Finite Group Databasebeta

← C53×#9079 #27216591

Subgroups — C53×#9079 #27216591

Order 13568.

1136 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C8×C53C211yesmaximal, characteristic, complemented
2C2×C53×#386C211yesmaximal, characteristic
3C53×#1779C211yesmaximal, characteristic, complemented
4C2×C53×#323C212yesmaximal, complemented
5C53×#1280C212yesmaximal, complemented
6C23×D4×C53C411yescharacteristic
7C24×C4×C53C2211yescharacteristic
8C23×C8×C53—21no—
9C22×C22⋊C4×C53C411yescharacteristic
10C22×C22⋊C4×C53C411yescharacteristic
11C22×C22⋊C4×C53C2211yescharacteristic
12C2×C22⋊C8×C53—21no—
13C2×C22⋊C8×C53—24no—
14C2×C22⋊C8×C53—21no—
15C2×C22⋊C8×C53—21no—
16C2×C22⋊C8×C53C2212yes—
17C2×C22⋊C8×C53C2212yes—
18C53×#386—21no—
19C53×#386—21no—
20C53×#386—21no—
21C53×#386—21no—
22C2×#95×C53C411yescharacteristic
23C2×#95×C53C2211yescharacteristic
24C53×#323—24no—
25C25×C53C2×C411yescharacteristic
26C25×C53C2×C411yescharacteristic
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53—22no—
30C22×D4×C53—42no—
31C22×D4×C53—42no—
32C22×D4×C53C814yescomplemented
33C23×C4×C53—22no—
34C23×C4×C53—22no—
35C23×C4×C53—42no—
36C23×C4×C53—42no—
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53C2×C411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46C22×C8×C53—24no—
47C22×C8×C53—22no—
48C22×C8×C53—22no—
49C22×C8×C53—42no—
50C22×C8×C53—42no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—42no—
60C2×C22⋊C4×C53—42no—
61C22⋊C8×C53—24no—
62C22⋊C8×C53—22no—
63C22⋊C8×C53—22no—
64C22⋊C8×C53—22no—
65C22⋊C8×C53—42no—
66C22⋊C8×C53—42no—
67C22⋊C8×C53D414yescomplemented
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53—22no—
79C24×C53—22no—
80C24×C53—41no—
81C24×C53—41no—
82C24×C53—41no—
83C24×C53—41no—
84C24×C53—41no—
85C24×C53—41no—
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22⋊C411yescharacteristic
88C24×C53C2×C812yes—
89C24×C53C2×C812yescomplemented
90C24×C53M1612yes—
91C24×C53C22×C411yescharacteristic
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—48no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—42no—
98C2×D4×C53—42no—
99C2×D4×C53—48no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—48no—
126C22×C4×C53—48no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C22⋊C411yescharacteristic
134C22×C4×C53C22⋊C411yescharacteristic
135C22×C4×C53C2×C812yescomplemented
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D412yes—
139C22×C4×C53#4112yes—
140C2×C8×C53—24no—
141C2×C8×C53—48no—
142C2×C8×C53—42no—
143C2×C8×C53—42no—
144C2×C8×C53—42no—
145C2×C8×C53—42no—
146C2×C8×C53—42no—
147C22⋊C4×C53—42no—
148C22⋊C4×C53—42no—
149C22⋊C4×C53—42no—
150C22⋊C4×C53—42no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—42no—
153C22⋊C4×C53—41no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—24no—
163C23×C53—24no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—24no—
171C23×C53—24no—
172C23×C53—22no—
173C23×C53—24no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—44no—
177C23×C53—44no—
178C23×C53—44no—
179C23×C53—44no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—42no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53—82no—
194C23×C53—82no—
195C23×C53C22⋊C812yes—
196C23×C53C22⋊C812yes—
197C23×C53#9912yes—
198C23×C53#10012yes—
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53C4×D412yes—
201C23×C53C22×C811yescharacteristic
202C23×C53C2×M1611yescharacteristic
203C23×C53#13112yes—
204D4×C53—48no—
205D4×C53—48no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—44no—
221C2×C4×C53—44no—
222C2×C4×C53—44no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—82no—
236C2×C4×C53—82no—
237C2×C4×C53#9912yes—
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53#12112yes—
240C2×C4×C53#12311yescharacteristic
241C8×C53—44no—
242C8×C53—82no—
243#9079C5311yesmaximal, characteristic, direct product factor, Sylow
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—48no—
262C22×C53—48no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—84no—
271C22×C53—84no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53#32314yescomplemented
277C22×C53#38611yescharacteristic
278C22×C53C2×C22⋊C811yescharacteristic
279C22×C53#40811yescharacteristic
280C22×C53C2×#9911yescharacteristic
281C22×C53C2×#9911yescharacteristic
282C22×C53C2×#10011yescharacteristic
283C22×C53#41311yescharacteristic
284C22×C53C8×D412yes—
285C22×C53#43512yes—
286C4×C53—21no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—21no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53—81no—
294C22×C22⋊C8C2×C5311yescharacteristic, complemented
295C2×#386C2×C5311yescharacteristic
296#1779C2×C5311yescharacteristic, complemented
297C2×#323C2×C5312yescomplemented
298#1280C2×C5312yescomplemented
299C2×C53—22no—
300C2×C53—22no—
301C2×C53—24no—
302C2×C53—42no—
303C2×C53—82no—
304C2×C53C2×#32311yescharacteristic
305C2×C53#145911yescharacteristic
306C2×C53#146811yescharacteristic
307C2×C53#147111yescharacteristic
308C2×C53#186911yescharacteristic
309C2×C53#189811yescharacteristic
310C2×C53#190011yescharacteristic
311C23×D4C4×C5311yescharacteristic
312C24×C4C22×C5311yescharacteristic
313C23×C8—21no—
314C22×C22⋊C4C4×C5311yescharacteristic
315C22×C22⋊C4C4×C5311yescharacteristic
316C22×C22⋊C4C22×C5311yescharacteristic
317C2×C22⋊C8—21no—
318C2×C22⋊C8—24no—
319C2×C22⋊C8—21no—
320C2×C22⋊C8—21no—
321C2×C22⋊C8C22×C5312yes—
322C2×C22⋊C8C22×C5312yes—
323#386—21no—
324#386—21no—
325#386—21no—
326#386—21no—
327C2×#95C4×C5311yescharacteristic
328C2×#95C22×C5311yescharacteristic
329#323—24no—
330C53#907911yescharacteristic, direct product factor, Sylow
331C25C2×C4×C5311yescharacteristic
332C25C2×C4×C5311yescharacteristic
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—42no—
337C22×D4—42no—
338C22×D4C8×C5314yescomplemented
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4C2×C4×C5311yescharacteristic
344C23×C4C2×C4×C5311yescharacteristic
345C23×C4C2×C4×C5311yescharacteristic
346C23×C4C2×C4×C5311yescharacteristic
347C23×C4D4×C5311yescharacteristic
348C23×C4D4×C5311yescharacteristic
349C23×C4D4×C5311yescharacteristic
350C23×C4D4×C5311yescharacteristic
351C23×C4C23×C5311yescharacteristic
352C22×C8—24no—
353C22×C8—22no—
354C22×C8—22no—
355C22×C8—42no—
356C22×C8—42no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—42no—
372C22⋊C8—42no—
373C22⋊C8D4×C5314yescomplemented
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24C22⋊C4×C5311yescharacteristic
393C24C22⋊C4×C5311yescharacteristic
394C24C2×C8×C5312yes—
395C24C2×C8×C5312yescomplemented
396C24M16×C5312yes—
397C24C22×C4×C5311yescharacteristic
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C22⋊C4×C5311yescharacteristic
440C22×C4C22⋊C4×C5311yescharacteristic
441C22×C4C2×C8×C5312yescomplemented
442C22×C4C2×D4×C5311yescharacteristic
443C22×C4C2×D4×C5311yescharacteristic
444C22×C4C2×D4×C5312yes—
445C22×C4#41×C5312yes—
446C2×C8—24no—
447C2×C8—48no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—24no—
469C23—24no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23C22⋊C8×C5312yes—
502C23C22⋊C8×C5312yes—
503C23#99×C5312yes—
504C23#100×C5312yes—
505C23C2×C22⋊C4×C5311yescharacteristic
506C23C4×D4×C5312yes—
507C23C22×C8×C5311yescharacteristic
508C23C2×M16×C5311yescharacteristic
509C23#131×C5312yes—
510D4—48no—
511D4—48no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#99×C5312yes—
544C2×C4#120×C5311yescharacteristic
545C2×C4#121×C5312yes—
546C2×C4#123×C5311yescharacteristic
547C8—44no—
548C8—82no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—48no—
567C22—48no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—84no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C53×#32314yescomplemented
582C22C53×#38611yescharacteristic
583C22C2×C22⋊C8×C5311yescharacteristic
584C22C53×#40811yescharacteristic
585C22C2×#99×C5311yescharacteristic
586C22C2×#99×C5311yescharacteristic
587C22C2×#100×C5311yescharacteristic
588C22C53×#41311yescharacteristic
589C22C8×D4×C5312yes—
590C22C53×#43512yes—
591C4—21no—
592C4—22no—
593C4—22no—
594C4—21no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2—82no—
604C2C2×C53×#32311yescharacteristic
605C2C53×#145911yescharacteristic
606C2C53×#146811yescharacteristic
607C2C53×#147111yescharacteristic
608C2C53×#186911yescharacteristic
609C2C53×#189811yescharacteristic
610C2C53×#190011yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC23C22×C8×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC23×D4×C53C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C24×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8×C53C2semidirect
3. C53×#1779C2semidirect
4. C2×C53×#323C2semidirect
5. C53×#1280C2semidirect
32. C22×D4×C53C8semidirect
67. C22⋊C8×C53D4semidirect
89. C24×C53C2×C8semidirect
135. C22×C4×C53C2×C8semidirect
243. #9079C53direct (N × H)
276. C22×C53#323semidirect
294. C22×C22⋊C8C2×C53semidirect
296. #1779C2×C53semidirect
297. C2×#323C2×C53semidirect
298. #1280C2×C53semidirect
330. C53#9079direct (N × H)
338. C22×D4C8×C53semidirect
373. C22⋊C8D4×C53semidirect
395. C24C2×C8×C53semidirect
441. C22×C4C2×C8×C53semidirect
581. C22C53×#323semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C8×C532
C2×C53×#3862
C53×#17792
C2×C53×#3232
C53×#12802
#907953