Finite Group Databasebeta

← C53×#13348 #27220860

Subgroups — C53×#13348 #27220860

Order 13568.

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

normal subgroups only

This group has 900 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×B(4,𝔽2)C211yesmaximal, characteristic
2C2×C53×#455C211yesmaximal, characteristic, complemented
3C53×#2684C211yesmaximal, characteristic, complemented
4C2×C53×#354C211yesmaximal, characteristic, complemented
5C2×C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6C53×#1790C211yesmaximal, characteristic, complemented
7C53×#1390C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C53C2211yescharacteristic
9C2×#137×C53—21no—
10C2×#124×C53C2211yescharacteristic
11C2×#120×C53—21no—
12C2×#120×C53C2211yescharacteristic
13C53×B(4,𝔽2)—22no—
14C53×B(4,𝔽2)—22no—
15C53×#455—22no—
16C53×#455—22no—
17C2×#104×C53—21no—
18C2×#100×C53C2211yescharacteristic
19C2×#99×C53—21no—
20C2×#99×C53—21no—
21C2×#99×C53C2211yescharacteristic
22C2×#99×C53C2211yescharacteristic
23C53×#397—21no—
24C53×#394—21no—
25C53×#393—21no—
26C53×#388—21no—
27C53×#386—21no—
28C53×#386—21no—
29C53×#386C2211yescharacteristic
30C53×#379—21no—
31C53×#354—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×Q8×C53D411yescharacteristic, complemented
42C22×D4×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53—21no—
45C22×D4×C53—41no—
46C22×D4×C53—41no—
47C22×D4×C53C2311yescharacteristic
48C23×C4×C53D411yescharacteristic, complemented
49#137×C53—24no—
50#137×C53—42no—
51C2×Q16×C53—41no—
52C2×SD16×C53—41no—
53C2×M16×C53—21no—
54#124×C53—21no—
55#124×C53—21no—
56#124×C53—21no—
57#124×C53—21no—
58#120×C53—24no—
59#120×C53—21no—
60#120×C53—21no—
61#120×C53—21no—
62#120×C53—21no—
63#120×C53—42no—
64C2×C4⋊C4×C53—21no—
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—24no—
81#100×C53—21no—
82#100×C53D412yescomplemented
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×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×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—41no—
117C2×Q8×C53—41no—
118C2×D4×C53—21no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—42no—
126C2×D4×C53—42no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—41no—
130C2×D4×C53—41no—
131C2×D4×C53—42no—
132C2×D4×C53—42no—
133C2×D4×C53—41no—
134C2×D4×C53—41no—
135C2×D4×C53—41no—
136C2×D4×C53—44no—
137C2×D4×C53—42no—
138C2×D4×C53—41no—
139C2×D4×C53—41no—
140C2×D4×C53—81no—
141C2×D4×C53—82no—
142C2×D4×C53—82no—
143C2×D4×C53C2×D411yescharacteristic
144C2×D4×C53C2×D411yescharacteristic
145C2×D4×C53C2×D411yescharacteristic
146C2×D4×C53#4111yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—41no—
155C22×C4×C53—41no—
156C22×C4×C53—41no—
157C22×C4×C53—41no—
158C22×C4×C53—81no—
159C22×C4×C53C2×D411yescharacteristic
160Q16×C53—44no—
161SD16×C53—44no—
162M16×C53—22no—
163M16×C53—41no—
164C2×C8×C53—41no—
165C4⋊C4×C53—81no—
166C22⋊C4×C53—21no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—44no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—41no—
172C22⋊C4×C53—41no—
173C22⋊C4×C53—41no—
174C22⋊C4×C53—41no—
175C22⋊C4×C53—41no—
176C22⋊C4×C53—41no—
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—82no—
186C42×C53—21no—
187C42×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—81no—
212C23×C53—82no—
213C23×C53—81no—
214C23×C53—82no—
215C23×C53—84no—
216C23×C53—81no—
217C23×C53—81no—
218C23×C53—81no—
219C23×C53—81no—
220C23×C53—81no—
221C23×C53#12011yescharacteristic
222C23×C53#12011yescharacteristic
223C23×C53#12111yescharacteristic
224C23×C53#12111yescharacteristic
225C23×C53#12711yescharacteristic
226Q8×C53—22no—
227Q8×C53—41no—
228Q8×C53—42no—
229Q8×C53—81no—
230D4×C53—22no—
231D4×C53—41no—
232D4×C53—41no—
233D4×C53—41no—
234D4×C53—41no—
235D4×C53—41no—
236D4×C53—42no—
237D4×C53—42no—
238D4×C53—42no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—42no—
242D4×C53—82no—
243D4×C53—82no—
244D4×C53—88no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
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—41no—
261C2×C4×C53—41no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—81no—
272C2×C4×C53—81no—
273C2×C4×C53—81no—
274C2×C4×C53—81no—
275C2×C4×C53—81no—
276C2×C4×C53—81no—
277C2×C4×C53#12011yescharacteristic
278C2×C4×C53#12111yescharacteristic
279C8×C53—42no—
280#13348C5311yesmaximal, characteristic, direct product factor, Sylow
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—44no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53—82no—
307C22×C53—82no—
308C22×C53—82no—
309C22×C53—81no—
310C22×C53—81no—
311C22×C53—81no—
312C22×C53—81no—
313C22×C53—161no—
314C22×C53—161no—
315C22×C53#39211yescharacteristic
316C22×C53B(4,𝔽2)11yescharacteristic
317C22×C53B(4,𝔽2)11yescharacteristic
318C4×C53—21no—
319C4×C53—21no—
320C4×C53—42no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53—81no—
324C4×C53—82no—
325C4×C53—81no—
326C4×C53—81no—
327C4×C53—161no—
328C2×B(4,𝔽2)C2×C5311yescharacteristic
329C2×#455C2×C5311yescharacteristic, complemented
330#2684C2×C5311yescharacteristic, complemented
331C2×#354C2×C5311yescharacteristic, complemented
332C2×AΣL(1,𝔽16)C2×C5311yescharacteristic, complemented
333#1790C2×C5311yescharacteristic, complemented
334#1390C2×C5311yescharacteristic, complemented
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—41no—
338C2×C53—41no—
339C2×C53—41no—
340C2×C53—41no—
341C2×C53—82no—
342C2×C53—82no—
343C2×C53#202111yescharacteristic
344C2×C53#219711yescharacteristic
345C2×C53#219911yescharacteristic
346C2×H(5,𝔽2)C22×C5311yescharacteristic
347C2×#137—21no—
348C2×#124C22×C5311yescharacteristic
349C2×#120—21no—
350C2×#120C22×C5311yescharacteristic
351B(4,𝔽2)—22no—
352B(4,𝔽2)—22no—
353#455—22no—
354#455—22no—
355C2×#104—21no—
356C2×#100C22×C5311yescharacteristic
357C2×#99—21no—
358C2×#99—21no—
359C2×#99C22×C5311yescharacteristic
360C2×#99C22×C5311yescharacteristic
361#397—21no—
362#394—21no—
363#393—21no—
364#388—21no—
365#386—21no—
366#386—21no—
367#386C22×C5311yescharacteristic
368#379—21no—
369#354—22no—
370AΣL(1,𝔽16)—22no—
371C53#1334811yescharacteristic, direct product factor, Sylow
372C25D4×C5311yescharacteristic
373H(5,𝔽2)—21no—
374H(5,𝔽2)—21no—
375H(5,𝔽2)—21no—
376H(5,𝔽2)—42no—
377H(5,𝔽2)D4×C5312yes—
378C2×#41—21no—
379C2×#41—41no—
380C22×Q8D4×C5311yescharacteristic, complemented
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—21no—
384C22×D4—41no—
385C22×D4—41no—
386C22×D4C23×C5311yescharacteristic
387C23×C4D4×C5311yescharacteristic, complemented
388#137—24no—
389#137—42no—
390C2×Q16—41no—
391C2×SD16—41no—
392C2×M16—21no—
393#124—21no—
394#124—21no—
395#124—21no—
396#124—21no—
397#120—24no—
398#120—21no—
399#120—21no—
400#120—21no—
401#120—21no—
402#120—42no—
403C2×C4⋊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—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—41no—
415C2×C22⋊C4—41no—
416C2×C22⋊C4D4×C5311yescharacteristic
417C2×C22⋊C4D4×C5311yescharacteristic, complemented
418C2×C42D4×C5311yescharacteristic, complemented
419#104—24no—
420#100—21no—
421#100D4×C5312yescomplemented
422#99—24no—
423#99—21no—
424#99—21no—
425#99—21no—
426#99—42no—
427#99D4×C5312yes—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—41no—
433C24—21no—
434C24—21no—
435C24—21no—
436C24—21no—
437C24—21no—
438C24—41no—
439C24—41no—
440C24—42no—
441C24—81no—
442C24C2×D4×C5311yescharacteristic
443C24C2×D4×C5311yescharacteristic
444#41—22no—
445#41—22no—
446#41—42no—
447#41—41no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—41no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—41no—
456C2×Q8—41no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—44no—
476C2×D4—42no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—81no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4C2×D4×C5311yescharacteristic
483C2×D4C2×D4×C5311yescharacteristic
484C2×D4C2×D4×C5311yescharacteristic
485C2×D4#41×C5311yescharacteristic
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—81no—
498C22×C4C2×D4×C5311yescharacteristic
499Q16—44no—
500SD16—44no—
501M16—22no—
502M16—41no—
503C2×C8—41no—
504C4⋊C4—81no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C42—21no—
526C42—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—81no—
551C23—82no—
552C23—81no—
553C23—82no—
554C23—84no—
555C23—81no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23#120×C5311yescharacteristic
561C23#120×C5311yescharacteristic
562C23#121×C5311yescharacteristic
563C23#121×C5311yescharacteristic
564C23#127×C5311yescharacteristic
565Q8—22no—
566Q8—41no—
567Q8—42no—
568Q8—81no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—82no—
582D4—82no—
583D4—88no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
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—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4#120×C5311yescharacteristic
617C2×C4#121×C5311yescharacteristic
618C8—42no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—161no—
652C22—161no—
653C22C53×#39211yescharacteristic
654C22C53×B(4,𝔽2)11yescharacteristic
655C22C53×B(4,𝔽2)11yescharacteristic
656C4—21no—
657C4—21no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4—161no—
666C2—21no—
667C2—21no—
668C2—41no—
669C2—41no—
670C2—41no—
671C2—41no—
672C2—82no—
673C2—82no—
674C2C53×#202111yescharacteristic
675C2C53×#219711yescharacteristic
676C2C53×#219911yescharacteristic
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
2. C2×C53×#455C2semidirect
3. C53×#2684C2semidirect
4. C2×C53×#354C2semidirect
5. C2×C53×AΣL(1,𝔽16)C2semidirect
6. C53×#1790C2semidirect
7. C53×#1390C2semidirect
41. C22×Q8×C53D4semidirect
48. C23×C4×C53D4semidirect
78. C2×C22⋊C4×C53D4semidirect
79. C2×C42×C53D4semidirect
82. #100×C53D4semidirect
280. #13348C53direct (N × H)
329. C2×#455C2×C53semidirect
330. #2684C2×C53semidirect
331. C2×#354C2×C53semidirect
332. C2×AΣL(1,𝔽16)C2×C53semidirect
333. #1790C2×C53semidirect
334. #1390C2×C53semidirect
371. C53#13348direct (N × H)
380. C22×Q8D4×C53semidirect
387. C23×C4D4×C53semidirect
417. C2×C22⋊C4D4×C53semidirect
418. C2×C42D4×C53semidirect
421. #100D4×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×B(4,𝔽2)2
C2×C53×#4552
C53×#26842
C2×C53×#3542
C2×C53×AΣL(1,𝔽16)2
C53×#17902
C53×#13902
#1334853