Finite Group Databasebeta

← C53×#18343 #27225855

Subgroups — C53×#18343 #27225855

Order 13568.

1034 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3459C211yesmaximal, characteristic
2C53×#3239C212yesmaximal, complemented
3C53×#2542C211yesmaximal, characteristic, complemented
4C53×#1618C214yesmaximal, complemented
5C53×#1614C212yesmaximal, complemented
6C53×#1613C212yesmaximal, complemented
7C53×#1509C212yesmaximal, complemented
8C53×#1466C211yesmaximal, characteristic, complemented
9C53×#543C2211yescharacteristic
10C53×#539—22no—
11C53×#539—21no—
12C53×#538—22no—
13C53×#538C2212yes—
14C53×#534—21no—
15C53×#534C2211yescharacteristic
16C2×C4⋊Q8×C53C2211yescharacteristic, complemented
17C53×#529—22no—
18C53×#529C2212yes—
19C53×#525—21no—
20C53×#525—22no—
21C2×C22⋊Q8×C53—21no—
22C2×C22⋊Q8×C53—21no—
23C2×C4×D4×C53—21no—
24C2×C4×D4×C53C2211yescharacteristic
25C53×#501—22no—
26C53×#489—22no—
27C53×#484C2214yescomplemented
28C53×#481C2214yes—
29C53×#475—22no—
30C53×#474—22no—
31C53×#432C2212yescomplemented
32C53×#399C2212yes—
33C53×#399C2212yescomplemented
34C53×#390—21no—
35C53×#384—21no—
36C4×C4⋊C4×C53C2211yescharacteristic, complemented
37C53×#328C2214yescomplemented
38C53×#327C2214yes—
39C53×#323C2212yes—
40C53×#323C2212yescomplemented
41C2×#41×C53—21no—
42C22×Q8×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53D411yescharacteristic
45C23×C4×C53—21no—
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C4⋊Q8×C53—22no—
49C4⋊Q8×C53—22no—
50#126×C53—22no—
51#126×C53—22no—
52#126×C53—22no—
53#125×C53—22no—
54#125×C53D412yes—
55#124×C53—22no—
56#124×C53D412yes—
57#123×C53—22no—
58#123×C53—22no—
59#123×C53—22no—
60#123×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—44no—
69C22⋊Q8×C53—44no—
70C22⋊Q8×C53C2312yes—
71#121×C53—22no—
72#121×C53—21no—
73#121×C53—22no—
74#121×C53—22no—
75#121×C53—21no—
76#121×C53C2312yes—
77#120×C53—22no—
78#120×C53—22no—
79#120×C53—22no—
80C4×Q8×C53D412yes—
81C4×D4×C53—22no—
82C4×D4×C53—22no—
83C4×D4×C53—22no—
84C4×D4×C53—22no—
85C4×D4×C53—22no—
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88C4×D4×C53D412yes—
89#117×C53—21no—
90#117×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C4⋊C4×C53C2312yes—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53C2311yescharacteristic
104C2×C42×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107#107×C53—24no—
108#106×C53—24no—
109C4⋊C8×C53—22no—
110#103×C53—24no—
111#102×C53—24no—
112C22⋊C8×C53C2314yes—
113#97×C53—22no—
114#95×C53C2312yes—
115C24×C53—21no—
116C24×C53C2×D411yescharacteristic
117C24×C53C2×D411yescharacteristic
118#41×C53—82no—
119C2×Q8×C53—21no—
120C2×Q8×C53—42no—
121C2×Q8×C53—42no—
122C2×Q8×C53—41no—
123C2×Q8×C53—41no—
124C2×Q8×C53C2×D412yes—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—22no—
132C2×D4×C53—41no—
133C2×D4×C53—42no—
134C2×D4×C53—42no—
135C2×D4×C53C2×D412yes—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—82no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C2×C8×C53—24no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
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—24no—
167C4⋊C4×C53—44no—
168C4⋊C4×C53—44no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53C2×D412yes—
171C4⋊C4×C53C2×D412yes—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
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—44no—
185C22⋊C4×C53—44no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—42no—
189C42×C53C2×D412yes—
190C42×C53C2×D412yes—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—22no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—81no—
203C23×C53—81no—
204C23×C53#13712yes—
205C23×C53C22×D411yescharacteristic
206Q8×C53—42no—
207Q8×C53—82no—
208Q8×C53—81no—
209Q8×C53—82no—
210D4×C53—44no—
211D4×C53—42no—
212D4×C53—44no—
213D4×C53—42no—
214D4×C53—42no—
215D4×C53—82no—
216D4×C53—81no—
217D4×C53—82no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—44no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—81no—
240C2×C4×C53—81no—
241C2×C4×C53—81no—
242C2×C4×C53—82no—
243C2×C4×C53—81no—
244C2×C4×C53#12011yescharacteristic
245C2×C4×C53#12012yes—
246C2×C4×C53#12011yescharacteristic
247C2×C4×C53C22×D412yes—
248C8×C53—44no—
249#18343C5311yesmaximal, characteristic, direct product factor, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—22no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—42no—
259C22×C53—41no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—41no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—84no—
266C22×C53C2×#12011yescharacteristic
267C22×C53C2×#13711yescharacteristic
268C22×C53#57511yescharacteristic
269C4×C53—22no—
270C4×C53—21no—
271C4×C53—22no—
272C4×C53—44no—
273C4×C53—42no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—42no—
277C4×C53—42no—
278C4×C53—82no—
279C4×C53#45312yes—
280#3459C2×C5311yescharacteristic
281#3239C2×C5312yescomplemented
282#2542C2×C5311yescharacteristic, complemented
283#1618C2×C5314yescomplemented
284#1614C2×C5312yescomplemented
285#1613C2×C5312yescomplemented
286#1509C2×C5312yescomplemented
287#1466C2×C5311yescharacteristic, complemented
288C2×C53—21no—
289C2×C53—21no—
290C2×C53—42no—
291C2×C53—81no—
292C2×C53—82no—
293C2×C53#300711yescharacteristic
294C2×C53C2×#45311yescharacteristic
295C2×C53#302811yescharacteristic
296#543C22×C5311yescharacteristic
297#539—22no—
298#539—21no—
299#538—22no—
300#538C22×C5312yes—
301#534—21no—
302#534C22×C5311yescharacteristic
303C2×C4⋊Q8C22×C5311yescharacteristic, complemented
304#529—22no—
305#529C22×C5312yes—
306#525—21no—
307#525—22no—
308C2×C22⋊Q8—21no—
309C2×C22⋊Q8—21no—
310C2×C4×D4—21no—
311C2×C4×D4C22×C5311yescharacteristic
312#501—22no—
313#489—22no—
314#484C22×C5314yescomplemented
315#481C22×C5314yes—
316#475—22no—
317#474—22no—
318#432C22×C5312yescomplemented
319#399C22×C5312yes—
320#399C22×C5312yescomplemented
321#390—21no—
322#384—21no—
323C4×C4⋊C4C22×C5311yescharacteristic, complemented
324#328C22×C5314yescomplemented
325#327C22×C5314yes—
326#323C22×C5312yes—
327#323C22×C5312yescomplemented
328C53#1834311yescharacteristic, direct product factor, Sylow
329C2×#41—21no—
330C22×Q8—21no—
331C22×D4—21no—
332C22×D4D4×C5311yescharacteristic
333C23×C4—21no—
334C23×C4D4×C5311yescharacteristic
335C23×C4D4×C5311yescharacteristic
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338#126—22no—
339#126—22no—
340#126—22no—
341#125—22no—
342#125D4×C5312yes—
343#124—22no—
344#124D4×C5312yes—
345#123—22no—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350#123—22no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8—44no—
357C22⋊Q8—44no—
358C22⋊Q8C23×C5312yes—
359#121—22no—
360#121—21no—
361#121—22no—
362#121—22no—
363#121—21no—
364#121C23×C5312yes—
365#120—22no—
366#120—22no—
367#120—22no—
368C4×Q8D4×C5312yes—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4D4×C5312yes—
377#117—21no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4D4×C5311yescharacteristic
385C2×C4⋊C4C23×C5312yes—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4C23×C5311yescharacteristic
391C2×C22⋊C4C23×C5311yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C23×C5311yescharacteristic
395#107—24no—
396#106—24no—
397C4⋊C8—22no—
398#103—24no—
399#102—24no—
400C22⋊C8C23×C5314yes—
401#97—22no—
402#95C23×C5312yes—
403C24—21no—
404C24C2×D4×C5311yescharacteristic
405C24C2×D4×C5311yescharacteristic
406#41—82no—
407C2×Q8—21no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—41no—
411C2×Q8—41no—
412C2×Q8C2×D4×C5312yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C2×D4×C5312yes—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4C2×D4×C5311yescharacteristic
439C22×C4C2×D4×C5311yescharacteristic
440C22×C4C2×D4×C5311yescharacteristic
441C22×C4C2×D4×C5311yescharacteristic
442C22×C4C24×C5311yescharacteristic
443C2×C8—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4C2×D4×C5312yes—
459C4⋊C4C2×D4×C5312yes—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C42—22no—
475C42—22no—
476C42—42no—
477C42C2×D4×C5312yes—
478C42C2×D4×C5312yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—81no—
491C23—81no—
492C23#137×C5312yes—
493C23C22×D4×C5311yescharacteristic
494Q8—42no—
495Q8—82no—
496Q8—81no—
497Q8—82no—
498D4—44no—
499D4—42no—
500D4—44no—
501D4—42no—
502D4—42no—
503D4—82no—
504D4—81no—
505D4—82no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4#120×C5312yes—
533C2×C4#120×C5311yescharacteristic
534C2×C4#120×C5311yescharacteristic
535C2×C4C22×D4×C5312yes—
536C8—44no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—84no—
553C22C2×#120×C5311yescharacteristic
554C22C2×#137×C5311yescharacteristic
555C22C53×#57511yescharacteristic
556C4—22no—
557C4—21no—
558C4—22no—
559C4—44no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4C53×#45312yes—
567C2—21no—
568C2—21no—
569C2—42no—
570C2—81no—
571C2—82no—
572C2C53×#300711yescharacteristic
573C2C2×C53×#45311yescharacteristic
574C2C53×#302811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3459C2
LCSG ▷ C22×C4 ▷ 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×#3239C2semidirect
3. C53×#2542C2semidirect
4. C53×#1618C2semidirect
5. C53×#1614C2semidirect
6. C53×#1613C2semidirect
7. C53×#1509C2semidirect
8. C53×#1466C2semidirect
16. C2×C4⋊Q8×C53C22semidirect
27. C53×#484C22semidirect
31. C53×#432C22semidirect
33. C53×#399C22semidirect
36. C4×C4⋊C4×C53C22semidirect
37. C53×#328C22semidirect
40. C53×#323C22semidirect
249. #18343C53direct (N × H)
281. #3239C2×C53semidirect
282. #2542C2×C53semidirect
283. #1618C2×C53semidirect
284. #1614C2×C53semidirect
285. #1613C2×C53semidirect
286. #1509C2×C53semidirect
287. #1466C2×C53semidirect
303. C2×C4⋊Q8C22×C53semidirect
314. #484C22×C53semidirect
318. #432C22×C53semidirect
320. #399C22×C53semidirect
323. C4×C4⋊C4C22×C53semidirect
324. #328C22×C53semidirect
327. #323C22×C53semidirect
328. C53#18343direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34592
C53×#32392
C53×#25422
C53×#16182
C53×#16142
C53×#16132
C53×#15092
C53×#14662
#1834353