Finite Group Databasebeta

← C53×#13344 #27220856

Subgroups — C53×#13344 #27220856

Order 13568.

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×C53×#458C211yesmaximal, characteristic, complemented
2C2×C53×B(4,𝔽2)C211yesmaximal, characteristic
3C53×#2697C211yesmaximal, characteristic, complemented
4C2×C53×#352C211yesmaximal, characteristic, complemented
5C2×C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6C53×#1792C211yesmaximal, characteristic, complemented
7C53×#1393C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C53C2211yescharacteristic
9C2×#123×C53—21no—
10C2×#123×C53C2211yescharacteristic
11C2×#120×C53—21no—
12C2×#120×C53C2211yescharacteristic
13C53×#458—22no—
14C53×#458—22no—
15C53×B(4,𝔽2)—22no—
16C53×B(4,𝔽2)—22no—
17C2×#104×C53—21no—
18C53×#417—21no—
19C2×#100×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C2×#99×C53—21no—
22C2×#99×C53C2211yescharacteristic
23C2×#99×C53C2211yescharacteristic
24C53×#397—21no—
25C53×#396—21no—
26C53×#394—21no—
27C53×#390C2211yescharacteristic
28C53×#388—21no—
29C53×#386—21no—
30C53×#379—21no—
31C53×#352—22no—
32C53×AΣL(1,𝔽16)—22no—
33C25×C53D411yescharacteristic
34H(5,𝔽2)×C53—21no—
35H(5,𝔽2)×C53—21no—
36H(5,𝔽2)×C53—21no—
37H(5,𝔽2)×C53—42no—
38H(5,𝔽2)×C53D412yes—
39C2×#41×C53—21no—
40C2×#41×C53—41no—
41C22×D4×C53—21no—
42C22×D4×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53—41no—
45C22×D4×C53—41no—
46C22×D4×C53C2311yescharacteristic
47C23×C4×C53D411yescharacteristic, complemented
48C2×M16×C53—21no—
49#123×C53—24no—
50#123×C53—21no—
51#123×C53—21no—
52#123×C53—21no—
53#123×C53—21no—
54#123×C53—42no—
55#120×C53—24no—
56#120×C53—21no—
57#120×C53—21no—
58#120×C53—21no—
59#120×C53—21no—
60#120×C53—42no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—41no—
64C2×C4⋊C4×C53D411yescharacteristic, complemented
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
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, complemented
79C2×C42×C53D411yescharacteristic, complemented
80#104×C53—42no—
81#103×C53—41no—
82#102×C53—41no—
83#100×C53—21no—
84#100×C53—21no—
85#99×C53—24no—
86#99×C53—24no—
87#99×C53—21no—
88#99×C53—21no—
89#99×C53D412yes—
90#99×C53D412yescomplemented
91#95×C53—21no—
92#95×C53—21no—
93#95×C53—21no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53—41no—
100C24×C53—41no—
101C24×C53—42no—
102C24×C53—81no—
103C24×C53C2×D411yescharacteristic
104C24×C53C2×D411yescharacteristic
105#41×C53—22no—
106#41×C53—22no—
107#41×C53—42no—
108#41×C53—41no—
109#41×C53—42no—
110#41×C53—42no—
111#41×C53—42no—
112#41×C53—41no—
113C2×Q8×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
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—44no—
133C2×D4×C53—42no—
134C2×D4×C53—41no—
135C2×D4×C53—41no—
136C2×D4×C53—81no—
137C2×D4×C53—82no—
138C2×D4×C53—82no—
139C2×D4×C53C2×D411yescharacteristic
140C2×D4×C53C2×D411yescharacteristic
141C2×D4×C53C2×D411yescharacteristic
142C2×D4×C53#4111yescharacteristic
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—21no—
152C22×C4×C53—41no—
153C22×C4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53—41no—
156C22×C4×C53—41no—
157C22×C4×C53C2×D411yescharacteristic
158M16×C53—41no—
159M16×C53—41no—
160C2×C8×C53—41no—
161C4⋊C4×C53—44no—
162C4⋊C4×C53—41no—
163C4⋊C4×C53—41no—
164C4⋊C4×C53—41no—
165C4⋊C4×C53—81no—
166C22⋊C4×C53—21no—
167C22⋊C4×C53—21no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—44no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—41no—
175C22⋊C4×C53—41no—
176C22⋊C4×C53—44no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53—41no—
179C22⋊C4×C53—41no—
180C22⋊C4×C53—41no—
181C22⋊C4×C53—81no—
182C22⋊C4×C53—81no—
183C22⋊C4×C53—81no—
184C22⋊C4×C53—81no—
185C22⋊C4×C53—81no—
186C42×C53—41no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—81no—
211C23×C53—82no—
212C23×C53—81no—
213C23×C53—82no—
214C23×C53—84no—
215C23×C53—81no—
216C23×C53—81no—
217C23×C53—81no—
218C23×C53—81no—
219C23×C53—81no—
220C23×C53#12011yescharacteristic
221C23×C53#12011yescharacteristic
222C23×C53#12011yescharacteristic
223C23×C53#12111yescharacteristic
224C23×C53#12111yescharacteristic
225Q8×C53—22no—
226Q8×C53—41no—
227D4×C53—22no—
228D4×C53—41no—
229D4×C53—41no—
230D4×C53—41no—
231D4×C53—41no—
232D4×C53—41no—
233D4×C53—42no—
234D4×C53—42no—
235D4×C53—42no—
236D4×C53—42no—
237D4×C53—42no—
238D4×C53—42no—
239D4×C53—82no—
240D4×C53—82no—
241D4×C53—88no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—81no—
270C2×C4×C53—81no—
271C2×C4×C53—81no—
272C2×C4×C53—81no—
273C2×C4×C53—81no—
274C2×C4×C53#12111yescharacteristic
275C2×C4×C53#12711yescharacteristic
276C8×C53—81no—
277#13344C5311yesmaximal, characteristic, direct product factor, Sylow
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—44no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—82no—
302C22×C53—82no—
303C22×C53—82no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53—81no—
310C22×C53—161no—
311C22×C53—161no—
312C22×C53#39211yescharacteristic
313C22×C53B(4,𝔽2)11yescharacteristic
314C22×C53B(4,𝔽2)11yescharacteristic
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—42no—
318C4×C53—81no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53—82no—
322C4×C53—82no—
323C4×C53—81no—
324C4×C53—81no—
325C2×#458C2×C5311yescharacteristic, complemented
326C2×B(4,𝔽2)C2×C5311yescharacteristic
327#2697C2×C5311yescharacteristic, complemented
328C2×#352C2×C5311yescharacteristic, complemented
329C2×AΣL(1,𝔽16)C2×C5311yescharacteristic, complemented
330#1792C2×C5311yescharacteristic, complemented
331#1393C2×C5311yescharacteristic, complemented
332C2×C53—21no—
333C2×C53—21no—
334C2×C53—41no—
335C2×C53—41no—
336C2×C53—41no—
337C2×C53—41no—
338C2×C53—82no—
339C2×C53—82no—
340C2×C53#202111yescharacteristic
341C2×C53#219711yescharacteristic
342C2×C53#220011yescharacteristic
343C2×H(5,𝔽2)C22×C5311yescharacteristic
344C2×#123—21no—
345C2×#123C22×C5311yescharacteristic
346C2×#120—21no—
347C2×#120C22×C5311yescharacteristic
348#458—22no—
349#458—22no—
350B(4,𝔽2)—22no—
351B(4,𝔽2)—22no—
352C2×#104—21no—
353#417—21no—
354C2×#100C22×C5311yescharacteristic
355C2×#99—21no—
356C2×#99—21no—
357C2×#99C22×C5311yescharacteristic
358C2×#99C22×C5311yescharacteristic
359#397—21no—
360#396—21no—
361#394—21no—
362#390C22×C5311yescharacteristic
363#388—21no—
364#386—21no—
365#379—21no—
366#352—22no—
367AΣL(1,𝔽16)—22no—
368C53#1334411yescharacteristic, direct product factor, Sylow
369C25D4×C5311yescharacteristic
370H(5,𝔽2)—21no—
371H(5,𝔽2)—21no—
372H(5,𝔽2)—21no—
373H(5,𝔽2)—42no—
374H(5,𝔽2)D4×C5312yes—
375C2×#41—21no—
376C2×#41—41no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—41no—
381C22×D4—41no—
382C22×D4C23×C5311yescharacteristic
383C23×C4D4×C5311yescharacteristic, 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×C5311yescharacteristic, 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×C5311yescharacteristic
414C2×C22⋊C4D4×C5311yescharacteristic, complemented
415C2×C42D4×C5311yescharacteristic, 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×C5312yes—
426#99D4×C5312yescomplemented
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×C5311yescharacteristic
440C24C2×D4×C5311yescharacteristic
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×C5311yescharacteristic
476C2×D4C2×D4×C5311yescharacteristic
477C2×D4C2×D4×C5311yescharacteristic
478C2×D4#41×C5311yescharacteristic
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×C5311yescharacteristic
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×C5311yescharacteristic
557C23#120×C5311yescharacteristic
558C23#120×C5311yescharacteristic
559C23#121×C5311yescharacteristic
560C23#121×C5311yescharacteristic
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×C5311yescharacteristic
611C2×C4#127×C5311yescharacteristic
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—
647C22C53×#39211yescharacteristic
648C22C53×B(4,𝔽2)11yescharacteristic
649C22C53×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—
668C2C53×#202111yescharacteristic
669C2C53×#219711yescharacteristic
670C2C53×#220011yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C53
prime radicalC2×C53×B(4,𝔽2)C2
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×#2697C2semidirect
4. C2×C53×#352C2semidirect
5. C2×C53×AΣL(1,𝔽16)C2semidirect
6. C53×#1792C2semidirect
7. C53×#1393C2semidirect
47. C23×C4×C53D4semidirect
64. C2×C4⋊C4×C53D4semidirect
78. C2×C22⋊C4×C53D4semidirect
79. C2×C42×C53D4semidirect
90. #99×C53D4semidirect
277. #13344C53direct (N × H)
325. C2×#458C2×C53semidirect
327. #2697C2×C53semidirect
328. C2×#352C2×C53semidirect
329. C2×AΣL(1,𝔽16)C2×C53semidirect
330. #1792C2×C53semidirect
331. #1393C2×C53semidirect
368. C53#13344direct (N × H)
383. C23×C4D4×C53semidirect
400. C2×C4⋊C4D4×C53semidirect
414. C2×C22⋊C4D4×C53semidirect
415. C2×C42D4×C53semidirect
426. #99D4×C53semidirect

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×B(4,𝔽2)2
C53×#26972
C2×C53×#3522
C2×C53×AΣL(1,𝔽16)2
C53×#17922
C53×#13932
#1334453