Finite Group Databasebeta

← C53×#13347 #27220859

Subgroups — C53×#13347 #27220859

Order 13568.

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

normal subgroups only

This group has 830 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×C53×#458C211yesmaximal, characteristic, complemented
2C2×C53×#453C211yesmaximal, characteristic
3C53×#2709C211yesmaximal, characteristic, complemented
4C2×C53×#353C211yesmaximal, characteristic, complemented
5C2×C53×#352C211yesmaximal, characteristic, complemented
6C53×#1790C211yesmaximal, characteristic, complemented
7C53×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C53C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C53—21no—
10C2×#127×C53C2211yescharacteristic
11C2×#123×C53—21no—
12C2×#123×C53C2211yescharacteristic
13C53×#458—22no—
14C53×#458—22no—
15C53×#453—22no—
16C53×#453—22no—
17C2×#104×C53—21no—
18C2×#100×C53C2211yescharacteristic
19C2×#99×C53—21no—
20C2×#99×C53—21no—
21C2×#99×C53C2211yescharacteristic
22C2×#99×C53C2211yescharacteristic
23C53×#394—21no—
24C53×#394—21no—
25C53×#394—21no—
26C53×#387—21no—
27C53×#386—21no—
28C53×#386—21no—
29C53×#386—21no—
30C53×#386C2211yescharacteristic
31C53×#353—22no—
32C53×#352—22no—
33H(5,𝔽2)×C53—21no—
34H(5,𝔽2)×C53—21no—
35H(5,𝔽2)×C53—21no—
36H(5,𝔽2)×C53—42no—
37H(5,𝔽2)×C53D412yes—
38C2×#41×C53—21no—
39C2×#41×C53—41no—
40C22×D4×C53—21no—
41C22×D4×C53—21no—
42C22×D4×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53—41no—
45C22×D4×C53D411yescharacteristic
46C22×D4×C53C2311yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49AGL(1,ℤ/8)×C53—24no—
50AGL(1,ℤ/8)×C53—42no—
51C2×SD16×C53—41no—
52C2×D8×C53—41no—
53C2×M16×C53—21no—
54#127×C53—21no—
55#127×C53—21no—
56#127×C53—21no—
57#127×C53—21no—
58#123×C53—24no—
59#123×C53—21no—
60#123×C53—21no—
61#123×C53—21no—
62#123×C53—21no—
63#123×C53—42no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—41no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—41no—
76C2×C22⋊C4×C53—41no—
77C2×C22⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53D411yescharacteristic
79C2×C42×C53D411yescharacteristic
80#104×C53—24no—
81#100×C53—21no—
82#100×C53D412yes—
83#99×C53—24no—
84#99×C53—21no—
85#99×C53—21no—
86#99×C53—21no—
87#99×C53—42no—
88#99×C53D412yes—
89#95×C53—21no—
90#95×C53—21no—
91#95×C53—21no—
92#95×C53—21no—
93#95×C53—41no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53C2×D411yescharacteristic
98C24×C53C2×D411yescharacteristic
99#41×C53—22no—
100#41×C53—22no—
101#41×C53—42no—
102#41×C53—41no—
103#41×C53—42no—
104#41×C53—42no—
105#41×C53—42no—
106#41×C53—41no—
107C2×Q8×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—41no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53—41no—
130C2×D4×C53—41no—
131C2×D4×C53—41no—
132C2×D4×C53—41no—
133C2×D4×C53—41no—
134C2×D4×C53—41no—
135C2×D4×C53—41no—
136C2×D4×C53—81no—
137C2×D4×C53—81no—
138C2×D4×C53C2×D411yescharacteristic
139C2×D4×C53C2×D411yescharacteristic
140C2×D4×C53C2×D411yescharacteristic
141C2×D4×C53#4111yescharacteristic
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—41no—
152C22×C4×C53—41no—
153C22×C4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53—41no—
156C22×C4×C53—41no—
157C22×C4×C53—81no—
158C22×C4×C53C2×D411yescharacteristic
159SD16×C53—44no—
160D8×C53—44no—
161M16×C53—22no—
162M16×C53—41no—
163C2×C8×C53—41no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—44no—
166C4⋊C4×C53—41no—
167C4⋊C4×C53—41no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—41no—
172C22⋊C4×C53—44no—
173C22⋊C4×C53—41no—
174C22⋊C4×C53—41no—
175C22⋊C4×C53—41no—
176C22⋊C4×C53—41no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53—81no—
179C22⋊C4×C53—81no—
180C22⋊C4×C53—81no—
181C22⋊C4×C53—81no—
182C22⋊C4×C53—81no—
183C22⋊C4×C53—82no—
184C42×C53—21no—
185C42×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—81no—
207C23×C53—81no—
208C23×C53—81no—
209C23×C53—81no—
210C23×C53#12011yescharacteristic
211C23×C53#12011yescharacteristic
212C23×C53#12111yescharacteristic
213C23×C53#12111yescharacteristic
214C23×C53#12711yescharacteristic
215Q8×C53—22no—
216Q8×C53—41no—
217D4×C53—22no—
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—42no—
224D4×C53—42no—
225D4×C53—42no—
226D4×C53—42no—
227D4×C53—42no—
228D4×C53—42no—
229D4×C53—42no—
230D4×C53—81no—
231D4×C53—82no—
232D4×C53—82no—
233D4×C53—82no—
234D4×C53—82no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—81no—
263C2×C4×C53—81no—
264C2×C4×C53—81no—
265C2×C4×C53—81no—
266C2×C4×C53—81no—
267C2×C4×C53—81no—
268C2×C4×C53—81no—
269C2×C4×C53#12011yescharacteristic
270C2×C4×C53#12111yescharacteristic
271C8×C53—42no—
272#13347C5311yesmaximal, characteristic, direct product factor, Sylow
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—42no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—82no—
295C22×C53—82no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—161no—
302C22×C53—161no—
303C22×C53#39211yescharacteristic
304C22×C53B(4,𝔽2)11yescharacteristic
305C22×C53B(4,𝔽2)11yescharacteristic
306C4×C53—21no—
307C4×C53—21no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53—82no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53—161no—
316C2×#458C2×C5311yescharacteristic, complemented
317C2×#453C2×C5311yescharacteristic
318#2709C2×C5311yescharacteristic, complemented
319C2×#353C2×C5311yescharacteristic, complemented
320C2×#352C2×C5311yescharacteristic, complemented
321#1790C2×C5311yescharacteristic, complemented
322#1390C2×C5311yescharacteristic, complemented
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—41no—
326C2×C53—41no—
327C2×C53—41no—
328C2×C53—41no—
329C2×C53—82no—
330C2×C53—161no—
331C2×C53#202111yescharacteristic
332C2×C53#219811yescharacteristic
333C2×C53#220011yescharacteristic
334C2×H(5,𝔽2)C22×C5311yescharacteristic
335C2×AGL(1,ℤ/8)—21no—
336C2×#127C22×C5311yescharacteristic
337C2×#123—21no—
338C2×#123C22×C5311yescharacteristic
339#458—22no—
340#458—22no—
341#453—22no—
342#453—22no—
343C2×#104—21no—
344C2×#100C22×C5311yescharacteristic
345C2×#99—21no—
346C2×#99—21no—
347C2×#99C22×C5311yescharacteristic
348C2×#99C22×C5311yescharacteristic
349#394—21no—
350#394—21no—
351#394—21no—
352#387—21no—
353#386—21no—
354#386—21no—
355#386—21no—
356#386C22×C5311yescharacteristic
357#353—22no—
358#352—22no—
359C53#1334711yescharacteristic, direct product factor, Sylow
360H(5,𝔽2)—21no—
361H(5,𝔽2)—21no—
362H(5,𝔽2)—21no—
363H(5,𝔽2)—42no—
364H(5,𝔽2)D4×C5312yes—
365C2×#41—21no—
366C2×#41—41no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—41no—
372C22×D4D4×C5311yescharacteristic
373C22×D4C23×C5311yescharacteristic
374C23×C4D4×C5311yescharacteristic
375C23×C4D4×C5311yescharacteristic
376AGL(1,ℤ/8)—24no—
377AGL(1,ℤ/8)—42no—
378C2×SD16—41no—
379C2×D8—41no—
380C2×M16—21no—
381#127—21no—
382#127—21no—
383#127—21no—
384#127—21no—
385#123—24no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—21no—
390#123—42no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—41no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—41no—
404C2×C22⋊C4D4×C5311yescharacteristic
405C2×C22⋊C4D4×C5311yescharacteristic
406C2×C42D4×C5311yescharacteristic
407#104—24no—
408#100—21no—
409#100D4×C5312yes—
410#99—24no—
411#99—21no—
412#99—21no—
413#99—21no—
414#99—42no—
415#99D4×C5312yes—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—41no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24C2×D4×C5311yescharacteristic
425C24C2×D4×C5311yescharacteristic
426#41—22no—
427#41—22no—
428#41—42no—
429#41—41no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—41no—
434C2×Q8—21no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4C2×D4×C5311yescharacteristic
466C2×D4C2×D4×C5311yescharacteristic
467C2×D4C2×D4×C5311yescharacteristic
468C2×D4#41×C5311yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—81no—
485C22×C4C2×D4×C5311yescharacteristic
486SD16—44no—
487D8—44no—
488M16—22no—
489M16—41no—
490C2×C8—41no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—44no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C42—21no—
512C42—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23#120×C5311yescharacteristic
538C23#120×C5311yescharacteristic
539C23#121×C5311yescharacteristic
540C23#121×C5311yescharacteristic
541C23#127×C5311yescharacteristic
542Q8—22no—
543Q8—41no—
544D4—22no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—81no—
558D4—82no—
559D4—82no—
560D4—82no—
561D4—82no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4#120×C5311yescharacteristic
597C2×C4#121×C5311yescharacteristic
598C8—42no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—161no—
628C22—161no—
629C22C53×#39211yescharacteristic
630C22C53×B(4,𝔽2)11yescharacteristic
631C22C53×B(4,𝔽2)11yescharacteristic
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—161no—
642C2—21no—
643C2—21no—
644C2—41no—
645C2—41no—
646C2—41no—
647C2—41no—
648C2—82no—
649C2—161no—
650C2C53×#202111yescharacteristic
651C2C53×#219811yescharacteristic
652C2C53×#220011yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C53
prime radicalC2×C53×#453C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#458C2semidirect
3. C53×#2709C2semidirect
4. C2×C53×#353C2semidirect
5. C2×C53×#352C2semidirect
6. C53×#1790C2semidirect
7. C53×#1390C2semidirect
272. #13347C53direct (N × H)
316. C2×#458C2×C53semidirect
318. #2709C2×C53semidirect
319. C2×#353C2×C53semidirect
320. C2×#352C2×C53semidirect
321. #1790C2×C53semidirect
322. #1390C2×C53semidirect
359. C53#13347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#4582
C2×C53×#4532
C53×#27092
C2×C53×#3532
C2×C53×#3522
C53×#17902
C53×#13902
#1334753