Finite Group Databasebeta

← C53×#8534 #27216046

Subgroups — C53×#8534 #27216046

Order 13568.

1408 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3484C211yesmaximal, characteristic
2C53×#1898C214yesmaximal, complemented
3C53×#1781C211yesmaximal, characteristic, complemented
4C53×#1438C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C53C411yescharacteristic, complemented
6C23×D4×C53C2211yescharacteristic
7C53×#546—24no—
8D42×C53—24no—
9C53×#535—21no—
10C53×#534—21no—
11C53×#534C412yescomplemented
12C53×#525—21no—
13C2×#120×C53—21no—
14C2×#120×C53C2212yes—
15C53×#518C411yescharacteristic, complemented
16C22×C22⋊C4×C53—21no—
17C22×C22⋊C4×C53C2212yescomplemented
18C2×#99×C53—21no—
19C2×#99×C53—21no—
20C2×#99×C53—24no—
21C53×#379C2212yescomplemented
22C2×#95×C53—21no—
23C53×#342—24no—
24C25×C53D411yescharacteristic
25C25×C53C2311yescharacteristic
26H(5,𝔽2)×C53—42no—
27H(5,𝔽2)×C53—42no—
28C2×#41×C53—21no—
29C2×#41×C53—22no—
30C22×D4×C53—21no—
31C22×D4×C53—22no—
32C22×D4×C53—24no—
33C22×D4×C53—22no—
34C22×D4×C53—42no—
35C22×D4×C53—42no—
36C22×D4×C53C2×C411yescharacteristic
37C22×D4×C53C2×C411yescharacteristic
38C22×D4×C53C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C22×D4×C53D414yes—
42C23×C4×C53—22no—
43C23×C4×C53D411yescharacteristic
44#127×C53—22no—
45#124×C53—22no—
46#123×C53—22no—
47#123×C53—22no—
48#123×C53—22no—
49C22⋊Q8×C53—22no—
50#121×C53—22no—
51#121×C53—24no—
52#121×C53—24no—
53#121×C53—24no—
54#120×C53—24no—
55#120×C53—22no—
56#120×C53—22no—
57#120×C53—22no—
58#120×C53—22no—
59#120×C53—22no—
60#120×C53—24no—
61C4×D4×C53—24no—
62C4×D4×C53—24no—
63#117×C53—21no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—24no—
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—44no—
70C2×C22⋊C4×C53—44no—
71C2×C22⋊C4×C53—42no—
72C2×C22⋊C4×C53C2×C412yes—
73C2×C22⋊C4×C53D414yes—
74#99×C53—42no—
75#99×C53—42no—
76#99×C53—48no—
77#95×C53—42no—
78C24×C53—21no—
79C24×C53—22no—
80C24×C53—22no—
81C24×C53—42no—
82C24×C53—41no—
83C24×C53—41no—
84C24×C53—41no—
85C24×C53—42no—
86C24×C53—42no—
87C24×C53—42no—
88C24×C53C22⋊C411yescharacteristic
89C24×C53C22⋊C411yescharacteristic
90C24×C53C22⋊C412yescomplemented
91C24×C53C22×C411yescharacteristic
92C24×C53C2×D411yescharacteristic
93C24×C53C2×D414yes—
94C24×C53C2×D411yescharacteristic
95#41×C53—42no—
96#41×C53—48no—
97#41×C53—42no—
98C2×Q8×C53—21no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—21no—
103C2×D4×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—24no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—24no—
111C2×D4×C53—24no—
112C2×D4×C53—22no—
113C2×D4×C53—22no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—416no—
119C2×D4×C53—48no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53—44no—
125C2×D4×C53—44no—
126C2×D4×C53—82no—
127C2×D4×C53—82no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—41no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53C22⋊C411yescharacteristic
143C22×C4×C53C22⋊C411yescharacteristic
144C22×C4×C53C22⋊C414yescomplemented
145C22×C4×C53C22⋊C412yescomplemented
146C4⋊C4×C53—22no—
147C4⋊C4×C53—24no—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—24no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—24no—
155C22⋊C4×C53—28no—
156C22⋊C4×C53—44no—
157C22⋊C4×C53—48no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—44no—
161C22⋊C4×C53—44no—
162C42×C53—22no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—24no—
174C23×C53—24no—
175C23×C53—22no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—44no—
180C23×C53—42no—
181C23×C53—44no—
182C23×C53—42no—
183C23×C53—44no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—44no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—44no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—82no—
195C23×C53—81no—
196C23×C53—82no—
197C23×C53—82no—
198C23×C53—81no—
199C23×C53—81no—
200C23×C53—82no—
201C23×C53—81no—
202C23×C53—81no—
203C23×C53—81no—
204C23×C53—82no—
205C23×C53#9912yescomplemented
206C23×C53C2×C22⋊C412yes—
207C23×C53C2×C22⋊C411yescharacteristic
208C23×C53#12012yes—
209C23×C53#12012yes—
210Q8×C53—42no—
211D4×C53—28no—
212D4×C53—44no—
213D4×C53—42no—
214D4×C53—44no—
215D4×C53—44no—
216D4×C53—44no—
217D4×C53—44no—
218D4×C53—42no—
219D4×C53—42no—
220D4×C53—44no—
221D4×C53—44no—
222D4×C53—84no—
223D4×C53—84no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—44no—
232C2×C4×C53—44no—
233C2×C4×C53—42no—
234C2×C4×C53—44no—
235C2×C4×C53—42no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—44no—
239C2×C4×C53—82no—
240C2×C4×C53—82no—
241C2×C4×C53—82no—
242C2×C4×C53—82no—
243C2×C4×C53—82no—
244C2×C4×C53—82no—
245C2×C4×C53—84no—
246C2×C4×C53#9912yescomplemented
247#8534C5311yesmaximal, characteristic, direct product factor, Sylow
248C22×C53—21no—
249C22×C53—22no—
250C22×C53—21no—
251C22×C53—22no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—41no—
267C22×C53—44no—
268C22×C53—42no—
269C22×C53—84no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—84no—
276C22×C53—84no—
277C22×C53—84no—
278C22×C53—81no—
279C22×C53—162no—
280C22×C53—162no—
281C22×C53#37911yescharacteristic
282C22×C53C2×#9911yescharacteristic
283C22×C53C2×#9911yescharacteristic
284C22×C53B(4,𝔽2)12yes—
285C22×C53#45812yes—
286C4×C53—22no—
287C4×C53—41no—
288C4×C53—42no—
289C4×C53—44no—
290C4×C53—84no—
291C4×C53—162no—
292#3484C2×C5311yescharacteristic
293#1898C2×C5314yescomplemented
294#1781C2×C5311yescharacteristic, complemented
295#1438C2×C5311yescharacteristic, complemented
296C2×C53—22no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—22no—
300C2×C53—42no—
301C2×C53—42no—
302C2×C53—42no—
303C2×C53—81no—
304C2×C53—82no—
305C2×C53#178111yescharacteristic
306C2×C53#179012yes—
307C2×H(5,𝔽2)C4×C5311yescharacteristic, complemented
308C23×D4C22×C5311yescharacteristic
309#546—24no—
310D42—24no—
311#535—21no—
312#534—21no—
313#534C4×C5312yescomplemented
314#525—21no—
315C2×#120—21no—
316C2×#120C22×C5312yes—
317#518C4×C5311yescharacteristic, complemented
318C22×C22⋊C4—21no—
319C22×C22⋊C4C22×C5312yescomplemented
320C2×#99—24no—
321C2×#99—21no—
322C2×#99—21no—
323#379C22×C5312yescomplemented
324C2×#95—21no—
325#342—24no—
326C53#853411yescharacteristic, direct product factor, Sylow
327C25D4×C5311yescharacteristic
328C25C23×C5311yescharacteristic
329H(5,𝔽2)—42no—
330H(5,𝔽2)—42no—
331C2×#41—21no—
332C2×#41—22no—
333C22×D4—24no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—42no—
338C22×D4—42no—
339C22×D4C2×C4×C5311yescharacteristic
340C22×D4C2×C4×C5311yescharacteristic
341C22×D4C2×C4×C5312yes—
342C22×D4D4×C5311yescharacteristic
343C22×D4D4×C5311yescharacteristic
344C22×D4D4×C5314yes—
345C23×C4—22no—
346C23×C4D4×C5311yescharacteristic
347#127—22no—
348#124—22no—
349#123—22no—
350#123—22no—
351#123—22no—
352C22⋊Q8—22no—
353#121—22no—
354#121—24no—
355#121—24no—
356#121—24no—
357#120—24no—
358#120—22no—
359#120—22no—
360#120—22no—
361#120—24no—
362#120—22no—
363#120—22no—
364C4×D4—24no—
365C4×D4—24no—
366#117—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4C2×C4×C5312yes—
376C2×C22⋊C4D4×C5314yes—
377#99—48no—
378#99—42no—
379#99—42no—
380#95—42no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22⋊C4×C5311yescharacteristic
392C24C22⋊C4×C5311yescharacteristic
393C24C22⋊C4×C5312yescomplemented
394C24C22×C4×C5311yescharacteristic
395C24C2×D4×C5311yescharacteristic
396C24C2×D4×C5314yes—
397C24C2×D4×C5311yescharacteristic
398#41—42no—
399#41—48no—
400#41—42no—
401C2×Q8—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C22⋊C4×C5311yescharacteristic
446C22×C4C22⋊C4×C5311yescharacteristic
447C22×C4C22⋊C4×C5314yescomplemented
448C22×C4C22⋊C4×C5312yescomplemented
449C4⋊C4—22no—
450C4⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—82no—
498C23—81no—
499C23—82no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—82no—
508C23#99×C5312yescomplemented
509C23C2×C22⋊C4×C5312yes—
510C23C2×C22⋊C4×C5311yescharacteristic
511C23#120×C5312yes—
512C23#120×C5312yes—
513Q8—42no—
514D4—28no—
515D4—44no—
516D4—42no—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—84no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4#99×C5312yescomplemented
550C22—21no—
551C22—22no—
552C22—21no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—41no—
569C22—44no—
570C22—42no—
571C22—84no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—81no—
581C22—162no—
582C22—162no—
583C22C53×#37911yescharacteristic
584C22C2×#99×C5311yescharacteristic
585C22C2×#99×C5311yescharacteristic
586C22C53×B(4,𝔽2)12yes—
587C22C53×#45812yes—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—44no—
592C4—84no—
593C4—162no—
594C2—22no—
595C2—21no—
596C2—21no—
597C2—22no—
598C2—42no—
599C2—42no—
600C2—42no—
601C2—81no—
602C2—82no—
603C2C53×#178111yescharacteristic
604C2C53×#179012yes—
subgroupG/H
center, socleC22×C53#379
commutatorC24C22×C4×C53
Frattini, cosocleC25C23×C53
prime radicalC53×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1898C2semidirect
3. C53×#1781C2semidirect
4. C53×#1438C2semidirect
5. C2×H(5,𝔽2)×C53C4semidirect
11. C53×#534C4semidirect
15. C53×#518C4semidirect
17. C22×C22⋊C4×C53C22semidirect
21. C53×#379C22semidirect
90. C24×C53C22⋊C4semidirect
144. C22×C4×C53C22⋊C4semidirect
145. C22×C4×C53C22⋊C4semidirect
205. C23×C53#99semidirect
246. C2×C4×C53#99semidirect
247. #8534C53direct (N × H)
293. #1898C2×C53semidirect
294. #1781C2×C53semidirect
295. #1438C2×C53semidirect
307. C2×H(5,𝔽2)C4×C53semidirect
313. #534C4×C53semidirect
317. #518C4×C53semidirect
319. C22×C22⋊C4C22×C53semidirect
323. #379C22×C53semidirect
326. C53#8534direct (N × H)
393. C24C22⋊C4×C53semidirect
447. C22×C4C22⋊C4×C53semidirect
448. C22×C4C22⋊C4×C53semidirect
508. C23#99×C53semidirect
549. C2×C4#99×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#34842
C53×#18982
C53×#17812
C53×#14382
#853453