Finite Group Databasebeta

← C53×#9922 #27217434

Subgroups — C53×#9922 #27217434

Order 13568.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#450C211yesmaximal, characteristic, complemented
2C53×#2626C211yesmaximal, characteristic, complemented
3C2×C53×#392C211yesmaximal, characteristic
4C53×#1786C211yesmaximal, characteristic, complemented
5C2×C53×#328C211yesmaximal, characteristic, complemented
6C2×C53×#323C211yesmaximal, characteristic, complemented
7C53×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C53—21no—
9C23×D4×C53C2211yescharacteristic
10C22×SD16×C53—21no—
11C2×C22⋊Q8×C53C2211yescharacteristic
12C22×C22⋊C4×C53—21no—
13C22×C22⋊C4×C53C2211yescharacteristic
14C53×#450—22no—
15C53×#450—22no—
16C2×#102×C53—21no—
17C2×#102×C53—22no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2211yescharacteristic
20C53×#399—21no—
21C53×#396—22no—
22C53×#393—21no—
23C53×#392—22no—
24C53×#392—22no—
25C53×#385—21no—
26C53×#380C2211yescharacteristic
27C53×#379—21no—
28C2×#95×C53C2211yescharacteristic
29C53×#328—22no—
30C53×#323—22no—
31C25×C53—21no—
32C25×C53D411yescharacteristic
33C22×Q8×C53D411yescharacteristic, complemented
34C22×D4×C53—24no—
35C22×D4×C53—21no—
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53—21no—
39C22×D4×C53—21no—
40C22×D4×C53—44no—
41C22×D4×C53—44no—
42C22×D4×C53—44no—
43C22×D4×C53—42no—
44C22×D4×C53—42no—
45C22×D4×C53D412yes—
46C23×C4×C53—21no—
47C23×C4×C53D411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49C2×SD16×C53—24no—
50C2×SD16×C53—44no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53D411yescharacteristic, complemented
60C2×C4⋊C4×C53D412yescomplemented
61C2×C22⋊C4×C53—24no—
62C2×C22⋊C4×C53—22no—
63C2×C22⋊C4×C53—22no—
64C2×C22⋊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—44no—
73#102×C53—24no—
74#102×C53—44no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53D412yescomplemented
79#95×C53—22no—
80#95×C53—21no—
81#95×C53—21no—
82#95×C53D412yes—
83C24×C53—24no—
84C24×C53—21no—
85C24×C53—22no—
86C24×C53—21no—
87C24×C53—21no—
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—42no—
92C24×C53—42no—
93C24×C53—42no—
94C24×C53—42no—
95C24×C53—42no—
96C24×C53—41no—
97C24×C53—41no—
98C24×C53—41no—
99C24×C53—41no—
100C24×C53—41no—
101C24×C53—41no—
102C24×C53SD1612yescomplemented
103C24×C53C2×D411yescharacteristic
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—42no—
107C2×D4×C53—24no—
108C2×D4×C53—24no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—42no—
112C2×D4×C53—42no—
113C2×D4×C53—48no—
114C2×D4×C53—44no—
115C2×D4×C53—44no—
116C2×D4×C53—44no—
117C2×D4×C53—44no—
118C2×D4×C53—48no—
119C2×D4×C53—48no—
120C2×D4×C53—416no—
121C2×D4×C53—416no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C22×C4×C53—24no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53—42no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53#4111yescharacteristic
143SD16×C53—48no—
144C2×C8×C53—22no—
145C2×C8×C53—42no—
146C2×C8×C53—42no—
147C2×C8×C53—41no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—42no—
151C4⋊C4×C53—41no—
152C4⋊C4×C53—41no—
153C4⋊C4×C53—82no—
154C22⋊C4×C53—48no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—82no—
164C22⋊C4×C53—82no—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—24no—
168C23×C53—22no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—44no—
172C23×C53—44no—
173C23×C53—44no—
174C23×C53—44no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—44no—
180C23×C53—44no—
181C23×C53—44no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—44no—
191C23×C53—42no—
192C23×C53—41no—
193C23×C53—82no—
194C23×C53—82no—
195C23×C53—81no—
196C23×C53—81no—
197C23×C53—82no—
198C23×C53—82no—
199C23×C53—82no—
200C23×C53—82no—
201C23×C53—82no—
202C23×C53—81no—
203C23×C53—81no—
204C23×C53—82no—
205C23×C53—81no—
206C23×C53—81no—
207C23×C53#12011yescharacteristic
208C23×C53#12011yescharacteristic
209C23×C53#12111yescharacteristic
210C23×C53C2×SD1611yescharacteristic
211C23×C53#13511yescharacteristic
212C23×C53AGL(1,ℤ/8)11yescharacteristic
213C23×C53#13711yescharacteristic
214Q8×C53—42no—
215Q8×C53—81no—
216D4×C53—24no—
217D4×C53—24no—
218D4×C53—42no—
219D4×C53—44no—
220D4×C53—42no—
221D4×C53—44no—
222D4×C53—432no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—44no—
231C2×C4×C53—44no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—81no—
245C2×C4×C53#12011yescharacteristic
246C2×C4×C53#12111yescharacteristic
247C2×C4×C53#12111yescharacteristic
248C2×C4×C53#12711yescharacteristic
249C8×C53—42no—
250C8×C53—81no—
251#9922C5311yesmaximal, characteristic, direct product factor, Sylow
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—44no—
261C22×C53—44no—
262C22×C53—44no—
263C22×C53—44no—
264C22×C53—42no—
265C22×C53—44no—
266C22×C53—82no—
267C22×C53—82no—
268C22×C53—82no—
269C22×C53—82no—
270C22×C53—84no—
271C22×C53—82no—
272C22×C53—84no—
273C22×C53—84no—
274C22×C53—82no—
275C22×C53—84no—
276C22×C53—82no—
277C22×C53—81no—
278C22×C53—82no—
279C22×C53—81no—
280C22×C53#39211yescharacteristic
281C22×C53#44811yescharacteristic
282C22×C53#44911yescharacteristic
283C22×C53#45011yescharacteristic
284C22×C53#45211yescharacteristic
285C22×C53#45312yes—
286C22×C53B(4,𝔽2)12yes—
287C22×C53#46511yescharacteristic
288C22×C53#46811yescharacteristic
289C4×C53—21no—
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—44no—
294C4×C53—82no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53—81no—
298C2×#450C2×C5311yescharacteristic, complemented
299#2626C2×C5311yescharacteristic, complemented
300C2×#392C2×C5311yescharacteristic
301#1786C2×C5311yescharacteristic, complemented
302C2×#328C2×C5311yescharacteristic, complemented
303C2×#323C2×C5311yescharacteristic, complemented
304#1294C2×C5311yescharacteristic, complemented
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—82no—
310C2×C53—84no—
311C2×C53#159612yes—
312C2×C53#160012yes—
313C2×C53#200011yescharacteristic
314C2×C53#200211yescharacteristic
315C2×C53#202111yescharacteristic
316C23×D4—21no—
317C23×D4C22×C5311yescharacteristic
318C22×SD16—21no—
319C2×C22⋊Q8C22×C5311yescharacteristic
320C22×C22⋊C4—21no—
321C22×C22⋊C4C22×C5311yescharacteristic
322#450—22no—
323#450—22no—
324C2×#102—21no—
325C2×#102—22no—
326C2×C22⋊C8C22×C5311yescharacteristic
327C2×C22⋊C8C22×C5311yescharacteristic
328#399—21no—
329#396—22no—
330#393—21no—
331#392—22no—
332#392—22no—
333#385—21no—
334#380C22×C5311yescharacteristic
335#379—21no—
336C2×#95C22×C5311yescharacteristic
337#328—22no—
338#323—22no—
339C53#992211yescharacteristic, direct product factor, Sylow
340C25—21no—
341C25D4×C5311yescharacteristic
342C22×Q8D4×C5311yescharacteristic, complemented
343C22×D4—24no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4—44no—
350C22×D4—44no—
351C22×D4—44no—
352C22×D4—42no—
353C22×D4—42no—
354C22×D4D4×C5312yes—
355C23×C4—21no—
356C23×C4D4×C5311yescharacteristic
357C23×C4C23×C5311yescharacteristic
358C2×SD16—24no—
359C2×SD16—44no—
360C22×C8—21no—
361C22×C8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4D4×C5311yescharacteristic, complemented
369C2×C4⋊C4D4×C5312yescomplemented
370C2×C22⋊C4—24no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—44no—
382#102—24no—
383#102—44no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—21no—
387C22⋊C8D4×C5312yescomplemented
388#95—22no—
389#95—21no—
390#95—21no—
391#95D4×C5312yes—
392C24—24no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24SD16×C5312yescomplemented
412C24C2×D4×C5311yescharacteristic
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—41no—
432C2×D4—41no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4C2×D4×C5311yescharacteristic
447C22×C4C2×D4×C5311yescharacteristic
448C22×C4C2×D4×C5311yescharacteristic
449C22×C4C2×D4×C5311yescharacteristic
450C22×C4C2×D4×C5311yescharacteristic
451C22×C4#41×C5311yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—41no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—82no—
463C22⋊C4—48no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—41no—
502C23—82no—
503C23—81no—
504C23—81no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23#120×C5311yescharacteristic
517C23#120×C5311yescharacteristic
518C23#121×C5311yescharacteristic
519C23C2×SD16×C5311yescharacteristic
520C23#135×C5311yescharacteristic
521C23AGL(1,ℤ/8)×C5311yescharacteristic
522C23#137×C5311yescharacteristic
523Q8—42no—
524Q8—81no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—44no—
531D4—432no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4#120×C5311yescharacteristic
555C2×C4#121×C5311yescharacteristic
556C2×C4#121×C5311yescharacteristic
557C2×C4#127×C5311yescharacteristic
558C8—42no—
559C8—81no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22—84no—
581C22—82no—
582C22—81no—
583C22—82no—
584C22—81no—
585C22—84no—
586C22—82no—
587C22—84no—
588C22C53×#39211yescharacteristic
589C22C53×#44811yescharacteristic
590C22C53×#44911yescharacteristic
591C22C53×#45011yescharacteristic
592C22C53×#45211yescharacteristic
593C22C53×#45312yes—
594C22C53×B(4,𝔽2)12yes—
595C22C53×#46511yescharacteristic
596C22C53×#46811yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—82no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—84no—
612C2C53×#159612yes—
613C2C53×#160012yes—
614C2C53×#200011yescharacteristic
615C2C53×#200211yescharacteristic
616C2C53×#202111yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC2×C53×#392C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#450C2semidirect
2. C53×#2626C2semidirect
4. C53×#1786C2semidirect
5. C2×C53×#328C2semidirect
6. C2×C53×#323C2semidirect
7. C53×#1294C2semidirect
33. C22×Q8×C53D4semidirect
59. C2×C4⋊C4×C53D4semidirect
60. C2×C4⋊C4×C53D4semidirect
78. C22⋊C8×C53D4semidirect
102. C24×C53SD16semidirect
251. #9922C53direct (N × H)
298. C2×#450C2×C53semidirect
299. #2626C2×C53semidirect
301. #1786C2×C53semidirect
302. C2×#328C2×C53semidirect
303. C2×#323C2×C53semidirect
304. #1294C2×C53semidirect
339. C53#9922direct (N × H)
342. C22×Q8D4×C53semidirect
368. C2×C4⋊C4D4×C53semidirect
369. C2×C4⋊C4D4×C53semidirect
387. C22⋊C8D4×C53semidirect
411. C24SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4502
C53×#26262
C2×C53×#3922
C53×#17862
C2×C53×#3282
C2×C53×#3232
C53×#12942
#992253