databasebeta

← C59×#13344 #28425453

Subgroups — C59×#13344 #28425453

Order 15104.

892 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#458C211yesmaximal, characteristic, complemented
2C2×C59×B(4,𝔽2)C211yesmaximal, characteristic
3C59×#2697C211yesmaximal, characteristic, complemented
4C2×C59×#352C211yesmaximal, characteristic, complemented
5C2×C59×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6C59×#1792C211yesmaximal, characteristic, complemented
7C59×#1393C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C59C2211yescharacteristic
9C2×#123×C59—21no—
10C2×#123×C59C2211yescharacteristic
11C2×#120×C59—21no—
12C2×#120×C59C2211yescharacteristic
13C59×#458—22no—
14C59×#458—22no—
15C59×B(4,𝔽2)—22no—
16C59×B(4,𝔽2)—22no—
17C2×#104×C59—21no—
18C59×#417—21no—
19C2×#100×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C2×#99×C59—21no—
22C2×#99×C59C2211yescharacteristic
23C2×#99×C59C2211yescharacteristic
24C59×#397—21no—
25C59×#396—21no—
26C59×#394—21no—
27C59×#390C2211yescharacteristic
28C59×#388—21no—
29C59×#386—21no—
30C59×#379—21no—
31C59×#352—22no—
32C59×AΣL(1,𝔽16)—22no—
33C25×C59D411yescharacteristic
34H(5,𝔽2)×C59—21no—
35H(5,𝔽2)×C59—21no—
36H(5,𝔽2)×C59—21no—
37H(5,𝔽2)×C59—42no—
38H(5,𝔽2)×C59D412yes—
39C2×#41×C59—21no—
40C2×#41×C59—41no—
41C22×D4×C59—21no—
42C22×D4×C59—21no—
43C22×D4×C59—21no—
44C22×D4×C59—41no—
45C22×D4×C59—41no—
46C22×D4×C59C2311yescharacteristic
47C23×C4×C59D411yescharacteristic, complemented
48C2×M16×C59—21no—
49#123×C59—24no—
50#123×C59—21no—
51#123×C59—21no—
52#123×C59—21no—
53#123×C59—21no—
54#123×C59—42no—
55#120×C59—24no—
56#120×C59—21no—
57#120×C59—21no—
58#120×C59—21no—
59#120×C59—21no—
60#120×C59—42no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—41no—
64C2×C4⋊C4×C59D411yescharacteristic, complemented
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—21no—
68C2×C22⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72C2×C22⋊C4×C59—21no—
73C2×C22⋊C4×C59—21no—
74C2×C22⋊C4×C59—21no—
75C2×C22⋊C4×C59—41no—
76C2×C22⋊C4×C59—41no—
77C2×C22⋊C4×C59D411yescharacteristic
78C2×C22⋊C4×C59D411yescharacteristic, complemented
79C2×C42×C59D411yescharacteristic, complemented
80#104×C59—42no—
81#103×C59—41no—
82#102×C59—41no—
83#100×C59—21no—
84#100×C59—21no—
85#99×C59—24no—
86#99×C59—24no—
87#99×C59—21no—
88#99×C59—21no—
89#99×C59D412yes—
90#99×C59D412yescomplemented
91#95×C59—21no—
92#95×C59—21no—
93#95×C59—21no—
94C24×C59—21no—
95C24×C59—21no—
96C24×C59—21no—
97C24×C59—21no—
98C24×C59—21no—
99C24×C59—41no—
100C24×C59—41no—
101C24×C59—42no—
102C24×C59—81no—
103C24×C59C2×D411yescharacteristic
104C24×C59C2×D411yescharacteristic
105#41×C59—22no—
106#41×C59—22no—
107#41×C59—42no—
108#41×C59—41no—
109#41×C59—42no—
110#41×C59—42no—
111#41×C59—42no—
112#41×C59—41no—
113C2×Q8×C59—21no—
114C2×D4×C59—21no—
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—22no—
119C2×D4×C59—41no—
120C2×D4×C59—41no—
121C2×D4×C59—42no—
122C2×D4×C59—42no—
123C2×D4×C59—41no—
124C2×D4×C59—41no—
125C2×D4×C59—41no—
126C2×D4×C59—41no—
127C2×D4×C59—42no—
128C2×D4×C59—42no—
129C2×D4×C59—41no—
130C2×D4×C59—41no—
131C2×D4×C59—41no—
132C2×D4×C59—44no—
133C2×D4×C59—42no—
134C2×D4×C59—41no—
135C2×D4×C59—41no—
136C2×D4×C59—81no—
137C2×D4×C59—82no—
138C2×D4×C59—82no—
139C2×D4×C59C2×D411yescharacteristic
140C2×D4×C59C2×D411yescharacteristic
141C2×D4×C59C2×D411yescharacteristic
142C2×D4×C59#4111yescharacteristic
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—41no—
153C22×C4×C59—41no—
154C22×C4×C59—41no—
155C22×C4×C59—41no—
156C22×C4×C59—41no—
157C22×C4×C59C2×D411yescharacteristic
158M16×C59—41no—
159M16×C59—41no—
160C2×C8×C59—41no—
161C4⋊C4×C59—44no—
162C4⋊C4×C59—41no—
163C4⋊C4×C59—41no—
164C4⋊C4×C59—41no—
165C4⋊C4×C59—81no—
166C22⋊C4×C59—21no—
167C22⋊C4×C59—21no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—21no—
171C22⋊C4×C59—21no—
172C22⋊C4×C59—44no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—41no—
175C22⋊C4×C59—41no—
176C22⋊C4×C59—44no—
177C22⋊C4×C59—41no—
178C22⋊C4×C59—41no—
179C22⋊C4×C59—41no—
180C22⋊C4×C59—41no—
181C22⋊C4×C59—81no—
182C22⋊C4×C59—81no—
183C22⋊C4×C59—81no—
184C22⋊C4×C59—81no—
185C22⋊C4×C59—81no—
186C42×C59—41no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—22no—
192C23×C59—22no—
193C23×C59—22no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—42no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—42no—
205C23×C59—42no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—81no—
211C23×C59—82no—
212C23×C59—81no—
213C23×C59—82no—
214C23×C59—84no—
215C23×C59—81no—
216C23×C59—81no—
217C23×C59—81no—
218C23×C59—81no—
219C23×C59—81no—
220C23×C59#12011yescharacteristic
221C23×C59#12011yescharacteristic
222C23×C59#12011yescharacteristic
223C23×C59#12111yescharacteristic
224C23×C59#12111yescharacteristic
225Q8×C59—22no—
226Q8×C59—41no—
227D4×C59—22no—
228D4×C59—41no—
229D4×C59—41no—
230D4×C59—41no—
231D4×C59—41no—
232D4×C59—41no—
233D4×C59—42no—
234D4×C59—42no—
235D4×C59—42no—
236D4×C59—42no—
237D4×C59—42no—
238D4×C59—42no—
239D4×C59—82no—
240D4×C59—82no—
241D4×C59—88no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—41no—
268C2×C4×C59—41no—
269C2×C4×C59—81no—
270C2×C4×C59—81no—
271C2×C4×C59—81no—
272C2×C4×C59—81no—
273C2×C4×C59—81no—
274C2×C4×C59#12111yescharacteristic
275C2×C4×C59#12711yescharacteristic
276C8×C59—81no—
277#13344C5911yesmaximal, characteristic, direct product factor, Sylow
278C22×C59—21no—
279C22×C59—21no—
280C22×C59—21no—
281C22×C59—21no—
282C22×C59—21no—
283C22×C59—21no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—44no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—82no—
302C22×C59—82no—
303C22×C59—82no—
304C22×C59—82no—
305C22×C59—82no—
306C22×C59—81no—
307C22×C59—81no—
308C22×C59—81no—
309C22×C59—81no—
310C22×C59—161no—
311C22×C59—161no—
312C22×C59#39211yescharacteristic
313C22×C59B(4,𝔽2)11yescharacteristic
314C22×C59B(4,𝔽2)11yescharacteristic
315C4×C59—21no—
316C4×C59—21no—
317C4×C59—42no—
318C4×C59—81no—
319C4×C59—81no—
320C4×C59—81no—
321C4×C59—82no—
322C4×C59—82no—
323C4×C59—81no—
324C4×C59—81no—
325C2×#458C2×C5911yescharacteristic, complemented
326C2×B(4,𝔽2)C2×C5911yescharacteristic
327#2697C2×C5911yescharacteristic, complemented
328C2×#352C2×C5911yescharacteristic, complemented
329C2×AΣL(1,𝔽16)C2×C5911yescharacteristic, complemented
330#1792C2×C5911yescharacteristic, complemented
331#1393C2×C5911yescharacteristic, complemented
332C2×C59—21no—
333C2×C59—21no—
334C2×C59—41no—
335C2×C59—41no—
336C2×C59—41no—
337C2×C59—41no—
338C2×C59—82no—
339C2×C59—82no—
340C2×C59#202111yescharacteristic
341C2×C59#219711yescharacteristic
342C2×C59#220011yescharacteristic
343C2×H(5,𝔽2)C22×C5911yescharacteristic
344C2×#123—21no—
345C2×#123C22×C5911yescharacteristic
346C2×#120—21no—
347C2×#120C22×C5911yescharacteristic
348#458—22no—
349#458—22no—
350B(4,𝔽2)—22no—
351B(4,𝔽2)—22no—
352C2×#104—21no—
353#417—21no—
354C2×#100C22×C5911yescharacteristic
355C2×#99—21no—
356C2×#99—21no—
357C2×#99C22×C5911yescharacteristic
358C2×#99C22×C5911yescharacteristic
359#397—21no—
360#396—21no—
361#394—21no—
362#390C22×C5911yescharacteristic
363#388—21no—
364#386—21no—
365#379—21no—
366#352—22no—
367AΣL(1,𝔽16)—22no—
368C59#1334411yescharacteristic, direct product factor, Sylow
369C25D4×C5911yescharacteristic
370H(5,𝔽2)—21no—
371H(5,𝔽2)—21no—
372H(5,𝔽2)—21no—
373H(5,𝔽2)—42no—
374H(5,𝔽2)D4×C5912yes—
375C2×#41—21no—
376C2×#41—41no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—41no—
381C22×D4—41no—
382C22×D4C23×C5911yescharacteristic
383C23×C4D4×C5911yescharacteristic, complemented
384C2×M16—21no—
385#123—24no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—21no—
390#123—42no—
391#120—24no—
392#120—21no—
393#120—21no—
394#120—21no—
395#120—21no—
396#120—42no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4D4×C5911yescharacteristic, complemented
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—41no—
412C2×C22⋊C4—41no—
413C2×C22⋊C4D4×C5911yescharacteristic
414C2×C22⋊C4D4×C5911yescharacteristic, complemented
415C2×C42D4×C5911yescharacteristic, complemented
416#104—42no—
417#103—41no—
418#102—41no—
419#100—21no—
420#100—21no—
421#99—24no—
422#99—24no—
423#99—21no—
424#99—21no—
425#99D4×C5912yes—
426#99D4×C5912yescomplemented
427#95—21no—
428#95—21no—
429#95—21no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—41no—
436C24—41no—
437C24—42no—
438C24—81no—
439C24C2×D4×C5911yescharacteristic
440C24C2×D4×C5911yescharacteristic
441#41—22no—
442#41—22no—
443#41—42no—
444#41—41no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—41no—
449C2×Q8—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—81no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4C2×D4×C5911yescharacteristic
476C2×D4C2×D4×C5911yescharacteristic
477C2×D4C2×D4×C5911yescharacteristic
478C2×D4#41×C5911yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×D4×C5911yescharacteristic
494M16—41no—
495M16—41no—
496C2×C8—41no—
497C4⋊C4—41no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—81no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C42—41no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—81no—
547C23—82no—
548C23—81no—
549C23—82no—
550C23—84no—
551C23—81no—
552C23—81no—
553C23—81no—
554C23—81no—
555C23—81no—
556C23#120×C5911yescharacteristic
557C23#120×C5911yescharacteristic
558C23#120×C5911yescharacteristic
559C23#121×C5911yescharacteristic
560C23#121×C5911yescharacteristic
561Q8—22no—
562Q8—41no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—82no—
576D4—82no—
577D4—88no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4#121×C5911yescharacteristic
611C2×C4#127×C5911yescharacteristic
612C8—81no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—161no—
646C22—161no—
647C22C59×#39211yescharacteristic
648C22C59×B(4,𝔽2)11yescharacteristic
649C22C59×B(4,𝔽2)11yescharacteristic
650C4—21no—
651C4—21no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—81no—
660C2—21no—
661C2—21no—
662C2—41no—
663C2—41no—
664C2—41no—
665C2—41no—
666C2—82no—
667C2—82no—
668C2C59×#202111yescharacteristic
669C2C59×#219711yescharacteristic
670C2C59×#220011yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C59
prime radicalC2×C59×B(4,𝔽2)C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C22×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C59×#458C2semidirect
3. C59×#2697C2semidirect
4. C2×C59×#352C2semidirect
5. C2×C59×AΣL(1,𝔽16)C2semidirect
6. C59×#1792C2semidirect
7. C59×#1393C2semidirect
47. C23×C4×C59D4semidirect
64. C2×C4⋊C4×C59D4semidirect
78. C2×C22⋊C4×C59D4semidirect
79. C2×C42×C59D4semidirect
90. #99×C59D4semidirect
277. #13344C59direct (N × H)
325. C2×#458C2×C59semidirect
327. #2697C2×C59semidirect
328. C2×#352C2×C59semidirect
329. C2×AΣL(1,𝔽16)C2×C59semidirect
330. #1792C2×C59semidirect
331. #1393C2×C59semidirect
368. C59#13344direct (N × H)
383. C23×C4D4×C59semidirect
400. C2×C4⋊C4D4×C59semidirect
414. C2×C22⋊C4D4×C59semidirect
415. C2×C42D4×C59semidirect
426. #99D4×C59semidirect

Maximal subgroups

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

subgroupindex
C2×C59×#4582
C2×C59×B(4,𝔽2)2
C59×#26972
C2×C59×#3522
C2×C59×AΣL(1,𝔽16)2
C59×#17922
C59×#13932
#1334459