Finite Group Databasebeta

← C53×#18987 #27226499

Subgroups — C53×#18987 #27226499

Order 13568.

904 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3529C211yesmaximal, characteristic
2C53×#3172C212yesmaximal, complemented
3C53×#3135C211yesmaximal, characteristic, complemented
4C53×#1664C212yesmaximal, complemented
5C53×#1662C212yesmaximal, complemented
6C53×#1661C212yesmaximal, complemented
7C53×#1659C212yesmaximal, complemented
8C53×#1525C211yesmaximal, characteristic, complemented
9C53×#1524C212yesmaximal, complemented
10C53×#563—21no—
11C53×#563C2211yescharacteristic
12C53×#562—22no—
13C53×#562C2212yes—
14C53×H(3,𝔽4)C2211yescharacteristic
15C53×#560—21no—
16C53×#560—22no—
17C53×#533C2211yescharacteristic, complemented
18C53×#532—21no—
19C53×#532C2211yescharacteristic
20C53×#529—22no—
21C53×#529C2212yes—
22C53×#525—21no—
23C53×#525—22no—
24C53×#501C2212yescomplemented
25C53×#497C2212yes—
26C53×#491—21no—
27C53×#491C2212yescomplemented
28C53×#490—21no—
29C53×#489C2212yes—
30C53×#489C2212yes—
31C53×#489C2212yescomplemented
32C53×#470—22no—
33C53×#468—22no—
34C4×Q16×C53—22no—
35C4×SD16×C53—22no—
36C53×#416—21no—
37C53×#416—21no—
38C4×M16×C53C2212yescomplemented
39C2×#97×C53C2211yescharacteristic, complemented
40C53×#330C2212yescomplemented
41C53×#330C2212yescomplemented
42C53×#329C2212yes—
43C53×#329C2212yes—
44C53×#329C2212yes—
45C53×#329C2211yescharacteristic, complemented
46C53×#329C2211yescharacteristic, complemented
47C2×#41×C53—21no—
48C2×#41×C53D411yescharacteristic
49C2×#41×C53D411yescharacteristic
50C23×C4×C53—21no—
51C2×Q16×C53—22no—
52C2×SD16×C53—22no—
53C2×M16×C53—22no—
54C22×C8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53D411yescharacteristic
57C4⋊Q8×C53C2312yes—
58#127×C53—21no—
59#127×C53D411yescharacteristic
60#126×C53—22no—
61#126×C53—22no—
62#126×C53—22no—
63#125×C53—22no—
64#125×C53—21no—
65#125×C53C2312yes—
66#124×C53—21no—
67#124×C53—22no—
68#124×C53—21no—
69#124×C53C2311yescharacteristic
70#124×C53C2312yes—
71#124×C53C2311yescharacteristic
72#123×C53—22no—
73#123×C53—21no—
74#123×C53—22no—
75#123×C53—22no—
76#123×C53—22no—
77#123×C53—21no—
78#123×C53—22no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53D412yes—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91#121×C53—22no—
92#121×C53D412yes—
93#120×C53—22no—
94#120×C53—21no—
95#120×C53—21no—
96#120×C53—22no—
97C4×Q8×C53—22no—
98C4×Q8×C53D412yes—
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53D412yes—
104#117×C53—21no—
105#117×C53—22no—
106#117×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109#107×C53—22no—
110#106×C53—22no—
111#103×C53—22no—
112#103×C53—22no—
113#103×C53—22no—
114#103×C53—22no—
115#103×C53—22no—
116#102×C53—22no—
117#102×C53—22no—
118#102×C53—22no—
119#97×C53—21no—
120#97×C53C2312yes—
121#97×C53C2312yes—
122#97×C53C2312yes—
123C4×C8×C53—22no—
124C24×C53—21no—
125#41×C53—42no—
126#41×C53—42no—
127#41×C53—82no—
128C2×Q8×C53—21no—
129C2×Q8×C53—22no—
130C2×Q8×C53C2×D411yescharacteristic
131C2×Q8×C53C2×D411yescharacteristic
132C2×Q8×C53C2×D412yes—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—21no—
136C2×D4×C53—22no—
137C2×D4×C53—22no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—22no—
141C2×D4×C53C2×D411yescharacteristic
142C2×D4×C53C2×D411yescharacteristic
143C2×D4×C53C2×D412yes—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—22no—
150C22×C4×C53—82no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D412yes—
153Q16×C53—44no—
154SD16×C53—44no—
155M16×C53—44no—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C2×C8×C53—22no—
159C2×C8×C53—42no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53C2×D412yes—
172C4⋊C4×C53C2×D412yes—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—41no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—22no—
190C42×C53—41no—
191C42×C53C2×D411yescharacteristic
192C42×C53C2×D412yes—
193C42×C53C2411yescharacteristic
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—22no—
197C23×C53—81no—
198C23×C53—82no—
199C23×C53#12011yescharacteristic
200Q8×C53—41no—
201Q8×C53—41no—
202Q8×C53—42no—
203Q8×C53—42no—
204Q8×C53—42no—
205Q8×C53—81no—
206D4×C53—41no—
207D4×C53—41no—
208D4×C53—42no—
209D4×C53—82no—
210D4×C53—82no—
211D4×C53—81no—
212D4×C53—81no—
213D4×C53—82no—
214D4×C53—82no—
215D4×C53—82no—
216D4×C53—81no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—44no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—82no—
229C2×C4×C53—82no—
230C2×C4×C53—81no—
231C2×C4×C53#12012yes—
232C2×C4×C53#12011yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×D412yes—
235C8×C53—24no—
236C8×C53—42no—
237C8×C53—42no—
238#18987C5311yesmaximal, characteristic, direct product factor, Sylow
239C22×C53—21no—
240C22×C53—22no—
241C22×C53—41no—
242C22×C53—42no—
243C22×C53—41no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—84no—
249C22×C53C2×#12011yescharacteristic
250C4×C53—21no—
251C4×C53—22no—
252C4×C53—22no—
253C4×C53—41no—
254C4×C53—42no—
255C4×C53—41no—
256C4×C53—42no—
257C4×C53—42no—
258C4×C53—82no—
259C4×C53#45512yes—
260#3529C2×C5311yescharacteristic
261#3172C2×C5312yescomplemented
262#3135C2×C5311yescharacteristic, complemented
263#1664C2×C5312yescomplemented
264#1662C2×C5312yescomplemented
265#1661C2×C5312yescomplemented
266#1659C2×C5312yescomplemented
267#1525C2×C5311yescharacteristic, complemented
268#1524C2×C5312yescomplemented
269C2×C53—41no—
270C2×C53—81no—
271C2×C53—81no—
272C2×C53—82no—
273C2×C53C2×#45511yescharacteristic
274C2×C53#301812yes—
275#563—21no—
276#563C22×C5311yescharacteristic
277#562—22no—
278#562C22×C5312yes—
279H(3,𝔽4)C22×C5311yescharacteristic
280#560—21no—
281#560—22no—
282#533C22×C5311yescharacteristic, complemented
283#532—21no—
284#532C22×C5311yescharacteristic
285#529—22no—
286#529C22×C5312yes—
287#525—21no—
288#525—22no—
289#501C22×C5312yescomplemented
290#497C22×C5312yes—
291#491—21no—
292#491C22×C5312yescomplemented
293#490—21no—
294#489C22×C5312yes—
295#489C22×C5312yes—
296#489C22×C5312yescomplemented
297#470—22no—
298#468—22no—
299C4×Q16—22no—
300C4×SD16—22no—
301#416—21no—
302#416—21no—
303C4×M16C22×C5312yescomplemented
304C2×#97C22×C5311yescharacteristic, complemented
305#330C22×C5312yescomplemented
306#330C22×C5312yescomplemented
307#329C22×C5312yes—
308#329C22×C5312yes—
309#329C22×C5312yes—
310#329C22×C5311yescharacteristic, complemented
311#329C22×C5311yescharacteristic, complemented
312C53#1898711yescharacteristic, direct product factor, Sylow
313C2×#41—21no—
314C2×#41D4×C5311yescharacteristic
315C2×#41D4×C5311yescharacteristic
316C23×C4—21no—
317C2×Q16—22no—
318C2×SD16—22no—
319C2×M16—22no—
320C22×C8—21no—
321C4⋊Q8—21no—
322C4⋊Q8D4×C5311yescharacteristic
323C4⋊Q8C23×C5312yes—
324#127—21no—
325#127D4×C5311yescharacteristic
326#126—22no—
327#126—22no—
328#126—22no—
329#125—22no—
330#125—21no—
331#125C23×C5312yes—
332#124—21no—
333#124—22no—
334#124—21no—
335#124C23×C5312yes—
336#124C23×C5311yescharacteristic
337#124C23×C5311yescharacteristic
338#123—22no—
339#123—21no—
340#123—22no—
341#123—21no—
342#123—22no—
343#123—22no—
344#123—22no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8D4×C5312yes—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121D4×C5312yes—
359#120—21no—
360#120—22no—
361#120—21no—
362#120—22no—
363C4×Q8—22no—
364C4×Q8D4×C5312yes—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4D4×C5312yes—
370#117—21no—
371#117—22no—
372#117—21no—
373C2×C42—21no—
374C2×C42C23×C5311yescharacteristic
375#107—22no—
376#106—22no—
377#103—22no—
378#103—22no—
379#103—22no—
380#103—22no—
381#103—22no—
382#102—22no—
383#102—22no—
384#102—22no—
385#97—21no—
386#97C23×C5312yes—
387#97C23×C5312yes—
388#97C23×C5312yes—
389C4×C8—22no—
390C24—21no—
391#41—42no—
392#41—42no—
393#41—82no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8C2×D4×C5311yescharacteristic
397C2×Q8C2×D4×C5311yescharacteristic
398C2×Q8C2×D4×C5312yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4C2×D4×C5311yescharacteristic
408C2×D4C2×D4×C5311yescharacteristic
409C2×D4C2×D4×C5312yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—82no—
417C22×C4C2×D4×C5311yescharacteristic
418C22×C4C2×D4×C5312yes—
419Q16—44no—
420SD16—44no—
421M16—44no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—42no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4C2×D4×C5312yes—
438C4⋊C4C2×D4×C5312yes—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—41no—
457C42C2×D4×C5312yes—
458C42C2×D4×C5311yescharacteristic
459C42C24×C5311yescharacteristic
460C23—21no—
461C23—22no—
462C23—21no—
463C23—81no—
464C23—82no—
465C23#120×C5311yescharacteristic
466Q8—42no—
467Q8—41no—
468Q8—41no—
469Q8—42no—
470Q8—42no—
471Q8—81no—
472D4—42no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—82no—
477D4—81no—
478D4—81no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—81no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4#120×C5312yes—
498C2×C4#120×C5311yescharacteristic
499C2×C4C22×D4×C5312yes—
500C2×C4C22×D4×C5311yescharacteristic
501C8—24no—
502C8—42no—
503C8—42no—
504C22—21no—
505C22—22no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22C2×#120×C5311yescharacteristic
515C4—21no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4C53×#45512yes—
525C2—41no—
526C2—81no—
527C2—82no—
528C2—81no—
529C2C2×C53×#45511yescharacteristic
530C2C53×#301812yes—
subgroupG/H
centerC2×C4×C53#120
commutator, Frattini, cosocleC42C24×C53
socleC22×C53C2×#120
prime radicalC53×#3529C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3172C2semidirect
3. C53×#3135C2semidirect
4. C53×#1664C2semidirect
5. C53×#1662C2semidirect
6. C53×#1661C2semidirect
7. C53×#1659C2semidirect
8. C53×#1525C2semidirect
9. C53×#1524C2semidirect
17. C53×#533C22semidirect
24. C53×#501C22semidirect
27. C53×#491C22semidirect
31. C53×#489C22semidirect
38. C4×M16×C53C22semidirect
39. C2×#97×C53C22semidirect
40. C53×#330C22semidirect
41. C53×#330C22semidirect
45. C53×#329C22semidirect
46. C53×#329C22semidirect
238. #18987C53direct (N × H)
261. #3172C2×C53semidirect
262. #3135C2×C53semidirect
263. #1664C2×C53semidirect
264. #1662C2×C53semidirect
265. #1661C2×C53semidirect
266. #1659C2×C53semidirect
267. #1525C2×C53semidirect
268. #1524C2×C53semidirect
282. #533C22×C53semidirect
289. #501C22×C53semidirect
292. #491C22×C53semidirect
296. #489C22×C53semidirect
303. C4×M16C22×C53semidirect
304. C2×#97C22×C53semidirect
305. #330C22×C53semidirect
306. #330C22×C53semidirect
310. #329C22×C53semidirect
311. #329C22×C53semidirect
312. C53#18987direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35292
C53×#31722
C53×#31352
C53×#16642
C53×#16622
C53×#16612
C53×#16592
C53×#15252
C53×#15242
#1898753