Finite Group Databasebeta

← C53×#9117 #27216629

Subgroups — C53×#9117 #27216629

Order 13568.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#407C211yesmaximal, characteristic, complemented
2C53×#2847C211yesmaximal, characteristic
3C2×C53×#327C212yesmaximal, complemented
4C53×C22⋊#95C211yesmaximal, characteristic, complemented
5C53×#1294C212yesmaximal, complemented
6C24×C4×C53C2211yescharacteristic
7C22×M16×C53—21no—
8C2×#121×C53C411yescharacteristic, complemented
9C2×#121×C53C411yescharacteristic, complemented
10C2×#121×C53C2211yescharacteristic
11C2×#120×C53—21no—
12C22×C22⋊C4×C53—21no—
13C53×#407—24no—
14C2×C22⋊C8×C53—21no—
15C2×C22⋊C8×C53C2212yes—
16C53×#386—21no—
17C53×#386—21no—
18C53×#386—21no—
19C53×#380C411yescharacteristic, complemented
20C53×#380C411yescharacteristic, complemented
21C53×#380C2211yescharacteristic
22C2×#95×C53—21no—
23C2×#95×C53C2212yes—
24C53×#327—24no—
25C25×C53D411yescharacteristic
26C22×D4×C53—21no—
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53C2×C411yescharacteristic
30C23×C4×C53—22no—
31C23×C4×C53—22no—
32C23×C4×C53—22no—
33C23×C4×C53—42no—
34C23×C4×C53—42no—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C2×M16×C53—24no—
42C2×M16×C53—42no—
43C2×M16×C53—42no—
44C22×C8×C53—22no—
45#121×C53—22no—
46#121×C53—22no—
47#121×C53—42no—
48#121×C53—42no—
49#120×C53—44no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—24no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—42no—
61C2×C22⋊C4×C53—42no—
62C22⋊C8×C53—24no—
63C22⋊C8×C53—22no—
64C22⋊C8×C53D414yescomplemented
65#95×C53—21no—
66#95×C53—21no—
67#95×C53—21no—
68#95×C53—22no—
69#95×C53—22no—
70#95×C53—42no—
71C24×C53—22no—
72C24×C53—22no—
73C24×C53—21no—
74C24×C53—21no—
75C24×C53—22no—
76C24×C53—22no—
77C24×C53—21no—
78C24×C53—41no—
79C24×C53—41no—
80C24×C53C22⋊C411yescharacteristic
81C24×C53C22⋊C411yescharacteristic
82C24×C53C2×D411yescharacteristic
83C2×D4×C53—22no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—42no—
93C2×D4×C53—42no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—24no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—48no—
120C22×C4×C53—48no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53C22⋊C411yescharacteristic
128C22×C4×C53C22⋊C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D412yes—
132C22×C4×C53#4112yes—
133M16×C53—48no—
134C2×C8×C53—24no—
135C2×C8×C53—42no—
136C2×C8×C53—42no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C22⋊C4×C53—42no—
141C22⋊C4×C53—48no—
142C22⋊C4×C53—42no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—42no—
145C22⋊C4×C53—42no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—24no—
153C23×C53—24no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—44no—
167C23×C53—44no—
168C23×C53—44no—
169C23×C53—44no—
170C23×C53—42no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—42no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53#9912yes—
182C23×C53#10412yes—
183C23×C53C2×C22⋊C411yescharacteristic
184C23×C53#12112yes—
185C23×C53AGL(1,ℤ/8)11yescharacteristic
186C23×C53AGL(1,ℤ/8)12yes—
187C23×C53#13711yescharacteristic
188D4×C53—82no—
189D4×C53—84no—
190D4×C53—84no—
191D4×C53—82no—
192D4×C53—82no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—44no—
207C2×C4×C53—44no—
208C2×C4×C53—44no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—44no—
212C2×C4×C53—44no—
213C2×C4×C53—44no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—82no—
224C2×C4×C53—82no—
225C2×C4×C53—82no—
226C2×C4×C53—82no—
227C2×C4×C53#9912yes—
228C2×C4×C53C4×D412yes—
229C2×C4×C53#12011yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C8×C53—44no—
232#9117C5311yesmaximal, characteristic, direct product factor, Sylow
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—48no—
248C22×C53—48no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53#38611yescharacteristic
265C22×C53C2×#9911yescharacteristic
266C22×C53C2×#9911yescharacteristic
267C22×C53#41711yescharacteristic
268C22×C53#41811yescharacteristic
269C22×C53C2×#10411yescharacteristic
270C22×C53#42111yescharacteristic
271C22×C53#46812yes—
272C22×C53#46912yes—
273C4×C53—21no—
274C4×C53—22no—
275C4×C53—21no—
276C4×C53—22no—
277C4×C53—41no—
278C4×C53—84no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53—81no—
282C2×#407C2×C5311yescharacteristic, complemented
283#2847C2×C5311yescharacteristic
284C2×#327C2×C5312yescomplemented
285C22⋊#95C2×C5311yescharacteristic, complemented
286#1294C2×C5312yescomplemented
287C2×C53—22no—
288C2×C53—22no—
289C2×C53—42no—
290C2×C53—42no—
291C2×C53—161no—
292C2×C53#150411yescharacteristic
293C2×C53#150711yescharacteristic
294C2×C53#151011yescharacteristic
295C2×C53#151311yescharacteristic
296C2×C53#187111yescharacteristic
297C2×C53#189511yescharacteristic
298C2×C53#189811yescharacteristic
299C24×C4C22×C5311yescharacteristic
300C22×M16—21no—
301C2×#121C4×C5311yescharacteristic, complemented
302C2×#121C4×C5311yescharacteristic, complemented
303C2×#121C22×C5311yescharacteristic
304C2×#120—21no—
305C22×C22⋊C4—21no—
306#407—24no—
307C2×C22⋊C8—21no—
308C2×C22⋊C8C22×C5312yes—
309#386—21no—
310#386—21no—
311#386—21no—
312#380C4×C5311yescharacteristic, complemented
313#380C4×C5311yescharacteristic, complemented
314#380C22×C5311yescharacteristic
315C2×#95—21no—
316C2×#95C22×C5312yes—
317#327—24no—
318C53#911711yescharacteristic, direct product factor, Sylow
319C25D4×C5311yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C2×C4×C5311yescharacteristic
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C2×C4×C5311yescharacteristic
330C23×C4C2×C4×C5311yescharacteristic
331C23×C4D4×C5311yescharacteristic
332C23×C4D4×C5311yescharacteristic
333C23×C4D4×C5311yescharacteristic
334C23×C4C23×C5311yescharacteristic
335C2×M16—24no—
336C2×M16—42no—
337C2×M16—42no—
338C22×C8—22no—
339#121—22no—
340#121—22no—
341#121—42no—
342#121—42no—
343#120—44no—
344C2×C4⋊C4C2×C4×C5311yescharacteristic
345C2×C4⋊C4C2×C4×C5311yescharacteristic
346C2×C4⋊C4C2×C4×C5311yescharacteristic
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8D4×C5314yescomplemented
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—42no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—41no—
373C24—41no—
374C24C22⋊C4×C5311yescharacteristic
375C24C22⋊C4×C5311yescharacteristic
376C24C2×D4×C5311yescharacteristic
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C5311yescharacteristic
422C22×C4C22⋊C4×C5311yescharacteristic
423C22×C4C22×C4×C5311yescharacteristic
424C22×C4C2×D4×C5311yescharacteristic
425C22×C4C2×D4×C5312yes—
426C22×C4#41×C5312yes—
427M16—48no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—48no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#99×C5312yes—
476C23#104×C5312yes—
477C23C2×C22⋊C4×C5311yescharacteristic
478C23#121×C5312yes—
479C23AGL(1,ℤ/8)×C5311yescharacteristic
480C23AGL(1,ℤ/8)×C5312yes—
481C23#137×C5311yescharacteristic
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#99×C5312yes—
522C2×C4C4×D4×C5312yes—
523C2×C4#120×C5311yescharacteristic
524C2×C4#123×C5311yescharacteristic
525C8—44no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C53×#38611yescharacteristic
558C22C2×#99×C5311yescharacteristic
559C22C2×#99×C5311yescharacteristic
560C22C53×#41711yescharacteristic
561C22C53×#41811yescharacteristic
562C22C2×#104×C5311yescharacteristic
563C22C53×#42111yescharacteristic
564C22C53×#46812yes—
565C22C53×#46912yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2C53×#150411yescharacteristic
581C2C53×#150711yescharacteristic
582C2C53×#151011yescharacteristic
583C2C53×#151311yescharacteristic
584C2C53×#187111yescharacteristic
585C2C53×#189511yescharacteristic
586C2C53×#189811yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC53×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C24×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#407C2semidirect
3. C2×C53×#327C2semidirect
4. C53×C22⋊#95C2semidirect
5. C53×#1294C2semidirect
8. C2×#121×C53C4semidirect
9. C2×#121×C53C4semidirect
19. C53×#380C4semidirect
20. C53×#380C4semidirect
64. C22⋊C8×C53D4semidirect
232. #9117C53direct (N × H)
282. C2×#407C2×C53semidirect
284. C2×#327C2×C53semidirect
285. C22⋊#95C2×C53semidirect
286. #1294C2×C53semidirect
301. C2×#121C4×C53semidirect
302. C2×#121C4×C53semidirect
312. #380C4×C53semidirect
313. #380C4×C53semidirect
318. C53#9117direct (N × H)
358. C22⋊C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4072
C53×#28472
C2×C53×#3272
C53×C22⋊#952
C53×#12942
#911753