Finite Group Databasebeta

← C53×#8381 #27215893

Subgroups — C53×#8381 #27215893

Order 13568.

1200 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#546C211yesmaximal, characteristic
2C53×#1799C211yesmaximal, characteristic, complemented
3C53×#1786C212yesmaximal, complemented
4C53×#1520C212yesmaximal, complemented
5C53×#1459C211yesmaximal, characteristic, complemented
6C22×#41×C53C411yescharacteristic, complemented
7C23×D4×C53C2211yescharacteristic
8C53×#546—24no—
9C53×#546—24no—
10C2×#124×C53C411yescharacteristic, complemented
11C2×#123×C53—21no—
12C2×C22⋊Q8×C53C2211yescharacteristic
13C2×#121×C53—21no—
14C2×#121×C53C411yescharacteristic, complemented
15C2×#120×C53—21no—
16C2×C4×D4×C53C411yescharacteristic, complemented
17C2×C4×D4×C53C2211yescharacteristic
18C22×C22⋊C4×C53—21no—
19C2×#102×C53—24no—
20C2×#99×C53—21no—
21C2×C22⋊C8×C53C2212yes—
22C53×#380C2212yes—
23C53×#342—21no—
24C53×#328—24no—
25C53×#324—21no—
26C53×#323—21no—
27C25×C53—21no—
28C2×#41×C53—22no—
29C2×#41×C53—22no—
30C2×#41×C53—44no—
31C22×Q8×C53C2×C411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—21no—
35C22×D4×C53—22no—
36C22×D4×C53—44no—
37C22×D4×C53C2×C411yescharacteristic
38C22×D4×C53C2×C411yescharacteristic
39C22×D4×C53D411yescharacteristic
40C22×D4×C53D414yes—
41C23×C4×C53—21no—
42C23×C4×C53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C22×C8×C53—22no—
46#124×C53—22no—
47#124×C53—22no—
48#123×C53—28no—
49C22⋊Q8×C53—22no—
50C22⋊Q8×C53D414yes—
51#121×C53—28no—
52#121×C53—22no—
53#121×C53—22no—
54#120×C53—28no—
55C4×D4×C53—22no—
56C4×D4×C53—22no—
57C4×D4×C53—22no—
58C4×D4×C53—22no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—22no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—28no—
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—22no—
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C42×C53C2×C411yescharacteristic
73#102×C53—48no—
74#99×C53—42no—
75C22⋊C8×C53—22no—
76C22⋊C8×C53—22no—
77#95×C53—22no—
78C24×C53—21no—
79C24×C53—21no—
80C24×C53—24no—
81C24×C53—44no—
82C24×C53—42no—
83C24×C53—41no—
84C24×C53—41no—
85C24×C53—82no—
86C24×C53C22⋊C411yescharacteristic
87C24×C53C2×D411yescharacteristic
88#41×C53—48no—
89#41×C53—44no—
90#41×C53—44no—
91C2×Q8×C53—21no—
92C2×Q8×C53—42no—
93C2×Q8×C53C22⋊C412yes—
94C2×D4×C53—24no—
95C2×D4×C53—28no—
96C2×D4×C53—24no—
97C2×D4×C53—22no—
98C2×D4×C53—22no—
99C2×D4×C53—21no—
100C2×D4×C53—21no—
101C2×D4×C53—24no—
102C2×D4×C53—21no—
103C2×D4×C53—21no—
104C2×D4×C53—24no—
105C2×D4×C53—42no—
106C2×D4×C53—44no—
107C2×D4×C53—48no—
108C2×D4×C53—44no—
109C2×D4×C53—44no—
110C2×D4×C53—82no—
111C2×D4×C53C22⋊C412yes—
112C2×D4×C53C22⋊C412yes—
113C22×C4×C53—24no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—24no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—44no—
122C22×C4×C53—42no—
123C22×C4×C53—82no—
124C22×C4×C53C22⋊C411yescharacteristic
125C22×C4×C53C22⋊C411yescharacteristic
126C22×C4×C53C22⋊C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D411yescharacteristic
131C2×C8×C53—44no—
132C2×C8×C53—42no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—24no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—44no—
137C4⋊C4×C53C22⋊C412yes—
138C22⋊C4×C53—24no—
139C22⋊C4×C53—24no—
140C22⋊C4×C53—24no—
141C22⋊C4×C53—24no—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—24no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—42no—
146C22⋊C4×C53—42no—
147C42×C53—21no—
148C42×C53—21no—
149C23×C53—24no—
150C23×C53—22no—
151C23×C53—21no—
152C23×C53—22no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—48no—
158C23×C53—42no—
159C23×C53—42no—
160C23×C53—42no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—44no—
168C23×C53—44no—
169C23×C53—84no—
170C23×C53—88no—
171C23×C53—84no—
172C23×C53—84no—
173C23×C53C2×C22⋊C411yescharacteristic
174C23×C53#12012yes—
175C23×C53#13512yes—
176C23×C53AGL(1,ℤ/8)11yescharacteristic
177C23×C53#13711yescharacteristic
178Q8×C53—42no—
179Q8×C53—42no—
180D4×C53—24no—
181D4×C53—22no—
182D4×C53—24no—
183D4×C53—48no—
184D4×C53—42no—
185D4×C53—48no—
186D4×C53—42no—
187D4×C53—42no—
188D4×C53—84no—
189D4×C53#9914yescomplemented
190C2×C4×C53—22no—
191C2×C4×C53—24no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—48no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—84no—
209C2×C4×C53—82no—
210C2×C4×C53—82no—
211C2×C4×C53—82no—
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53C2×C22⋊C411yescharacteristic
214C2×C4×C53#12012yes—
215C8×C53—82no—
216#8381C5311yesmaximal, characteristic, direct product factor, Sylow
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—21no—
220C22×C53—22no—
221C22×C53—24no—
222C22×C53—42no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—44no—
226C22×C53—44no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—41no—
230C22×C53—42no—
231C22×C53—41no—
232C22×C53—42no—
233C22×C53—41no—
234C22×C53—42no—
235C22×C53—88no—
236C22×C53—88no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—82no—
240C22×C53—82no—
241C22×C53#37911yescharacteristic
242C22×C53#41611yescharacteristic
243C22×C53#41711yescharacteristic
244C22×C53#44912yes—
245C22×C53#45212yes—
246C4×C53—22no—
247C4×C53—21no—
248C4×C53—42no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53—42no—
252C4×C53—41no—
253C4×C53—162no—
254C4×C53C2×#9912yes—
255C2×#546C2×C5311yescharacteristic
256#1799C2×C5311yescharacteristic, complemented
257#1786C2×C5312yescomplemented
258#1520C2×C5312yescomplemented
259#1459C2×C5311yescharacteristic, complemented
260C2×C53—22no—
261C2×C53—21no—
262C2×C53—21no—
263C2×C53—22no—
264C2×C53—44no—
265C2×C53—81no—
266C2×C53—82no—
267C2×C53#178111yescharacteristic
268C2×C53#178811yescharacteristic
269C2×C53#179511yescharacteristic
270C22×#41C4×C5311yescharacteristic, complemented
271C23×D4C22×C5311yescharacteristic
272#546—24no—
273#546—24no—
274C2×#124C4×C5311yescharacteristic, complemented
275C2×#123—21no—
276C2×C22⋊Q8C22×C5311yescharacteristic
277C2×#121—21no—
278C2×#121C4×C5311yescharacteristic, complemented
279C2×#120—21no—
280C2×C4×D4C4×C5311yescharacteristic, complemented
281C2×C4×D4C22×C5311yescharacteristic
282C22×C22⋊C4—21no—
283C2×#102—24no—
284C2×#99—21no—
285C2×C22⋊C8C22×C5312yes—
286#380C22×C5312yes—
287#342—21no—
288#328—24no—
289#324—21no—
290#323—21no—
291C53#838111yescharacteristic, direct product factor, Sylow
292C25—21no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—44no—
296C22×Q8C2×C4×C5311yescharacteristic
297C22×D4—21no—
298C22×D4—21no—
299C22×D4—21no—
300C22×D4—22no—
301C22×D4—44no—
302C22×D4C2×C4×C5311yescharacteristic
303C22×D4C2×C4×C5311yescharacteristic
304C22×D4D4×C5314yes—
305C22×D4D4×C5311yescharacteristic
306C23×C4—21no—
307C23×C4—21no—
308C23×C4C2×C4×C5311yescharacteristic
309C23×C4C23×C5311yescharacteristic
310C22×C8—22no—
311#124—22no—
312#124—22no—
313#123—28no—
314C22⋊Q8—22no—
315C22⋊Q8D4×C5314yes—
316#121—28no—
317#121—22no—
318#121—22no—
319#120—28no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C2×C4×C5311yescharacteristic
327C2×C4⋊C4D4×C5311yescharacteristic
328C2×C22⋊C4—28no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4D4×C5311yescharacteristic
336C2×C22⋊C4D4×C5311yescharacteristic
337C2×C42C2×C4×C5311yescharacteristic
338#102—48no—
339#99—42no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342#95—22no—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—44no—
347C24—42no—
348C24—41no—
349C24—41no—
350C24—82no—
351C24C22⋊C4×C5311yescharacteristic
352C24C2×D4×C5311yescharacteristic
353#41—48no—
354#41—44no—
355#41—44no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8C22⋊C4×C5312yes—
359C2×D4—24no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—24no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—82no—
376C2×D4C22⋊C4×C5312yes—
377C2×D4C22⋊C4×C5312yes—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—82no—
389C22×C4C22⋊C4×C5311yescharacteristic
390C22×C4C22⋊C4×C5311yescharacteristic
391C22×C4C22⋊C4×C5311yescharacteristic
392C22×C4C22×C4×C5311yescharacteristic
393C22×C4C2×D4×C5312yes—
394C22×C4C2×D4×C5312yes—
395C22×C4C2×D4×C5311yescharacteristic
396C2×C8—44no—
397C2×C8—42no—
398C4⋊C4—21no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—44no—
402C4⋊C4C22⋊C4×C5312yes—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C42—21no—
413C42—21no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—41no—
421C23—41no—
422C23—48no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—44no—
433C23—44no—
434C23—84no—
435C23—88no—
436C23—84no—
437C23—84no—
438C23C2×C22⋊C4×C5311yescharacteristic
439C23#120×C5312yes—
440C23#135×C5312yes—
441C23AGL(1,ℤ/8)×C5311yescharacteristic
442C23#137×C5311yescharacteristic
443Q8—42no—
444Q8—42no—
445D4—24no—
446D4—22no—
447D4—24no—
448D4—48no—
449D4—42no—
450D4—48no—
451D4—42no—
452D4—42no—
453D4—84no—
454D4#99×C5314yescomplemented
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4C2×C22⋊C4×C5311yescharacteristic
478C2×C4C2×C22⋊C4×C5311yescharacteristic
479C2×C4#120×C5312yes—
480C8—82no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—24no—
486C22—42no—
487C22—41no—
488C22—41no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—42no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—42no—
499C22—88no—
500C22—88no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22C53×#37911yescharacteristic
506C22C53×#41611yescharacteristic
507C22C53×#41711yescharacteristic
508C22C53×#44912yes—
509C22C53×#45212yes—
510C4—22no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—162no—
518C4C2×#99×C5312yes—
519C2—22no—
520C2—21no—
521C2—21no—
522C2—22no—
523C2—44no—
524C2—82no—
525C2—81no—
526C2C53×#178111yescharacteristic
527C2C53×#178811yescharacteristic
528C2C53×#179511yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×C53×#546C2
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
2. C53×#1799C2semidirect
3. C53×#1786C2semidirect
4. C53×#1520C2semidirect
5. C53×#1459C2semidirect
6. C22×#41×C53C4semidirect
10. C2×#124×C53C4semidirect
14. C2×#121×C53C4semidirect
16. C2×C4×D4×C53C4semidirect
189. D4×C53#99semidirect
216. #8381C53direct (N × H)
256. #1799C2×C53semidirect
257. #1786C2×C53semidirect
258. #1520C2×C53semidirect
259. #1459C2×C53semidirect
270. C22×#41C4×C53semidirect
274. C2×#124C4×C53semidirect
278. C2×#121C4×C53semidirect
280. C2×C4×D4C4×C53semidirect
291. C53#8381direct (N × H)
454. D4#99×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5462
C53×#17992
C53×#17862
C53×#15202
C53×#14592
#838153