Finite Group Databasebeta

← C53×#8428 #27215940

Subgroups — C53×#8428 #27215940

Order 13568.

1020 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3496C211yesmaximal, characteristic
2C53×#1804C211yesmaximal, characteristic, complemented
3C53×#1540C211yesmaximal, characteristic, complemented
4C53×#1517C211yesmaximal, characteristic, complemented
5C53×#1517C211yesmaximal, characteristic, complemented
6C53×#1480C211yesmaximal, characteristic, complemented
7C53×#1478C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C53—21no—
9C53×#547—21no—
10C53×#547C412yescomplemented
11C53×#546—22no—
12C53×#546—22no—
13D42×C53—21no—
14D42×C53C412yescomplemented
15C53×#537—21no—
16C53×#534—22no—
17C53×#532C2211yescharacteristic
18C53×#532C2211yescharacteristic
19C2×#121×C53—22no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#431C2211yescharacteristic, complemented
22C53×#431C2211yescharacteristic, complemented
23C2×C4⋊C8×C53C2211yescharacteristic, complemented
24C2×#102×C53—21no—
25C53×#382C2211yescharacteristic, complemented
26C53×#340—21no—
27C53×#332—21no—
28C53×#331—21no—
29C53×#327—22no—
30C53×#327—22no—
31C53×D4⋊C8—22no—
32C53×D4⋊C8—22no—
33H(5,𝔽2)×C53—42no—
34H(5,𝔽2)×C53—42no—
35H(5,𝔽2)×C53—42no—
36H(5,𝔽2)×C53—42no—
37C2×#41×C53—22no—
38C2×#41×C53—22no—
39C2×#41×C53—21no—
40C2×#41×C53—21no—
41C22×D4×C53—24no—
42C22×D4×C53—22no—
43C22×D4×C53—22no—
44C22×D4×C53—22no—
45C22×D4×C53D411yescharacteristic
46C23×C4×C53—21no—
47C22×C8×C53—21no—
48C4⋊Q8×C53C2×C411yescharacteristic
49C4⋊Q8×C53D411yescharacteristic
50#127×C53C2×C411yescharacteristic
51#127×C53D411yescharacteristic
52#124×C53—21no—
53#124×C53—21no—
54#123×C53—24no—
55#123×C53—24no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58#121×C53—24no—
59#121×C53—24no—
60#121×C53—22no—
61#121×C53—22no—
62#121×C53—22no—
63#121×C53—22no—
64#121×C53—21no—
65#121×C53—21no—
66#121×C53D412yes—
67#121×C53D412yes—
68#120×C53—24no—
69#120×C53—24no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53C2×C412yes—
75C4×D4×C53C2×C412yes—
76C4×D4×C53D412yes—
77C4×D4×C53D412yes—
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53D411yescharacteristic
81C2×C22⋊C4×C53—24no—
82C2×C22⋊C4×C53—21no—
83C2×C42×C53C2311yescharacteristic
84C4⋊C8×C53—21no—
85C4⋊C8×C53—21no—
86C4⋊C8×C53—21no—
87C4⋊C8×C53—21no—
88#102×C53—42no—
89C22⋊C8×C53—21no—
90C22⋊C8×C53—21no—
91C4×C8×C53—21no—
92C4×C8×C53—21no—
93#95×C53—21no—
94#95×C53—21no—
95C24×C53—21no—
96C24×C53—24no—
97#41×C53—44no—
98#41×C53—44no—
99#41×C53—44no—
100#41×C53—44no—
101#41×C53—41no—
102#41×C53—41no—
103#41×C53—41no—
104#41×C53—41no—
105#41×C53—41no—
106#41×C53—41no—
107#41×C53—41no—
108#41×C53—41no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×D4×C53—24no—
112C2×D4×C53—24no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—24no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—44no—
124C2×D4×C53—44no—
125C2×D4×C53—44no—
126C2×D4×C53—48no—
127C2×D4×C53—48no—
128C2×D4×C53—44no—
129C2×D4×C53—84no—
130C2×D4×C53C22⋊C412yes—
131C2×D4×C53C22⋊C412yes—
132C2×D4×C53C22⋊C412yes—
133C2×D4×C53D812yes—
134C2×D4×C53SD1612yes—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—41no—
142C22×C4×C53—82no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C2×C8×C53—41no—
147C2×C8×C53—41no—
148C2×C8×C53—41no—
149C2×C8×C53—41no—
150C2×C8×C53—41no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53C22⋊C412yes—
158C4⋊C4×C53C22⋊C412yes—
159C4⋊C4×C53C22⋊C412yes—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C42×C53C22×C411yescharacteristic
169C42×C53C2×D411yescharacteristic
170C42×C53C2×D411yescharacteristic
171C42×C53C2×D411yescharacteristic
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—48no—
178C23×C53—81no—
179C23×C53—81no—
180C23×C53—84no—
181C23×C53—84no—
182C23×C53#12011yescharacteristic
183Q8×C53—41no—
184Q8×C53—41no—
185Q8×C53—41no—
186Q8×C53—41no—
187D4×C53—22no—
188D4×C53—22no—
189D4×C53—22no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—48no—
194D4×C53—48no—
195D4×C53—42no—
196D4×C53—42no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—81no—
204D4×C53—84no—
205D4×C53#10214yes—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—44no—
212C2×C4×C53—48no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—82no—
218C2×C4×C53—81no—
219C2×C4×C53—81no—
220C2×C4×C53—81no—
221C2×C4×C53—81no—
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53#12011yescharacteristic
226C2×C4×C53#12011yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53C2×D811yescharacteristic
229C2×C4×C53C2×SD1611yescharacteristic
230C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
231C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
232C8×C53—81no—
233C8×C53—81no—
234C8×C53—81no—
235#8428C5311yesmaximal, characteristic, direct product factor, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—24no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—82no—
248C22×C53—82no—
249C22×C53—168no—
250C22×C53#37911yescharacteristic
251C22×C53#44711yescharacteristic
252C22×C53#45011yescharacteristic
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—42no—
260C4×C53—161no—
261C4×C53C2×#10211yescharacteristic
262C4×C53#41811yescharacteristic
263C4×C53#44711yescharacteristic
264C4×C53#45011yescharacteristic
265#3496C2×C5311yescharacteristic
266#1804C2×C5311yescharacteristic, complemented
267#1540C2×C5311yescharacteristic, complemented
268#1517C2×C5311yescharacteristic, complemented
269#1517C2×C5311yescharacteristic, complemented
270#1480C2×C5311yescharacteristic, complemented
271#1478C2×C5311yescharacteristic, complemented
272C2×C53—21no—
273C2×C53—21no—
274C2×C53—44no—
275C2×C53—82no—
276C2×C53—82no—
277C2×C53#178611yescharacteristic
278C2×C53#179411yescharacteristic
279C2×C53#179511yescharacteristic
280C2×H(5,𝔽2)—21no—
281#547—21no—
282#547C4×C5312yescomplemented
283#546—22no—
284#546—22no—
285D42—21no—
286D42C4×C5312yescomplemented
287#537—21no—
288#534—22no—
289#532C22×C5311yescharacteristic
290#532C22×C5311yescharacteristic
291C2×#121—22no—
292C2×C4×D4C22×C5311yescharacteristic
293#431C22×C5311yescharacteristic, complemented
294#431C22×C5311yescharacteristic, complemented
295C2×C4⋊C8C22×C5311yescharacteristic, complemented
296C2×#102—21no—
297#382C22×C5311yescharacteristic, complemented
298#340—21no—
299#332—21no—
300#331—21no—
301#327—22no—
302#327—22no—
303D4⋊C8—22no—
304D4⋊C8—22no—
305C53#842811yescharacteristic, direct product factor, Sylow
306H(5,𝔽2)—42no—
307H(5,𝔽2)—42no—
308H(5,𝔽2)—42no—
309H(5,𝔽2)—42no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—21no—
313C2×#41—21no—
314C22×D4—24no—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—22no—
318C22×D4D4×C5311yescharacteristic
319C23×C4—21no—
320C22×C8—21no—
321C4⋊Q8C2×C4×C5311yescharacteristic
322C4⋊Q8D4×C5311yescharacteristic
323#127C2×C4×C5311yescharacteristic
324#127D4×C5311yescharacteristic
325#124—21no—
326#124—21no—
327#123—24no—
328#123—24no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331#121—24no—
332#121—24no—
333#121—22no—
334#121—22no—
335#121—21no—
336#121—21no—
337#121—22no—
338#121—22no—
339#121D4×C5312yes—
340#121D4×C5312yes—
341#120—24no—
342#120—24no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4C2×C4×C5312yes—
348C4×D4C2×C4×C5312yes—
349C4×D4D4×C5312yes—
350C4×D4D4×C5312yes—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4D4×C5311yescharacteristic
354C2×C22⋊C4—24no—
355C2×C22⋊C4—21no—
356C2×C42C23×C5311yescharacteristic
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361#102—42no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366#95—21no—
367#95—21no—
368C24—21no—
369C24—24no—
370#41—44no—
371#41—44no—
372#41—44no—
373#41—44no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4—84no—
403C2×D4C22⋊C4×C5312yes—
404C2×D4C22⋊C4×C5312yes—
405C2×D4C22⋊C4×C5312yes—
406C2×D4D8×C5312yes—
407C2×D4SD16×C5312yes—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—82no—
416C22×C4C2×D4×C5311yescharacteristic
417C22×C4C2×D4×C5311yescharacteristic
418C22×C4C2×D4×C5311yescharacteristic
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4C22⋊C4×C5312yes—
431C4⋊C4C22⋊C4×C5312yes—
432C4⋊C4C22⋊C4×C5312yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C42C22×C4×C5311yescharacteristic
442C42C2×D4×C5311yescharacteristic
443C42C2×D4×C5311yescharacteristic
444C42C2×D4×C5311yescharacteristic
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—48no—
451C23—81no—
452C23—81no—
453C23—84no—
454C23—84no—
455C23#120×C5311yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—48no—
467D4—48no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—81no—
477D4—84no—
478D4#102×C5314yes—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4C2×C22⋊C4×C5311yescharacteristic
496C2×C4C2×C22⋊C4×C5311yescharacteristic
497C2×C4C2×C22⋊C4×C5311yescharacteristic
498C2×C4#120×C5311yescharacteristic
499C2×C4#120×C5311yescharacteristic
500C2×C4#120×C5311yescharacteristic
501C2×C4C2×D8×C5311yescharacteristic
502C2×C4C2×SD16×C5311yescharacteristic
503C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
504C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
505C8—81no—
506C8—81no—
507C8—81no—
508C22—21no—
509C22—21no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—82no—
520C22—82no—
521C22—168no—
522C22C53×#37911yescharacteristic
523C22C53×#44711yescharacteristic
524C22C53×#45011yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—161no—
533C4C2×#102×C5311yescharacteristic
534C4C53×#41811yescharacteristic
535C4C53×#44711yescharacteristic
536C4C53×#45011yescharacteristic
537C2—21no—
538C2—21no—
539C2—44no—
540C2—82no—
541C2—82no—
542C2C53×#178611yescharacteristic
543C2C53×#179411yescharacteristic
544C2C53×#179511yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3496C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1804C2semidirect
3. C53×#1540C2semidirect
4. C53×#1517C2semidirect
5. C53×#1517C2semidirect
6. C53×#1480C2semidirect
7. C53×#1478C2semidirect
10. C53×#547C4semidirect
14. D42×C53C4semidirect
21. C53×#431C22semidirect
22. C53×#431C22semidirect
23. C2×C4⋊C8×C53C22semidirect
25. C53×#382C22semidirect
235. #8428C53direct (N × H)
266. #1804C2×C53semidirect
267. #1540C2×C53semidirect
268. #1517C2×C53semidirect
269. #1517C2×C53semidirect
270. #1480C2×C53semidirect
271. #1478C2×C53semidirect
282. #547C4×C53semidirect
286. D42C4×C53semidirect
293. #431C22×C53semidirect
294. #431C22×C53semidirect
295. C2×C4⋊C8C22×C53semidirect
297. #382C22×C53semidirect
305. C53#8428direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34962
C53×#18042
C53×#15402
C53×#15172
C53×#14802
C53×#14782
#842853