Finite Group Databasebeta

← C53×#8403 #27215915

Subgroups — C53×#8403 #27215915

Order 13568.

1364 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#534C211yesmaximal, characteristic
2C53×#1874C212yesmaximal, complemented
3C2×C53×#327C212yesmaximal, complemented
4C2×C53×#323C211yesmaximal, characteristic, complemented
5C53×#1439C211yesmaximal, characteristic, complemented
6C23×D4×C53—21no—
7C53×#534—24no—
8C53×#534—22no—
9C53×#534C414yescomplemented
10C2×#123×C53—22no—
11C2×#121×C53—21no—
12C2×#121×C53C2212yes—
13C2×#120×C53—22no—
14C22×C4⋊C4×C53C2211yescharacteristic, complemented
15C22×C22⋊C4×C53—21no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C2×#102×C53—22no—
18C2×C22⋊C8×C53C2212yescomplemented
19C2×#95×C53—21no—
20C2×#95×C53C2211yescharacteristic, complemented
21C53×#340—22no—
22C53×#327—24no—
23C53×#323—22no—
24C25×C53—21no—
25C25×C53D411yescharacteristic
26C22×D4×C53—22no—
27C22×D4×C53—24no—
28C22×D4×C53—24no—
29C22×D4×C53—22no—
30C22×D4×C53—21no—
31C22×D4×C53—42no—
32C22×D4×C53—42no—
33C22×D4×C53—42no—
34C22×D4×C53—42no—
35C22×D4×C53D412yes—
36C23×C4×C53—21no—
37C23×C4×C53—21no—
38C23×C4×C53—21no—
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C22×C8×C53—22no—
42#123×C53—28no—
43#123×C53—28no—
44#121×C53—24no—
45#121×C53—24no—
46#121×C53—22no—
47#121×C53—22no—
48#121×C53C2×C414yes—
49#121×C53D414yes—
50#120×C53—28no—
51#120×C53—28no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—42no—
55C2×C4⋊C4×C53D412yes—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53—42no—
61C2×C22⋊C4×C53—42no—
62C2×C22⋊C4×C53—42no—
63C2×C22⋊C4×C53C2×C412yes—
64C2×C22⋊C4×C53D412yes—
65#102×C53—44no—
66C22⋊C8×C53—22no—
67C22⋊C8×C53—22no—
68#95×C53—22no—
69#95×C53—42no—
70C24×C53—21no—
71C24×C53—21no—
72C24×C53—21no—
73C24×C53—21no—
74C24×C53—24no—
75C24×C53—22no—
76C24×C53—41no—
77C24×C53—41no—
78C24×C53—41no—
79C24×C53—41no—
80C24×C53—42no—
81C24×C53—42no—
82C24×C53—81no—
83C24×C53—81no—
84C24×C53—81no—
85C24×C53—81no—
86C24×C53C22⋊C412yes—
87C24×C53D812yes—
88C24×C53SD1612yes—
89C24×C53C2×D411yescharacteristic
90C2×D4×C53—28no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—22no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—48no—
98C2×D4×C53—48no—
99C2×D4×C53—48no—
100C2×D4×C53—44no—
101C2×D4×C53—44no—
102C2×D4×C53—48no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C2×D4×C53—42no—
106C2×D4×C53—42no—
107C2×D4×C53—81no—
108C2×D4×C53—81no—
109C2×D4×C53C22⋊C414yescomplemented
110C22×C4×C53—24no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
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×C53—41no—
134C22×C4×C53C22⋊C412yes—
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D411yescharacteristic
139C2×C8×C53—42no—
140C2×C8×C53—42no—
141C2×C8×C53—42no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—42no—
146C4⋊C4×C53C22⋊C414yescomplemented
147C22⋊C4×C53—28no—
148C22⋊C4×C53—24no—
149C22⋊C4×C53—24no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—24no—
152C22⋊C4×C53—48no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53—22no—
162C23×C53—22no—
163C23×C53—24no—
164C23×C53—42no—
165C23×C53—42no—
166C23×C53—42no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—48no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—81no—
182C23×C53—81no—
183C23×C53—81no—
184C23×C53—81no—
185C23×C53—81no—
186C23×C53—81no—
187C23×C53—84no—
188C23×C53—84no—
189C23×C53—84no—
190C23×C53—84no—
191C23×C53—84no—
192C23×C53—84no—
193C23×C53#10214yescomplemented
194C23×C53C2×C22⋊C411yescharacteristic
195C23×C53#12011yescharacteristic
196C23×C53#12011yescharacteristic
197C23×C53C2×D811yescharacteristic
198C23×C53C2×SD1611yescharacteristic
199C23×C53AGL(1,ℤ/8)11yescharacteristic
200C23×C53AGL(1,ℤ/8)11yescharacteristic
201D4×C53—44no—
202D4×C53—416no—
203D4×C53—44no—
204D4×C53—44no—
205D4×C53—44no—
206D4×C53—44no—
207D4×C53—82no—
208D4×C53—84no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—48no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—81no—
229C2×C4×C53—81no—
230C2×C4×C53—81no—
231C2×C4×C53—81no—
232C2×C4×C53—81no—
233C2×C4×C53—81no—
234C2×C4×C53—81no—
235C2×C4×C53—81no—
236C2×C4×C53—81no—
237C2×C4×C53C2×C22⋊C412yes—
238C2×C4×C53#12012yes—
239C8×C53—82no—
240#8403C5311yesmaximal, characteristic, direct product factor, Sylow
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—24no—
254C22×C53—44no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—44no—
262C22×C53—44no—
263C22×C53—44no—
264C22×C53—82no—
265C22×C53—82no—
266C22×C53—82no—
267C22×C53—816no—
268C22×C53—82no—
269C22×C53—82no—
270C22×C53—82no—
271C22×C53—82no—
272C22×C53#37911yescharacteristic
273C22×C53C2×#10211yescharacteristic
274C22×C53#41811yescharacteristic
275C22×C53#44711yescharacteristic
276C22×C53#44711yescharacteristic
277C22×C53#45011yescharacteristic
278C22×C53#45011yescharacteristic
279C22×C53#45311yescharacteristic
280C22×C53#45511yescharacteristic
281C22×C53B(4,𝔽2)11yescharacteristic
282C22×C53#45811yescharacteristic
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—44no—
287C4×C53—82no—
288C4×C53—82no—
289C2×#534C2×C5311yescharacteristic
290#1874C2×C5312yescomplemented
291C2×#327C2×C5312yescomplemented
292C2×#323C2×C5311yescharacteristic, complemented
293#1439C2×C5311yescharacteristic, complemented
294C2×C53—21no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—44no—
299C2×C53—84no—
300C2×C53#159511yescharacteristic
301C2×C53#160111yescharacteristic
302C2×C53#160711yescharacteristic
303C2×C53#161311yescharacteristic
304C2×C53#178611yescharacteristic
305C2×C53#179011yescharacteristic
306C2×C53#179211yescharacteristic
307C23×D4—21no—
308#534—24no—
309#534—22no—
310#534C4×C5314yescomplemented
311C2×#123—22no—
312C2×#121—21no—
313C2×#121C22×C5312yes—
314C2×#120—22no—
315C22×C4⋊C4C22×C5311yescharacteristic, complemented
316C22×C22⋊C4—21no—
317C22×C22⋊C4C22×C5311yescharacteristic
318C2×#102—22no—
319C2×C22⋊C8C22×C5312yescomplemented
320C2×#95—21no—
321C2×#95C22×C5311yescharacteristic, complemented
322#340—22no—
323#327—24no—
324#323—22no—
325C53#840311yescharacteristic, direct product factor, Sylow
326C25—21no—
327C25D4×C5311yescharacteristic
328C22×D4—22no—
329C22×D4—24no—
330C22×D4—22no—
331C22×D4—24no—
332C22×D4—21no—
333C22×D4—42no—
334C22×D4—42no—
335C22×D4—42no—
336C22×D4—42no—
337C22×D4D4×C5312yes—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4D4×C5311yescharacteristic
342C23×C4C23×C5311yescharacteristic
343C22×C8—22no—
344#123—28no—
345#123—28no—
346#121—24no—
347#121—22no—
348#121—22no—
349#121—24no—
350#121C2×C4×C5314yes—
351#121D4×C5314yes—
352#120—28no—
353#120—28no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4D4×C5312yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4C2×C4×C5312yes—
366C2×C22⋊C4D4×C5312yes—
367#102—44no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370#95—22no—
371#95—42no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—24no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—42no—
383C24—42no—
384C24—81no—
385C24—81no—
386C24—81no—
387C24—81no—
388C24C22⋊C4×C5312yes—
389C24D8×C5312yes—
390C24SD16×C5312yes—
391C24C2×D4×C5311yescharacteristic
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4C22⋊C4×C5314yescomplemented
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4C22⋊C4×C5312yes—
437C22×C4C22×C4×C5311yescharacteristic
438C22×C4C2×D4×C5312yes—
439C22×C4C2×D4×C5312yes—
440C22×C4C2×D4×C5311yescharacteristic
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4C22⋊C4×C5314yescomplemented
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—48no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—48no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23#102×C5314yescomplemented
496C23C2×C22⋊C4×C5311yescharacteristic
497C23#120×C5311yescharacteristic
498C23#120×C5311yescharacteristic
499C23C2×D8×C5311yescharacteristic
500C23C2×SD16×C5311yescharacteristic
501C23AGL(1,ℤ/8)×C5311yescharacteristic
502C23AGL(1,ℤ/8)×C5311yescharacteristic
503D4—44no—
504D4—416no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—82no—
510D4—84no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4C2×C22⋊C4×C5312yes—
540C2×C4#120×C5312yes—
541C8—82no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—24no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—816no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22C53×#37911yescharacteristic
574C22C2×#102×C5311yescharacteristic
575C22C53×#41811yescharacteristic
576C22C53×#44711yescharacteristic
577C22C53×#44711yescharacteristic
578C22C53×#45011yescharacteristic
579C22C53×#45011yescharacteristic
580C22C53×#45311yescharacteristic
581C22C53×#45511yescharacteristic
582C22C53×B(4,𝔽2)11yescharacteristic
583C22C53×#45811yescharacteristic
584C4—22no—
585C4—22no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—82no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—84no—
596C2C53×#159511yescharacteristic
597C2C53×#160111yescharacteristic
598C2C53×#160711yescharacteristic
599C2C53×#161311yescharacteristic
600C2C53×#178611yescharacteristic
601C2C53×#179011yescharacteristic
602C2C53×#179211yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×C53×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1874C2semidirect
3. C2×C53×#327C2semidirect
4. C2×C53×#323C2semidirect
5. C53×#1439C2semidirect
9. C53×#534C4semidirect
14. C22×C4⋊C4×C53C22semidirect
18. C2×C22⋊C8×C53C22semidirect
20. C2×#95×C53C22semidirect
109. C2×D4×C53C22⋊C4semidirect
146. C4⋊C4×C53C22⋊C4semidirect
193. C23×C53#102semidirect
240. #8403C53direct (N × H)
290. #1874C2×C53semidirect
291. C2×#327C2×C53semidirect
292. C2×#323C2×C53semidirect
293. #1439C2×C53semidirect
310. #534C4×C53semidirect
315. C22×C4⋊C4C22×C53semidirect
319. C2×C22⋊C8C22×C53semidirect
321. C2×#95C22×C53semidirect
325. C53#8403direct (N × H)
411. C2×D4C22⋊C4×C53semidirect
448. C4⋊C4C22⋊C4×C53semidirect
495. C23#102×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5342
C53×#18742
C2×C53×#3272
C2×C53×#3232
C53×#14392
#840353