Finite Group Databasebeta

← C2×C53×#1390 #27220084

Subgroups — C2×C53×#1390 #27220084

Order 13568.

1202 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#100×C53C211yesmaximal, characteristic
2C22×#99×C53C211yesmaximal, characteristic
3C2×C53×#386C211yesmaximal, characteristic
4C53×#1390C214yesmaximal, direct product factor
5C23×D4×C53C2211yescharacteristic
6C24×C4×C53C411yescharacteristic, complemented
7C22×M16×C53—21no—
8C22×C22⋊C4×C53—21no—
9C22×C22⋊C4×C53—21no—
10C22×C22⋊C4×C53C411yescharacteristic, complemented
11C2×#100×C53—21no—
12C2×#100×C53—22no—
13C2×#100×C53C414yescomplemented
14C2×#100×C53C2212yes—
15C2×#99×C53—21no—
16C2×#99×C53—22no—
17C2×#99×C53C414yescomplemented
18C2×#99×C53C2212yes—
19C53×#386—21no—
20C53×#386—21no—
21C53×#386C412yescomplemented
22C53×#386C2212yes—
23C2×#95×C53—21no—
24C25×C53C2×C411yescharacteristic
25C25×C53D411yescharacteristic
26C22×D4×C53—42no—
27C22×D4×C53—41no—
28C22×D4×C53—41no—
29C22×D4×C53C2×C411yescharacteristic
30C22×D4×C53C2×C412yes—
31C22×D4×C53C2×C412yes—
32C22×D4×C53D411yescharacteristic
33C22×D4×C53D412yes—
34C22×D4×C53D411yescharacteristic
35C22×D4×C53Q812yes—
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53—21no—
39C23×C4×C53—21no—
40C23×C4×C53—44no—
41C23×C4×C53—41no—
42C23×C4×C53—41no—
43C23×C4×C53—41no—
44C23×C4×C53C2×C412yescomplemented
45C23×C4×C53D411yescharacteristic
46C2×M16×C53—24no—
47C2×M16×C53—22no—
48C2×M16×C53—42no—
49C2×M16×C53—41no—
50C22×C8×C53—41no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—24no—
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—44no—
57C2×C22⋊C4×C53—42no—
58C2×C22⋊C4×C53—41no—
59C2×C22⋊C4×C53—42no—
60C2×C22⋊C4×C53C2×C412yescomplemented
61#100×C53—24no—
62#100×C53—22no—
63#100×C53C2×C414yescomplemented
64#99×C53—24no—
65#99×C53—22no—
66#99×C53C2×C414yescomplemented
67#95×C53—22no—
68#95×C53—22no—
69C24×C53—22no—
70C24×C53—21no—
71C24×C53—21no—
72C24×C53—21no—
73C24×C53—22no—
74C24×C53—21no—
75C24×C53—42no—
76C24×C53—42no—
77C24×C53—42no—
78C24×C53—42no—
79C24×C53C4212yes—
80C24×C53C22⋊C412yes—
81C24×C53C22⋊C411yescharacteristic
82C24×C53C22⋊C411yescharacteristic
83C24×C53C22⋊C411yescharacteristic
84C24×C53C4⋊C412yes—
85C24×C53C4⋊C412yes—
86C24×C53C22×C411yescharacteristic
87C24×C53C2×D411yescharacteristic
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—42no—
101C2×D4×C53—42no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53C4212yes—
105C2×D4×C53C22⋊C412yes—
106C2×D4×C53C22⋊C412yes—
107C2×D4×C53C4⋊C412yes—
108C2×D4×C53C4⋊C412yes—
109C2×D4×C53C4⋊C412yes—
110C2×D4×C53C22×C411yescharacteristic
111C2×D4×C53C22×C411yescharacteristic
112C2×D4×C53C2×D411yescharacteristic
113C2×D4×C53C2×Q811yescharacteristic
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—44no—
121C22×C4×C53—44no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—48no—
125C22×C4×C53—44no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—42no—
129C22×C4×C53—44no—
130C22×C4×C53—44no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—82no—
136C22×C4×C53—82no—
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C412yes—
139C22×C4×C53C4⋊C412yes—
140C22×C4×C53C2×D411yescharacteristic
141M16×C53—24no—
142M16×C53—44no—
143M16×C53—42no—
144C2×C8×C53—44no—
145C2×C8×C53—42no—
146C22⋊C4×C53—24no—
147C22⋊C4×C53—44no—
148C22⋊C4×C53—44no—
149C22⋊C4×C53—44no—
150C22⋊C4×C53—42no—
151C22⋊C4×C53—42no—
152C22⋊C4×C53—42no—
153C23×C53—22no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—22no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—22no—
168C23×C53—22no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—44no—
174C23×C53—41no—
175C23×C53—44no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—48no—
180C23×C53—41no—
181C23×C53—44no—
182C23×C53—41no—
183C23×C53—44no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—81no—
191C23×C53—81no—
192C23×C53—84no—
193C23×C53#9512yes—
194C23×C53#9512yes—
195C23×C53#9512yes—
196C23×C53#9911yescharacteristic
197C23×C53#9912yes—
198C23×C53#9911yescharacteristic
199C23×C53C2×C4211yescharacteristic
200C23×C53C2×C22⋊C411yescharacteristic
201C23×C53C2×C22⋊C411yescharacteristic
202C23×C53C2×C4⋊C411yescharacteristic
203C23×C53C2×C4⋊C411yescharacteristic
204D4×C53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—41no—
211D4×C53—41no—
212D4×C53—41no—
213D4×C53—41no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—24no—
223C2×C4×C53—42no—
224C2×C4×C53—44no—
225C2×C4×C53—48no—
226C2×C4×C53—44no—
227C2×C4×C53—44no—
228C2×C4×C53—42no—
229C2×C4×C53—44no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—44no—
233C2×C4×C53—42no—
234C2×C4×C53—84no—
235C2×C4×C53—82no—
236C2×C4×C53—82no—
237C2×C4×C53—88no—
238C2×C4×C53#9512yes—
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53C2×C4⋊C411yescharacteristic
241C8×C53—44no—
242C2×#1390C5311yesmaximal, characteristic, direct product factor, Sylow
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—24no—
248C22×C53—24no—
249C22×C53—21no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—44no—
264C22×C53—44no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—44no—
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—82no—
280C22×C53—82no—
281C22×C53#34212yes—
282C22×C53#34212yes—
283C22×C53AΣL(1,𝔽16)12yes—
284C22×C53#35212yes—
285C22×C53C2×#9511yescharacteristic
286C22×C53C2×#9911yescharacteristic
287C22×C53C2×#9911yescharacteristic
288C4×C53—21no—
289C4×C53—22no—
290C4×C53—21no—
291C4×C53—44no—
292C4×C53—82no—
293C4×C53—84no—
294C22×#100C2×C5311yescharacteristic
295C22×#99C2×C5311yescharacteristic
296C2×#386C2×C5311yescharacteristic
297#1390C2×C5314yesdirect product factor
298C2×C53—21no—
299C2×C53—21no—
300C2×C53—22no—
301C2×C53—41no—
302C2×C53—42no—
303C2×C53—41no—
304C2×C53—41no—
305C2×C53—42no—
306C2×C53—41no—
307C2×C53#139014yesdirect product factor
308C2×C53C2×#34211yescharacteristic
309C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
310C2×C53C2×#35211yescharacteristic
311C23×D4C22×C5311yescharacteristic
312C24×C4C4×C5311yescharacteristic, complemented
313C22×M16—21no—
314C22×C22⋊C4—21no—
315C22×C22⋊C4—21no—
316C22×C22⋊C4C4×C5311yescharacteristic, complemented
317C2×#100—22no—
318C2×#100—21no—
319C2×#100C4×C5314yescomplemented
320C2×#100C22×C5312yes—
321C2×#99—22no—
322C2×#99—21no—
323C2×#99C4×C5314yescomplemented
324C2×#99C22×C5312yes—
325#386—21no—
326#386—21no—
327#386C4×C5312yescomplemented
328#386C22×C5312yes—
329C2×#95—21no—
330C53C2×#139011yescharacteristic, direct product factor, Sylow
331C25C2×C4×C5311yescharacteristic
332C25D4×C5311yescharacteristic
333C22×D4—41no—
334C22×D4—42no—
335C22×D4—41no—
336C22×D4C2×C4×C5312yes—
337C22×D4C2×C4×C5311yescharacteristic
338C22×D4C2×C4×C5312yes—
339C22×D4D4×C5311yescharacteristic
340C22×D4D4×C5312yes—
341C22×D4D4×C5311yescharacteristic
342C22×D4Q8×C5312yes—
343C22×D4C23×C5311yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—44no—
348C23×C4—41no—
349C23×C4—41no—
350C23×C4—41no—
351C23×C4C2×C4×C5312yescomplemented
352C23×C4D4×C5311yescharacteristic
353C2×M16—22no—
354C2×M16—24no—
355C2×M16—42no—
356C2×M16—41no—
357C22×C8—41no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—44no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4C2×C4×C5312yescomplemented
368#100—22no—
369#100—24no—
370#100C2×C4×C5314yescomplemented
371#99—22no—
372#99—24no—
373#99C2×C4×C5314yescomplemented
374#95—22no—
375#95—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—21no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C42×C5312yes—
387C24C22⋊C4×C5311yescharacteristic
388C24C22⋊C4×C5312yes—
389C24C22⋊C4×C5311yescharacteristic
390C24C22⋊C4×C5311yescharacteristic
391C24C4⋊C4×C5312yes—
392C24C4⋊C4×C5312yes—
393C24C22×C4×C5311yescharacteristic
394C24C2×D4×C5311yescharacteristic
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C42×C5312yes—
412C2×D4C22⋊C4×C5312yes—
413C2×D4C22⋊C4×C5312yes—
414C2×D4C4⋊C4×C5312yes—
415C2×D4C4⋊C4×C5312yes—
416C2×D4C4⋊C4×C5312yes—
417C2×D4C22×C4×C5311yescharacteristic
418C2×D4C22×C4×C5311yescharacteristic
419C2×D4C2×D4×C5311yescharacteristic
420C2×D4C2×Q8×C5311yescharacteristic
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22⋊C4×C5311yescharacteristic
445C22×C4C22⋊C4×C5312yes—
446C22×C4C4⋊C4×C5312yes—
447C22×C4C2×D4×C5311yescharacteristic
448M16—24no—
449M16—44no—
450M16—42no—
451C2×C8—44no—
452C2×C8—42no—
453C22⋊C4—24no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—41no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—81no—
498C23—81no—
499C23—84no—
500C23#95×C5312yes—
501C23#95×C5312yes—
502C23#95×C5312yes—
503C23#99×C5311yescharacteristic
504C23#99×C5312yes—
505C23#99×C5311yescharacteristic
506C23C2×C42×C5311yescharacteristic
507C23C2×C22⋊C4×C5311yescharacteristic
508C23C2×C22⋊C4×C5311yescharacteristic
509C23C2×C4⋊C4×C5311yescharacteristic
510C23C2×C4⋊C4×C5311yescharacteristic
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—88no—
545C2×C4#95×C5312yes—
546C2×C4C2×C22⋊C4×C5311yescharacteristic
547C2×C4C2×C4⋊C4×C5311yescharacteristic
548C8—44no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—24no—
554C22—24no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22C53×#34212yes—
588C22C53×#34212yes—
589C22C53×AΣL(1,𝔽16)12yes—
590C22C53×#35212yes—
591C22C2×#95×C5311yescharacteristic
592C22C2×#99×C5311yescharacteristic
593C22C2×#99×C5311yescharacteristic
594C4—21no—
595C4—22no—
596C4—21no—
597C4—44no—
598C4—82no—
599C4—84no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—42no—
605C2—41no—
606C2—41no—
607C2—42no—
608C2—41no—
609C2C53×#139014yesdirect product factor
610C2C2×C53×#34211yescharacteristic
611C2C2×C53×AΣL(1,𝔽16)11yescharacteristic
612C2C2×C53×#35211yescharacteristic
subgroupG/H
center, socleC23×C53#99
commutatorC23C2×C42×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC23×D4×C53C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C24×C53 ◁ C23×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1390C2direct (N × H)
6. C24×C4×C53C4semidirect
10. C22×C22⋊C4×C53C4semidirect
13. C2×#100×C53C4semidirect
17. C2×#99×C53C4semidirect
21. C53×#386C4semidirect
44. C23×C4×C53C2×C4semidirect
60. C2×C22⋊C4×C53C2×C4semidirect
63. #100×C53C2×C4semidirect
66. #99×C53C2×C4semidirect
242. C2×#1390C53direct (N × H)
297. #1390C2×C53direct (N × H)
307. C2×C53#1390direct (N × H)
312. C24×C4C4×C53semidirect
316. C22×C22⋊C4C4×C53semidirect
319. C2×#100C4×C53semidirect
323. C2×#99C4×C53semidirect
327. #386C4×C53semidirect
330. C53C2×#1390direct (N × H)
351. C23×C4C2×C4×C53semidirect
367. C2×C22⋊C4C2×C4×C53semidirect
370. #100C2×C4×C53semidirect
373. #99C2×C4×C53semidirect
609. C2C53×#1390direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#100×C532
C22×#99×C532
C2×C53×#3862
C53×#13902
C2×#139053