Finite Group Databasebeta

← C53×#12690 #27220202

Subgroups — C53×#12690 #27220202

Order 13568.

1228 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#534C211yesmaximal, characteristic
2C2×C53×#352C212yesmaximal, complemented
3C2×C53×AΣL(1,𝔽16)C212yesmaximal, complemented
4C53×#1783C211yesmaximal, characteristic, complemented
5C53×#1781C211yesmaximal, characteristic, complemented
6C23×D4×C53C411yescharacteristic, complemented
7C23×D4×C53C2211yescharacteristic
8C53×#534—24no—
9C53×#534—24no—
10C2×#123×C53—21no—
11C2×#123×C53C411yescharacteristic, complemented
12C2×#123×C53C2211yescharacteristic
13C2×#121×C53—21no—
14C2×#121×C53—21no—
15C2×#120×C53—21no—
16C2×#120×C53C411yescharacteristic, complemented
17C2×#120×C53C2211yescharacteristic
18C22×C22⋊C4×C53C411yescharacteristic, complemented
19C2×#100×C53—21no—
20C2×#100×C53C2212yes—
21C2×#99×C53—21no—
22C2×#99×C53C2212yes—
23C53×#407—21no—
24C53×#379—21no—
25C53×#352—24no—
26C53×AΣL(1,𝔽16)—24no—
27C25×C53C2×C411yescharacteristic
28C25×C53C2×C411yescharacteristic
29C25×C53D411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53—22no—
33C22×D4×C53—21no—
34C22×D4×C53—21no—
35C22×D4×C53—22no—
36C22×D4×C53—22no—
37C22×D4×C53—22no—
38C22×D4×C53—42no—
39C22×D4×C53—42no—
40C22×D4×C53—42no—
41C22×D4×C53—42no—
42C22×D4×C53C2×C411yescharacteristic
43C22×D4×C53D411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C22×D4×C53C2311yescharacteristic
46C23×C4×C53—21no—
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—22no—
50#123×C53—28no—
51#123×C53—22no—
52#123×C53—22no—
53#123×C53—22no—
54#123×C53D414yes—
55#121×C53—28no—
56#121×C53—28no—
57#120×C53—28no—
58#120×C53—22no—
59#120×C53—22no—
60#120×C53—22no—
61#120×C53D414yes—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—42no—
73C2×C22⋊C4×C53C2×C411yescharacteristic
74C2×C22⋊C4×C53C2×C411yescharacteristic
75#100×C53—22no—
76#100×C53—22no—
77#100×C53—42no—
78#99×C53—22no—
79#99×C53—22no—
80#99×C53—42no—
81C22⋊C8×C53—42no—
82C24×C53—22no—
83C24×C53—22no—
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—21no—
87C24×C53—22no—
88C24×C53—21no—
89C24×C53—22no—
90C24×C53—41no—
91C24×C53—41no—
92C24×C53—41no—
93C24×C53—82no—
94C24×C53—82no—
95C24×C53—82no—
96C24×C53C22⋊C411yescharacteristic
97C24×C53C22⋊C411yescharacteristic
98C24×C53C22⋊C411yescharacteristic
99C24×C53C22⋊C412yescomplemented
100C24×C53C22×C411yescharacteristic
101C24×C53C2×D411yescharacteristic
102C2×D4×C53—24no—
103C2×D4×C53—24no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—22no—
109C2×D4×C53—24no—
110C2×D4×C53—24no—
111C2×D4×C53—22no—
112C2×D4×C53—41no—
113C2×D4×C53—41no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—48no—
119C2×D4×C53—48no—
120C2×D4×C53—48no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—48no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—82no—
127C2×D4×C53—82no—
128C2×D4×C53C2×D412yes—
129C2×D4×C53C2×D412yes—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—42no—
141C22×C4×C53—82no—
142C22×C4×C53C22⋊C411yescharacteristic
143C22×C4×C53C22⋊C412yescomplemented
144C22×C4×C53C2×D411yescharacteristic
145M16×C53—42no—
146M16×C53—42no—
147C2×C8×C53—42no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—24no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—24no—
152C22⋊C4×C53—24no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C22⋊C4×C53—24no—
156C22⋊C4×C53—24no—
157C22⋊C4×C53—42no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—84no—
160C22⋊C4×C53C22⋊C412yescomplemented
161C22⋊C4×C53C22⋊C412yescomplemented
162C23×C53—24no—
163C23×C53—24no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—22no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—42no—
177C23×C53—44no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—44no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—82no—
190C23×C53—82no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53—84no—
194C23×C53—82no—
195C23×C53—82no—
196C23×C53—82no—
197C23×C53—84no—
198C23×C53—84no—
199C23×C53—84no—
200C23×C53—84no—
201C23×C53#9912yes—
202C23×C53#9912yes—
203C23×C53C2×C22⋊C411yescharacteristic
204C23×C53C2×C22⋊C411yescharacteristic
205C23×C53C2×C22⋊C411yescharacteristic
206C23×C53#12012yes—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—44no—
210D4×C53—44no—
211D4×C53—42no—
212D4×C53—42no—
213D4×C53—48no—
214D4×C53—48no—
215D4×C53—84no—
216D4×C53—84no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—44no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—84no—
233C2×C4×C53—82no—
234C2×C4×C53—84no—
235C2×C4×C53#12012yes—
236C8×C53—82no—
237#12690C5311yesmaximal, characteristic, direct product factor, Sylow
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—42no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—41no—
262C22×C53—42no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—82no—
266C22×C53—84no—
267C22×C53—84no—
268C22×C53—88no—
269C22×C53—84no—
270C22×C53—84no—
271C22×C53—82no—
272C22×C53#37911yescharacteristic
273C22×C53C2×#9911yescharacteristic
274C22×C53C2×#9911yescharacteristic
275C4×C53—22no—
276C4×C53—42no—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—42no—
280C4×C53—162no—
281C2×#534C2×C5311yescharacteristic
282C2×#352C2×C5312yescomplemented
283C2×AΣL(1,𝔽16)C2×C5312yescomplemented
284#1783C2×C5311yescharacteristic, complemented
285#1781C2×C5311yescharacteristic, complemented
286C2×C53—21no—
287C2×C53—21no—
288C2×C53—42no—
289C2×C53—42no—
290C2×C53—42no—
291C2×C53—41no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53—82no—
295C2×C53#178111yescharacteristic
296C2×C53#212112yes—
297C23×D4C4×C5311yescharacteristic, complemented
298C23×D4C22×C5311yescharacteristic
299#534—24no—
300#534—24no—
301C2×#123—21no—
302C2×#123C4×C5311yescharacteristic, complemented
303C2×#123C22×C5311yescharacteristic
304C2×#121—21no—
305C2×#121—21no—
306C2×#120—21no—
307C2×#120C4×C5311yescharacteristic, complemented
308C2×#120C22×C5311yescharacteristic
309C22×C22⋊C4C4×C5311yescharacteristic, complemented
310C2×#100—21no—
311C2×#100C22×C5312yes—
312C2×#99—21no—
313C2×#99C22×C5312yes—
314#407—21no—
315#379—21no—
316#352—24no—
317AΣL(1,𝔽16)—24no—
318C53#1269011yescharacteristic, direct product factor, Sylow
319C25C2×C4×C5311yescharacteristic
320C25C2×C4×C5311yescharacteristic
321C25D4×C5311yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4—42no—
334C22×D4C2×C4×C5311yescharacteristic
335C22×D4D4×C5311yescharacteristic
336C22×D4D4×C5311yescharacteristic
337C22×D4C23×C5311yescharacteristic
338C23×C4—21no—
339C23×C4C2×C4×C5311yescharacteristic
340C23×C4D4×C5311yescharacteristic
341C2×M16—22no—
342#123—28no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123D4×C5314yes—
347#121—28no—
348#121—28no—
349#120—28no—
350#120—22no—
351#120—22no—
352#120—22no—
353#120D4×C5314yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4C2×C4×C5311yescharacteristic
366C2×C22⋊C4C2×C4×C5311yescharacteristic
367#100—22no—
368#100—22no—
369#100—42no—
370#99—22no—
371#99—22no—
372#99—42no—
373C22⋊C8—42no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24C22⋊C4×C5311yescharacteristic
389C24C22⋊C4×C5311yescharacteristic
390C24C22⋊C4×C5311yescharacteristic
391C24C22⋊C4×C5312yescomplemented
392C24C22×C4×C5311yescharacteristic
393C24C2×D4×C5311yescharacteristic
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×D4×C5312yes—
421C2×D4C2×D4×C5312yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4C22⋊C4×C5311yescharacteristic
435C22×C4C22⋊C4×C5312yescomplemented
436C22×C4C2×D4×C5311yescharacteristic
437M16—42no—
438M16—42no—
439C2×C8—42no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—84no—
452C22⋊C4C22⋊C4×C5312yescomplemented
453C22⋊C4C22⋊C4×C5312yescomplemented
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23#99×C5312yes—
494C23#99×C5312yes—
495C23C2×C22⋊C4×C5311yescharacteristic
496C23C2×C22⋊C4×C5311yescharacteristic
497C23C2×C22⋊C4×C5311yescharacteristic
498C23#120×C5312yes—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—48no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4#120×C5312yes—
528C8—82no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—88no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22C53×#37911yescharacteristic
564C22C2×#99×C5311yescharacteristic
565C22C2×#99×C5311yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—162no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—41no—
577C2—41no—
578C2—42no—
579C2—81no—
580C2—82no—
581C2C53×#178111yescharacteristic
582C2C53×#212112yes—
subgroupG/H
center, socleC22×C53C2×#99
commutatorC24C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC2×C53×#534C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C23×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×#352C2semidirect
3. C2×C53×AΣL(1,𝔽16)C2semidirect
4. C53×#1783C2semidirect
5. C53×#1781C2semidirect
6. C23×D4×C53C4semidirect
11. C2×#123×C53C4semidirect
16. C2×#120×C53C4semidirect
18. C22×C22⋊C4×C53C4semidirect
99. C24×C53C22⋊C4semidirect
143. C22×C4×C53C22⋊C4semidirect
160. C22⋊C4×C53C22⋊C4semidirect
161. C22⋊C4×C53C22⋊C4semidirect
237. #12690C53direct (N × H)
282. C2×#352C2×C53semidirect
283. C2×AΣL(1,𝔽16)C2×C53semidirect
284. #1783C2×C53semidirect
285. #1781C2×C53semidirect
297. C23×D4C4×C53semidirect
302. C2×#123C4×C53semidirect
307. C2×#120C4×C53semidirect
309. C22×C22⋊C4C4×C53semidirect
318. C53#12690direct (N × H)
391. C24C22⋊C4×C53semidirect
435. C22×C4C22⋊C4×C53semidirect
452. C22⋊C4C22⋊C4×C53semidirect
453. C22⋊C4C22⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5342
C2×C53×#3522
C2×C53×AΣL(1,𝔽16)2
C53×#17832
C53×#17812
#1269053