Finite Group Databasebeta

← C53×#9255 #27216767

Subgroups — C53×#9255 #27216767

Order 13568.

924 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#450C211yesmaximal, characteristic
2D4×C22⋊C4×C53C211yesmaximal, characteristic
3C53×#1996C211yesmaximal, characteristic, complemented
4C53×#1786C211yesmaximal, characteristic
5C53×#1520C211yesmaximal, characteristic, complemented
6C53×#1289C211yesmaximal, characteristic, complemented
7C53×#1282C211yesmaximal, characteristic, complemented
8C23×D4×C53C2211yescharacteristic
9C22×SD16×C53C412yescomplemented
10C2×C22⋊Q8×C53C2211yescharacteristic
11C2×C4×D4×C53—21no—
12C22×C22⋊C4×C53—21no—
13C53×#450—21no—
14C53×#450—22no—
15C53×#450—21no—
16C2×#102×C53—22no—
17C2×#102×C53C412yescomplemented
18C2×C22⋊C8×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2211yescharacteristic
20C53×#390—21no—
21C53×#386—22no—
22C53×#380C2211yescharacteristic
23C53×#380C2211yescharacteristic
24C53×#379—21no—
25C4×C22⋊C4×C53C2211yescharacteristic
26C53×#342—21no—
27C53×#342—21no—
28C53×#328—22no—
29C25×C53—21no—
30C22×Q8×C53C2×C411yescharacteristic, complemented
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—44no—
35C22×D4×C53C2×C412yes—
36C22×D4×C53D411yescharacteristic
37C22×D4×C53D412yes—
38C22×D4×C53D411yescharacteristic
39C23×C4×C53—22no—
40C23×C4×C53C2311yescharacteristic
41C2×SD16×C53—22no—
42C2×SD16×C53—22no—
43C2×SD16×C53—44no—
44C22×C8×C53—21no—
45C22×C8×C53C2×C412yescomplemented
46C22⋊Q8×C53—21no—
47C22⋊Q8×C53—21no—
48C22⋊Q8×C53—21no—
49C22⋊Q8×C53—21no—
50C4×D4×C53—44no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—42no—
62C2×C22⋊C4×C53—42no—
63C2×C22⋊C4×C53D411yescharacteristic
64C2×C22⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66#102×C53—22no—
67#102×C53—22no—
68#102×C53—44no—
69C22⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53D412yes—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53—22no—
79C24×C53—22no—
80C24×C53—44no—
81C24×C53—41no—
82C24×C53—41no—
83C24×C53—41no—
84C24×C53—41no—
85C24×C53—82no—
86C24×C53C2×D411yescharacteristic
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53—42no—
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—22no—
94C2×D4×C53—22no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—42no—
98C2×D4×C53—42no—
99C2×D4×C53—42no—
100C2×D4×C53—44no—
101C2×D4×C53—44no—
102C2×D4×C53—44no—
103C2×D4×C53C22⋊C412yes—
104C2×D4×C53C22⋊C412yes—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—42no—
111C22×C4×C53—44no—
112C22×C4×C53—44no—
113C22×C4×C53—41no—
114C22×C4×C53—82no—
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×C4×C53#4111yescharacteristic
121SD16×C53—44no—
122SD16×C53—44no—
123C2×C8×C53—22no—
124C2×C8×C53—42no—
125C2×C8×C53—42no—
126C2×C8×C53—41no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C22⋊C4×C53—21no—
134C22⋊C4×C53—21no—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—44no—
137C22⋊C4×C53—44no—
138C22⋊C4×C53—41no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—82no—
145C22⋊C4×C53SD1612yescomplemented
146C42×C53—42no—
147C23×C53—24no—
148C23×C53—22no—
149C23×C53—22no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—44no—
153C23×C53—44no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—42no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53—42no—
160C23×C53—42no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—42no—
164C23×C53—42no—
165C23×C53—82no—
166C23×C53—82no—
167C23×C53—84no—
168C23×C53—84no—
169C23×C53—84no—
170C23×C53—84no—
171C23×C53C4×D411yescharacteristic
172C23×C53#12011yescharacteristic
173C23×C53#12111yescharacteristic
174C23×C53C2×SD1611yescharacteristic
175C23×C53#13511yescharacteristic
176C23×C53AGL(1,ℤ/8)11yescharacteristic
177C23×C53#13711yescharacteristic
178Q8×C53—42no—
179Q8×C53—81no—
180D4×C53—24no—
181D4×C53—22no—
182D4×C53—22no—
183D4×C53—22no—
184D4×C53—44no—
185D4×C53—44no—
186D4×C53—42no—
187D4×C53#9914yes—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—42no—
204C2×C4×C53—41no—
205C2×C4×C53—88no—
206C2×C4×C53—81no—
207C2×C4×C53—81no—
208C2×C4×C53—81no—
209C2×C4×C53—81no—
210C2×C4×C53—81no—
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C2×C4×C53C4×D411yescharacteristic
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#12311yescharacteristic
215C8×C53—42no—
216C8×C53—81no—
217#9255C5311yesmaximal, characteristic, direct product factor, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—24no—
225C22×C53—41no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—44no—
230C22×C53—44no—
231C22×C53—42no—
232C22×C53—42no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—42no—
236C22×C53—84no—
237C22×C53—84no—
238C22×C53—84no—
239C22×C53—84no—
240C22×C53—82no—
241C22×C53—82no—
242C22×C53#38611yescharacteristic
243C22×C53C4×SD1611yescharacteristic
244C22×C53#44011yescharacteristic
245C22×C53#45011yescharacteristic
246C22×C53#45211yescharacteristic
247C22×C53#46111yescharacteristic
248C22×C53#46311yescharacteristic
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—42no—
253C4×C53—81no—
254C4×C53—82no—
255C4×C53—81no—
256C4×C53—81no—
257C4×C53—161no—
258C4×C53C2×#9912yes—
259C2×#450C2×C5311yescharacteristic
260D4×C22⋊C4C2×C5311yescharacteristic
261#1996C2×C5311yescharacteristic, complemented
262#1786C2×C5311yescharacteristic
263#1520C2×C5311yescharacteristic, complemented
264#1289C2×C5311yescharacteristic, complemented
265#1282C2×C5311yescharacteristic, complemented
266C2×C53—21no—
267C2×C53—21no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—44no—
273C2×C53—82no—
274C2×C53#187711yescharacteristic
275C2×C53#189811yescharacteristic
276C2×C53#190411yescharacteristic
277C23×D4C22×C5311yescharacteristic
278C22×SD16C4×C5312yescomplemented
279C2×C22⋊Q8C22×C5311yescharacteristic
280C2×C4×D4—21no—
281C22×C22⋊C4—21no—
282#450—21no—
283#450—22no—
284#450—21no—
285C2×#102—22no—
286C2×#102C4×C5312yescomplemented
287C2×C22⋊C8C22×C5311yescharacteristic
288C2×C22⋊C8C22×C5311yescharacteristic
289#390—21no—
290#386—22no—
291#380C22×C5311yescharacteristic
292#380C22×C5311yescharacteristic
293#379—21no—
294C4×C22⋊C4C22×C5311yescharacteristic
295#342—21no—
296#342—21no—
297#328—22no—
298C53#925511yescharacteristic, direct product factor, Sylow
299C25—21no—
300C22×Q8C2×C4×C5311yescharacteristic, complemented
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304C22×D4—44no—
305C22×D4C2×C4×C5312yes—
306C22×D4D4×C5311yescharacteristic
307C22×D4D4×C5312yes—
308C22×D4D4×C5311yescharacteristic
309C23×C4—22no—
310C23×C4C23×C5311yescharacteristic
311C2×SD16—22no—
312C2×SD16—22no—
313C2×SD16—44no—
314C22×C8—21no—
315C22×C8C2×C4×C5312yescomplemented
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C4×D4—44no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C2×C4×C5311yescharacteristic, complemented
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4D4×C5311yescharacteristic
334C2×C22⋊C4D4×C5311yescharacteristic
335C2×C42—21no—
336#102—22no—
337#102—22no—
338#102—44no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343#95—21no—
344#95—21no—
345#95D4×C5312yes—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—44no—
351C24—41no—
352C24—41no—
353C24—41no—
354C24—41no—
355C24—82no—
356C24C2×D4×C5311yescharacteristic
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—42no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4C22⋊C4×C5312yes—
374C2×D4C22⋊C4×C5312yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—41no—
384C22×C4—82no—
385C22×C4C22×C4×C5311yescharacteristic
386C22×C4C2×D4×C5311yescharacteristic
387C22×C4C2×D4×C5311yescharacteristic
388C22×C4C2×D4×C5311yescharacteristic
389C22×C4#41×C5311yescharacteristic
390C22×C4#41×C5311yescharacteristic
391SD16—44no—
392SD16—44no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—41no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—82no—
415C22⋊C4SD16×C5312yescomplemented
416C42—42no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—82no—
436C23—82no—
437C23—84no—
438C23—84no—
439C23—84no—
440C23—84no—
441C23C4×D4×C5311yescharacteristic
442C23#120×C5311yescharacteristic
443C23#121×C5311yescharacteristic
444C23C2×SD16×C5311yescharacteristic
445C23#135×C5311yescharacteristic
446C23AGL(1,ℤ/8)×C5311yescharacteristic
447C23#137×C5311yescharacteristic
448Q8—42no—
449Q8—81no—
450D4—24no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—44no—
455D4—44no—
456D4—42no—
457D4#99×C5314yes—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—88no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4C2×C22⋊C4×C5311yescharacteristic
482C2×C4C4×D4×C5311yescharacteristic
483C2×C4#121×C5311yescharacteristic
484C2×C4#123×C5311yescharacteristic
485C8—42no—
486C8—81no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—24no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—44no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—84no—
509C22—82no—
510C22—82no—
511C22C53×#38611yescharacteristic
512C22C4×SD16×C5311yescharacteristic
513C22C53×#44011yescharacteristic
514C22C53×#45011yescharacteristic
515C22C53×#45211yescharacteristic
516C22C53×#46111yescharacteristic
517C22C53×#46311yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—82no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—161no—
527C4C2×#99×C5312yes—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—44no—
535C2—82no—
536C2C53×#187711yescharacteristic
537C2C53×#189811yescharacteristic
538C2C53×#190411yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC23×D4×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1996C2semidirect
5. C53×#1520C2semidirect
6. C53×#1289C2semidirect
7. C53×#1282C2semidirect
9. C22×SD16×C53C4semidirect
17. C2×#102×C53C4semidirect
30. C22×Q8×C53C2×C4semidirect
45. C22×C8×C53C2×C4semidirect
54. C2×C4⋊C4×C53C2×C4semidirect
145. C22⋊C4×C53SD16semidirect
217. #9255C53direct (N × H)
261. #1996C2×C53semidirect
263. #1520C2×C53semidirect
264. #1289C2×C53semidirect
265. #1282C2×C53semidirect
278. C22×SD16C4×C53semidirect
286. C2×#102C4×C53semidirect
298. C53#9255direct (N × H)
300. C22×Q8C2×C4×C53semidirect
315. C22×C8C2×C4×C53semidirect
324. C2×C4⋊C4C2×C4×C53semidirect
415. C22⋊C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4502
D4×C22⋊C4×C532
C53×#19962
C53×#17862
C53×#15202
C53×#12892
C53×#12822
#925553