Finite Group Databasebeta

← C53×#24464 #27231976

Subgroups — C53×#24464 #27231976

Order 13568.

1600 conjugacy classes of subgroups.

The subgroup lattice has 1100 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3488C211yesmaximal, characteristic, complemented
2C53×#3055C211yesmaximal, characteristic, complemented
3C2×C53×#453C211yesmaximal, characteristic, complemented
4C53×#3006C211yesmaximal, characteristic, complemented
5C22×#104×C53C211yesmaximal, characteristic, complemented
6C53×#2856C211yesmaximal, characteristic, complemented
7C53×#2039C211yesmaximal, characteristic, complemented
8C53×#2037C211yesmaximal, characteristic, complemented
9C53×#2006C211yesmaximal, characteristic, complemented
10C53×#2002C211yesmaximal, characteristic, complemented
11C53×#1900C211yesmaximal, characteristic, complemented
12C53×#1896C211yesmaximal, characteristic, complemented
13C53×#1883C211yesmaximal, characteristic, complemented
14C53×#1859C211yesmaximal, characteristic, complemented
15C53×#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)×C53C2211yescharacteristic
17C22×#41×C53C2211yescharacteristic
18C2×AGL(1,ℤ/8)×C53C2211yescharacteristic
19C2×AGL(1,ℤ/8)×C53C2211yescharacteristic, complemented
20C2×AGL(1,ℤ/8)×C53C2211yescharacteristic, complemented
21C22×M16×C53C2211yescharacteristic, complemented
22C53×#550—21no—
23C53×#548—21no—
24C53×#546—21no—
25C53×#546—22no—
26D42×C53—22no—
27D42×C53—21no—
28C53×#537—21no—
29C53×#537C2211yescharacteristic
30C53×#535—21no—
31C53×#535C2211yescharacteristic
32C2×#127×C53C2211yescharacteristic, complemented
33C53×#525—21no—
34C53×#525C2211yescharacteristic
35C2×#121×C53—21no—
36C2×#121×C53—22no—
37C2×#121×C53C2211yescharacteristic
38C2×#121×C53C2211yescharacteristic, complemented
39C53×#520C2211yescharacteristic
40C22×C42×C53C2211yescharacteristic, complemented
41C53×#469—21no—
42C53×#468—21no—
43C53×#466—21no—
44C53×#465—21no—
45C53×#453—24no—
46C53×#449—21no—
47C53×#449—21no—
48C53×#448—21no—
49C53×#447—21no—
50C53×#428C2211yescharacteristic, complemented
51C2×#104×C53—24no—
52C2×#104×C53C2211yescharacteristic
53C2×#104×C53C2211yescharacteristic
54C2×#104×C53C2211yescharacteristic, complemented
55C2×#104×C53C2211yescharacteristic, complemented
56C2×#104×C53C2211yescharacteristic, complemented
57C2×#104×C53C2211yescharacteristic, complemented
58C53×#418C2211yescharacteristic, complemented
59C53×#417C2211yescharacteristic
60C53×#417C2211yescharacteristic, complemented
61C53×#417C2211yescharacteristic, complemented
62C2×#100×C53C2211yescharacteristic, complemented
63C2×#100×C53C2211yescharacteristic, complemented
64C53×#407C2211yescharacteristic, complemented
65C53×#390—22no—
66C53×#390C2211yescharacteristic, complemented
67C53×#390C2211yescharacteristic, complemented
68C53×#384C2211yescharacteristic, complemented
69C53×#343C2211yescharacteristic, complemented
70C53×#343C2211yescharacteristic, complemented
71C53×#339C2211yescharacteristic, complemented
72C53×#339C2211yescharacteristic, complemented
73H(5,𝔽2)×C53—22no—
74H(5,𝔽2)×C53—22no—
75H(5,𝔽2)×C53—41no—
76H(5,𝔽2)×C53—41no—
77C2×#41×C53—21no—
78C2×#41×C53—21no—
79C2×#41×C53—21no—
80C2×#41×C53—21no—
81C2×#41×C53—22no—
82C2×#41×C53—22no—
83C2×#41×C53—41no—
84C2×#41×C53—41no—
85C2×#41×C53D411yescharacteristic, complemented
86C2×#41×C53D411yescharacteristic, complemented
87C2×#41×C53C2311yescharacteristic
88C2×#41×C53C2311yescharacteristic
89C2×#41×C53C2311yescharacteristic
90C22×Q8×C53D411yescharacteristic, complemented
91C22×D4×C53—21no—
92C22×D4×C53—22no—
93C22×D4×C53—22no—
94C22×D4×C53—21no—
95C22×D4×C53—21no—
96C22×D4×C53—21no—
97C22×D4×C53—22no—
98C22×D4×C53D412yes—
99C22×D4×C53D411yescharacteristic, complemented
100C22×D4×C53C2311yescharacteristic
101C22×D4×C53C2311yescharacteristic
102C23×C4×C53—21no—
103C23×C4×C53—21no—
104C23×C4×C53C2311yescharacteristic
105AGL(1,ℤ/8)×C53—22no—
106AGL(1,ℤ/8)×C53—22no—
107AGL(1,ℤ/8)×C53—22no—
108AGL(1,ℤ/8)×C53—41no—
109AGL(1,ℤ/8)×C53—41no—
110AGL(1,ℤ/8)×C53—41no—
111C2×SD16×C53—21no—
112C2×SD16×C53—21no—
113C2×SD16×C53—21no—
114C2×D8×C53—21no—
115C2×D8×C53—21no—
116C2×D8×C53—21no—
117C2×M16×C53—22no—
118C2×M16×C53—21no—
119C2×M16×C53—41no—
120C2×M16×C53C2311yescharacteristic
121C2×M16×C53C2311yescharacteristic
122C2×M16×C53C2311yescharacteristic
123C2×M16×C53C2311yescharacteristic
124C22×C8×C53—21no—
125#127×C53—22no—
126#127×C53—21no—
127#127×C53—22no—
128#127×C53—41no—
129#124×C53—22no—
130#124×C53—21no—
131#123×C53—22no—
132#123×C53—21no—
133C22⋊Q8×C53—22no—
134C22⋊Q8×C53—21no—
135C22⋊Q8×C53—21no—
136C22⋊Q8×C53D411yescharacteristic, complemented
137C22⋊Q8×C53D411yescharacteristic, complemented
138#121×C53—22no—
139#121×C53—22no—
140#121×C53—22no—
141#121×C53—22no—
142#121×C53—22no—
143#121×C53—21no—
144#121×C53—21no—
145#121×C53—21no—
146#121×C53—21no—
147#121×C53—21no—
148#121×C53—21no—
149#121×C53—21no—
150#121×C53—21no—
151#121×C53—22no—
152#121×C53—21no—
153#121×C53—21no—
154#121×C53—21no—
155#121×C53—21no—
156#121×C53—48no—
157#121×C53D411yescharacteristic, complemented
158#121×C53D411yescharacteristic, complemented
159#120×C53—22no—
160#120×C53—22no—
161#120×C53—22no—
162#120×C53—21no—
163#120×C53—22no—
164C4×Q8×C53—21no—
165C4×D4×C53—22no—
166C4×D4×C53—22no—
167C4×D4×C53—21no—
168C4×D4×C53—22no—
169C4×D4×C53—21no—
170C4×D4×C53—21no—
171#117×C53—21no—
172#117×C53C2311yescharacteristic
173C2×C4⋊C4×C53—21no—
174C2×C4⋊C4×C53—22no—
175C2×C4⋊C4×C53C2311yescharacteristic
176C2×C4⋊C4×C53C2311yescharacteristic
177C2×C22⋊C4×C53—22no—
178C2×C22⋊C4×C53—22no—
179C2×C22⋊C4×C53—22no—
180C2×C22⋊C4×C53—21no—
181C2×C22⋊C4×C53D412yes—
182C2×C42×C53—21no—
183C2×C42×C53—22no—
184C2×C42×C53—41no—
185C2×C42×C53C2311yescharacteristic
186C2×C42×C53C2311yescharacteristic
187#107×C53—21no—
188#106×C53—21no—
189#104×C53—22no—
190#104×C53—22no—
191#104×C53—22no—
192#104×C53—22no—
193#104×C53—41no—
194#104×C53D414yescomplemented
195#103×C53—21no—
196#103×C53—21no—
197#103×C53—21no—
198#102×C53—21no—
199#102×C53—21no—
200#102×C53—21no—
201#102×C53—21no—
202#102×C53—21no—
203#100×C53—22no—
204#100×C53—41no—
205C22⋊C8×C53—21no—
206C22⋊C8×C53—21no—
207C24×C53—22no—
208C24×C53—21no—
209C24×C53—21no—
210C24×C53C2×D411yescharacteristic
211C24×C53C2×D412yes—
212#41×C53—21no—
213#41×C53—21no—
214#41×C53—22no—
215#41×C53—22no—
216#41×C53—21no—
217#41×C53—21no—
218#41×C53—21no—
219#41×C53—21no—
220#41×C53—21no—
221#41×C53—21no—
222#41×C53—21no—
223#41×C53—21no—
224#41×C53—42no—
225#41×C53—42no—
226#41×C53—41no—
227#41×C53—41no—
228#41×C53—42no—
229#41×C53—42no—
230#41×C53—42no—
231#41×C53—42no—
232#41×C53—42no—
233#41×C53—42no—
234#41×C53C2×D412yes—
235#41×C53#4112yes—
236C2×Q8×C53—21no—
237C2×Q8×C53—22no—
238C2×Q8×C53—41no—
239C2×Q8×C53C2×D411yescharacteristic
240C2×Q8×C53C2×D411yescharacteristic
241C2×Q8×C53C2×D411yescharacteristic
242C2×D4×C53—21no—
243C2×D4×C53—22no—
244C2×D4×C53—22no—
245C2×D4×C53—22no—
246C2×D4×C53—22no—
247C2×D4×C53—21no—
248C2×D4×C53—21no—
249C2×D4×C53—21no—
250C2×D4×C53—21no—
251C2×D4×C53—21no—
252C2×D4×C53—22no—
253C2×D4×C53—22no—
254C2×D4×C53—22no—
255C2×D4×C53—21no—
256C2×D4×C53—22no—
257C2×D4×C53—21no—
258C2×D4×C53—21no—
259C2×D4×C53—21no—
260C2×D4×C53—21no—
261C2×D4×C53—21no—
262C2×D4×C53—21no—
263C2×D4×C53—21no—
264C2×D4×C53—21no—
265C2×D4×C53—21no—
266C2×D4×C53—21no—
267C2×D4×C53—21no—
268C2×D4×C53—21no—
269C2×D4×C53—21no—
270C2×D4×C53—41no—
271C2×D4×C53—42no—
272C2×D4×C53—42no—
273C2×D4×C53—44no—
274C2×D4×C53—44no—
275C2×D4×C53—44no—
276C2×D4×C53—42no—
277C2×D4×C53—42no—
278C2×D4×C53—44no—
279C2×D4×C53—44no—
280C2×D4×C53—42no—
281C2×D4×C53—42no—
282C2×D4×C53—44no—
283C2×D4×C53—44no—
284C2×D4×C53—41no—
285C2×D4×C53—41no—
286C2×D4×C53—84no—
287C2×D4×C53C2×D411yescharacteristic
288C2×D4×C53C2×D411yescharacteristic
289C2×D4×C53C2×D411yescharacteristic
290C22×C4×C53—21no—
291C22×C4×C53—21no—
292C22×C4×C53—21no—
293C22×C4×C53—21no—
294C22×C4×C53—21no—
295C22×C4×C53—21no—
296C22×C4×C53—22no—
297C22×C4×C53—24no—
298C22×C4×C53—21no—
299C22×C4×C53—21no—
300C22×C4×C53—22no—
301C22×C4×C53—21no—
302C22×C4×C53—21no—
303C22×C4×C53—21no—
304C22×C4×C53—21no—
305C22×C4×C53—21no—
306C22×C4×C53—42no—
307C22×C4×C53—42no—
308C22×C4×C53—42no—
309C22×C4×C53—42no—
310C22×C4×C53C2×D411yescharacteristic
311C22×C4×C53C2×D411yescharacteristic
312C22×C4×C53C2×D411yescharacteristic
313C22×C4×C53C2×D411yescharacteristic
314C22×C4×C53C2×D411yescharacteristic
315C22×C4×C53C2×D412yes—
316C22×C4×C53C2411yescharacteristic
317SD16×C53—42no—
318SD16×C53—42no—
319SD16×C53—42no—
320D8×C53—42no—
321D8×C53—42no—
322D8×C53—42no—
323M16×C53—21no—
324M16×C53—21no—
325M16×C53—21no—
326M16×C53—21no—
327M16×C53—21no—
328M16×C53—21no—
329M16×C53—21no—
330M16×C53—42no—
331M16×C53C2×D412yescomplemented
332C2×C8×C53—21no—
333C2×C8×C53—21no—
334C2×C8×C53—21no—
335C2×C8×C53—21no—
336C2×C8×C53—42no—
337C4⋊C4×C53—22no—
338C4⋊C4×C53—22no—
339C4⋊C4×C53—21no—
340C4⋊C4×C53—21no—
341C4⋊C4×C53—21no—
342C4⋊C4×C53—21no—
343C4⋊C4×C53—21no—
344C4⋊C4×C53—21no—
345C4⋊C4×C53—44no—
346C4⋊C4×C53C2×D411yescharacteristic
347C4⋊C4×C53C2×D411yescharacteristic
348C22⋊C4×C53—22no—
349C22⋊C4×C53—22no—
350C22⋊C4×C53—22no—
351C22⋊C4×C53—22no—
352C22⋊C4×C53—22no—
353C22⋊C4×C53—21no—
354C22⋊C4×C53—21no—
355C22⋊C4×C53—21no—
356C22⋊C4×C53—21no—
357C22⋊C4×C53—22no—
358C22⋊C4×C53—22no—
359C22⋊C4×C53—21no—
360C22⋊C4×C53—44no—
361C22⋊C4×C53—44no—
362C22⋊C4×C53—41no—
363C22⋊C4×C53—41no—
364C42×C53—22no—
365C42×C53—21no—
366C42×C53—21no—
367C42×C53—21no—
368C42×C53—21no—
369C42×C53—42no—
370C42×C53C2×D412yescomplemented
371C23×C53—21no—
372C23×C53—21no—
373C23×C53—21no—
374C23×C53—22no—
375C23×C53—21no—
376C23×C53—22no—
377C23×C53—21no—
378C23×C53—22no—
379C23×C53—22no—
380C23×C53—41no—
381C23×C53—42no—
382C23×C53—44no—
383C23×C53—42no—
384C23×C53—42no—
385C23×C53—42no—
386C23×C53—42no—
387C23×C53—42no—
388C23×C53—41no—
389C23×C53—41no—
390C23×C53—41no—
391C23×C53—81no—
392C23×C53—82no—
393C23×C53—82no—
394C23×C53#12011yescharacteristic
395C23×C53#12011yescharacteristic
396C23×C53C22×D411yescharacteristic
397Q8×C53—21no—
398Q8×C53—21no—
399Q8×C53—21no—
400Q8×C53—21no—
401Q8×C53—21no—
402Q8×C53—42no—
403Q8×C53#12112yescomplemented
404D4×C53—22no—
405D4×C53—22no—
406D4×C53—21no—
407D4×C53—21no—
408D4×C53—21no—
409D4×C53—21no—
410D4×C53—21no—
411D4×C53—42no—
412D4×C53—42no—
413D4×C53—44no—
414D4×C53—42no—
415D4×C53—44no—
416D4×C53—41no—
417D4×C53—41no—
418D4×C53—41no—
419D4×C53—41no—
420D4×C53—44no—
421D4×C53—42no—
422D4×C53—41no—
423D4×C53—41no—
424D4×C53—41no—
425D4×C53—41no—
426D4×C53—41no—
427D4×C53—41no—
428D4×C53—41no—
429D4×C53—41no—
430D4×C53—88no—
431D4×C53—81no—
432D4×C53—82no—
433D4×C53—82no—
434D4×C53—82no—
435D4×C53—82no—
436D4×C53—82no—
437D4×C53—82no—
438D4×C53#12112yescomplemented
439C2×C4×C53—22no—
440C2×C4×C53—22no—
441C2×C4×C53—21no—
442C2×C4×C53—21no—
443C2×C4×C53—22no—
444C2×C4×C53—21no—
445C2×C4×C53—21no—
446C2×C4×C53—21no—
447C2×C4×C53—21no—
448C2×C4×C53—21no—
449C2×C4×C53—21no—
450C2×C4×C53—21no—
451C2×C4×C53—22no—
452C2×C4×C53—21no—
453C2×C4×C53—21no—
454C2×C4×C53—22no—
455C2×C4×C53—22no—
456C2×C4×C53—21no—
457C2×C4×C53—21no—
458C2×C4×C53—21no—
459C2×C4×C53—22no—
460C2×C4×C53—21no—
461C2×C4×C53—42no—
462C2×C4×C53—42no—
463C2×C4×C53—42no—
464C2×C4×C53—48no—
465C2×C4×C53—44no—
466C2×C4×C53—41no—
467C2×C4×C53—41no—
468C2×C4×C53—41no—
469C2×C4×C53—41no—
470C2×C4×C53—41no—
471C2×C4×C53—41no—
472C2×C4×C53—42no—
473C2×C4×C53—42no—
474C2×C4×C53—41no—
475C2×C4×C53—82no—
476C2×C4×C53#12011yescharacteristic
477C2×C4×C53#12011yescharacteristic
478C2×C4×C53C22×D411yescharacteristic
479C2×C4×C53C22×D411yescharacteristic
480C2×C4×C53C22×D411yescharacteristic
481C2×C4×C53C2×#4111yescharacteristic
482C2×C4×C53H(5,𝔽2)11yescharacteristic
483C2×C4×C53H(5,𝔽2)11yescharacteristic
484C8×C53—22no—
485C8×C53—41no—
486C8×C53—41no—
487C8×C53—41no—
488#24464C5311yesmaximal, characteristic, direct product factor, Sylow
489C22×C53—21no—
490C22×C53—21no—
491C22×C53—21no—
492C22×C53—21no—
493C22×C53—21no—
494C22×C53—22no—
495C22×C53—21no—
496C22×C53—21no—
497C22×C53—42no—
498C22×C53—42no—
499C22×C53—41no—
500C22×C53—41no—
501C22×C53—41no—
502C22×C53—41no—
503C22×C53—42no—
504C22×C53—42no—
505C22×C53—41no—
506C22×C53—41no—
507C22×C53—41no—
508C22×C53—42no—
509C22×C53—41no—
510C22×C53—84no—
511C22×C53—82no—
512C22×C53—82no—
513C22×C53—82no—
514C22×C53—82no—
515C22×C53—81no—
516C22×C53—81no—
517C22×C53—81no—
518C22×C53—164no—
519C22×C53#45312yescomplemented
520C22×C53C2×#12011yescharacteristic
521C22×C53D4211yescharacteristic
522C22×C53#54611yescharacteristic
523C4×C53—24no—
524C4×C53—21no—
525C4×C53—21no—
526C4×C53—22no—
527C4×C53—42no—
528C4×C53—41no—
529C4×C53—41no—
530C4×C53—42no—
531C4×C53—41no—
532C4×C53—41no—
533C4×C53—81no—
534C4×C53C2×#12111yescharacteristic
535C4×C53#53411yescharacteristic
536C4×C53D4211yescharacteristic
537C4×C53#54611yescharacteristic
538#3488C2×C5311yescharacteristic, complemented
539#3055C2×C5311yescharacteristic, complemented
540C2×#453C2×C5311yescharacteristic, complemented
541#3006C2×C5311yescharacteristic, complemented
542C22×#104C2×C5311yescharacteristic, complemented
543#2856C2×C5311yescharacteristic, complemented
544#2039C2×C5311yescharacteristic, complemented
545#2037C2×C5311yescharacteristic, complemented
546#2006C2×C5311yescharacteristic, complemented
547#2002C2×C5311yescharacteristic, complemented
548#1900C2×C5311yescharacteristic, complemented
549#1896C2×C5311yescharacteristic, complemented
550#1883C2×C5311yescharacteristic, complemented
551#1859C2×C5311yescharacteristic, complemented
552#1792C2×C5311yescharacteristic, complemented
553C2×C53—21no—
554C2×C53—21no—
555C2×C53—22no—
556C2×C53—41no—
557C2×C53—42no—
558C2×C53—81no—
559C2×C53—82no—
560C2×C53—81no—
561C2×C53—161no—
562C2×C53#241011yescharacteristic
563C2×C53C2×#45311yescharacteristic
564C2×C53#301811yescharacteristic
565C2×H(5,𝔽2)C22×C5311yescharacteristic
566C22×#41C22×C5311yescharacteristic
567C2×AGL(1,ℤ/8)C22×C5311yescharacteristic
568C2×AGL(1,ℤ/8)C22×C5311yescharacteristic, complemented
569C2×AGL(1,ℤ/8)C22×C5311yescharacteristic, complemented
570C22×M16C22×C5311yescharacteristic, complemented
571#550—21no—
572#548—21no—
573#546—21no—
574#546—22no—
575D42—22no—
576D42—21no—
577#537—21no—
578#537C22×C5311yescharacteristic
579#535—21no—
580#535C22×C5311yescharacteristic
581C2×#127C22×C5311yescharacteristic, complemented
582#525—21no—
583#525C22×C5311yescharacteristic
584C2×#121—21no—
585C2×#121—22no—
586C2×#121C22×C5311yescharacteristic
587C2×#121C22×C5311yescharacteristic, complemented
588#520C22×C5311yescharacteristic
589C22×C42C22×C5311yescharacteristic, complemented
590#469—21no—
591#468—21no—
592#466—21no—
593#465—21no—
594#453—24no—
595#449—21no—
596#449—21no—
597#448—21no—
598#447—21no—
599#428C22×C5311yescharacteristic, complemented
600C2×#104—24no—
601C2×#104C22×C5311yescharacteristic
602C2×#104C22×C5311yescharacteristic
603C2×#104C22×C5311yescharacteristic, complemented
604C2×#104C22×C5311yescharacteristic, complemented
605C2×#104C22×C5311yescharacteristic, complemented
606C2×#104C22×C5311yescharacteristic, complemented
607#418C22×C5311yescharacteristic, complemented
608#417C22×C5311yescharacteristic
609#417C22×C5311yescharacteristic, complemented
610#417C22×C5311yescharacteristic, complemented
611C2×#100C22×C5311yescharacteristic, complemented
612C2×#100C22×C5311yescharacteristic, complemented
613#407C22×C5311yescharacteristic, complemented
614#390—22no—
615#390C22×C5311yescharacteristic, complemented
616#390C22×C5311yescharacteristic, complemented
617#384C22×C5311yescharacteristic, complemented
618#343C22×C5311yescharacteristic, complemented
619#343C22×C5311yescharacteristic, complemented
620#339C22×C5311yescharacteristic, complemented
621#339C22×C5311yescharacteristic, complemented
622C53#2446411yescharacteristic, direct product factor, Sylow
623H(5,𝔽2)—22no—
624H(5,𝔽2)—22no—
625H(5,𝔽2)—41no—
626H(5,𝔽2)—41no—
627C2×#41—21no—
628C2×#41—21no—
629C2×#41—21no—
630C2×#41—21no—
631C2×#41—22no—
632C2×#41—22no—
633C2×#41—41no—
634C2×#41—41no—
635C2×#41D4×C5311yescharacteristic, complemented
636C2×#41D4×C5311yescharacteristic, complemented
637C2×#41C23×C5311yescharacteristic
638C2×#41C23×C5311yescharacteristic
639C2×#41C23×C5311yescharacteristic
640C22×Q8D4×C5311yescharacteristic, complemented
641C22×D4—21no—
642C22×D4—22no—
643C22×D4—22no—
644C22×D4—21no—
645C22×D4—21no—
646C22×D4—21no—
647C22×D4—22no—
648C22×D4D4×C5312yes—
649C22×D4D4×C5311yescharacteristic, complemented
650C22×D4C23×C5311yescharacteristic
651C22×D4C23×C5311yescharacteristic
652C23×C4—21no—
653C23×C4—21no—
654C23×C4C23×C5311yescharacteristic
655AGL(1,ℤ/8)—22no—
656AGL(1,ℤ/8)—22no—
657AGL(1,ℤ/8)—22no—
658AGL(1,ℤ/8)—41no—
659AGL(1,ℤ/8)—41no—
660AGL(1,ℤ/8)—41no—
661C2×SD16—21no—
662C2×SD16—21no—
663C2×SD16—21no—
664C2×D8—21no—
665C2×D8—21no—
666C2×D8—21no—
667C2×M16—21no—
668C2×M16—22no—
669C2×M16—41no—
670C2×M16C23×C5311yescharacteristic
671C2×M16C23×C5311yescharacteristic
672C2×M16C23×C5311yescharacteristic
673C2×M16C23×C5311yescharacteristic
674C22×C8—21no—
675#127—22no—
676#127—21no—
677#127—22no—
678#127—41no—
679#124—22no—
680#124—21no—
681#123—22no—
682#123—21no—
683C22⋊Q8—22no—
684C22⋊Q8—21no—
685C22⋊Q8—21no—
686C22⋊Q8D4×C5311yescharacteristic, complemented
687C22⋊Q8D4×C5311yescharacteristic, complemented
688#121—22no—
689#121—22no—
690#121—22no—
691#121—22no—
692#121—22no—
693#121—21no—
694#121—21no—
695#121—21no—
696#121—21no—
697#121—21no—
698#121—21no—
699#121—22no—
700#121—21no—
701#121—21no—
702#121—21no—
703#121—21no—
704#121—21no—
705#121—21no—
706#121—48no—
707#121D4×C5311yescharacteristic, complemented
708#121D4×C5311yescharacteristic, complemented
709#120—22no—
710#120—22no—
711#120—22no—
712#120—21no—
713#120—22no—
714C4×Q8—21no—
715C4×D4—22no—
716C4×D4—22no—
717C4×D4—21no—
718C4×D4—22no—
719C4×D4—21no—
720C4×D4—21no—
721#117—21no—
722#117C23×C5311yescharacteristic
723C2×C4⋊C4—21no—
724C2×C4⋊C4—22no—
725C2×C4⋊C4C23×C5311yescharacteristic
726C2×C4⋊C4C23×C5311yescharacteristic
727C2×C22⋊C4—22no—
728C2×C22⋊C4—22no—
729C2×C22⋊C4—22no—
730C2×C22⋊C4—21no—
731C2×C22⋊C4D4×C5312yes—
732C2×C42—22no—
733C2×C42—21no—
734C2×C42—41no—
735C2×C42C23×C5311yescharacteristic
736C2×C42C23×C5311yescharacteristic
737#107—21no—
738#106—21no—
739#104—22no—
740#104—22no—
741#104—22no—
742#104—22no—
743#104—41no—
744#104D4×C5314yescomplemented
745#103—21no—
746#103—21no—
747#103—21no—
748#102—21no—
749#102—21no—
750#102—21no—
751#102—21no—
752#102—21no—
753#100—22no—
754#100—41no—
755C22⋊C8—21no—
756C22⋊C8—21no—
757C24—22no—
758C24—21no—
759C24—21no—
760C24C2×D4×C5311yescharacteristic
761C24C2×D4×C5312yes—
762#41—21no—
763#41—21no—
764#41—22no—
765#41—22no—
766#41—21no—
767#41—21no—
768#41—21no—
769#41—21no—
770#41—21no—
771#41—21no—
772#41—21no—
773#41—21no—
774#41—42no—
775#41—42no—
776#41—41no—
777#41—41no—
778#41—42no—
779#41—42no—
780#41—42no—
781#41—42no—
782#41—42no—
783#41—42no—
784#41C2×D4×C5312yes—
785#41#41×C5312yes—
786C2×Q8—21no—
787C2×Q8—22no—
788C2×Q8—41no—
789C2×Q8C2×D4×C5311yescharacteristic
790C2×Q8C2×D4×C5311yescharacteristic
791C2×Q8C2×D4×C5311yescharacteristic
792C2×D4—21no—
793C2×D4—22no—
794C2×D4—22no—
795C2×D4—22no—
796C2×D4—22no—
797C2×D4—21no—
798C2×D4—21no—
799C2×D4—21no—
800C2×D4—21no—
801C2×D4—21no—
802C2×D4—22no—
803C2×D4—22no—
804C2×D4—22no—
805C2×D4—21no—
806C2×D4—22no—
807C2×D4—21no—
808C2×D4—21no—
809C2×D4—21no—
810C2×D4—21no—
811C2×D4—21no—
812C2×D4—21no—
813C2×D4—21no—
814C2×D4—21no—
815C2×D4—21no—
816C2×D4—21no—
817C2×D4—21no—
818C2×D4—21no—
819C2×D4—21no—
820C2×D4—41no—
821C2×D4—42no—
822C2×D4—42no—
823C2×D4—44no—
824C2×D4—44no—
825C2×D4—44no—
826C2×D4—42no—
827C2×D4—42no—
828C2×D4—44no—
829C2×D4—44no—
830C2×D4—42no—
831C2×D4—42no—
832C2×D4—44no—
833C2×D4—44no—
834C2×D4—41no—
835C2×D4—41no—
836C2×D4—84no—
837C2×D4C2×D4×C5311yescharacteristic
838C2×D4C2×D4×C5311yescharacteristic
839C2×D4C2×D4×C5311yescharacteristic
840C22×C4—21no—
841C22×C4—21no—
842C22×C4—21no—
843C22×C4—21no—
844C22×C4—21no—
845C22×C4—21no—
846C22×C4—22no—
847C22×C4—24no—
848C22×C4—21no—
849C22×C4—21no—
850C22×C4—22no—
851C22×C4—21no—
852C22×C4—21no—
853C22×C4—21no—
854C22×C4—21no—
855C22×C4—21no—
856C22×C4—42no—
857C22×C4—42no—
858C22×C4—42no—
859C22×C4—42no—
860C22×C4C2×D4×C5311yescharacteristic
861C22×C4C2×D4×C5311yescharacteristic
862C22×C4C2×D4×C5311yescharacteristic
863C22×C4C2×D4×C5311yescharacteristic
864C22×C4C2×D4×C5311yescharacteristic
865C22×C4C2×D4×C5312yes—
866C22×C4C24×C5311yescharacteristic
867SD16—42no—
868SD16—42no—
869SD16—42no—
870D8—42no—
871D8—42no—
872D8—42no—
873M16—21no—
874M16—21no—
875M16—21no—
876M16—21no—
877M16—21no—
878M16—21no—
879M16—21no—
880M16—42no—
881M16C2×D4×C5312yescomplemented
882C2×C8—21no—
883C2×C8—21no—
884C2×C8—21no—
885C2×C8—21no—
886C2×C8—42no—
887C4⋊C4—22no—
888C4⋊C4—22no—
889C4⋊C4—21no—
890C4⋊C4—21no—
891C4⋊C4—21no—
892C4⋊C4—21no—
893C4⋊C4—21no—
894C4⋊C4—21no—
895C4⋊C4—44no—
896C4⋊C4C2×D4×C5311yescharacteristic
897C4⋊C4C2×D4×C5311yescharacteristic
898C22⋊C4—22no—
899C22⋊C4—22no—
900C22⋊C4—22no—
901C22⋊C4—22no—
902C22⋊C4—22no—
903C22⋊C4—21no—
904C22⋊C4—21no—
905C22⋊C4—21no—
906C22⋊C4—21no—
907C22⋊C4—22no—
908C22⋊C4—22no—
909C22⋊C4—21no—
910C22⋊C4—44no—
911C22⋊C4—44no—
912C22⋊C4—41no—
913C22⋊C4—41no—
914C42—22no—
915C42—21no—
916C42—21no—
917C42—21no—
918C42—21no—
919C42—42no—
920C42C2×D4×C5312yescomplemented
921C23—21no—
922C23—21no—
923C23—21no—
924C23—22no—
925C23—22no—
926C23—21no—
927C23—21no—
928C23—22no—
929C23—22no—
930C23—41no—
931C23—42no—
932C23—44no—
933C23—42no—
934C23—42no—
935C23—42no—
936C23—42no—
937C23—42no—
938C23—41no—
939C23—41no—
940C23—41no—
941C23—81no—
942C23—82no—
943C23—82no—
944C23#120×C5311yescharacteristic
945C23#120×C5311yescharacteristic
946C23C22×D4×C5311yescharacteristic
947Q8—21no—
948Q8—21no—
949Q8—21no—
950Q8—21no—
951Q8—21no—
952Q8—42no—
953Q8#121×C5312yescomplemented
954D4—22no—
955D4—22no—
956D4—21no—
957D4—21no—
958D4—21no—
959D4—21no—
960D4—21no—
961D4—42no—
962D4—42no—
963D4—44no—
964D4—42no—
965D4—44no—
966D4—41no—
967D4—41no—
968D4—41no—
969D4—41no—
970D4—44no—
971D4—42no—
972D4—41no—
973D4—41no—
974D4—41no—
975D4—41no—
976D4—41no—
977D4—41no—
978D4—41no—
979D4—41no—
980D4—88no—
981D4—81no—
982D4—82no—
983D4—82no—
984D4—82no—
985D4—82no—
986D4—82no—
987D4—82no—
988D4#121×C5312yescomplemented
989C2×C4—22no—
990C2×C4—22no—
991C2×C4—21no—
992C2×C4—22no—
993C2×C4—21no—
994C2×C4—21no—
995C2×C4—21no—
996C2×C4—21no—
997C2×C4—21no—
998C2×C4—21no—
999C2×C4—21no—
1000C2×C4—21no—
1001C2×C4—22no—
1002C2×C4—21no—
1003C2×C4—21no—
1004C2×C4—22no—
1005C2×C4—22no—
1006C2×C4—21no—
1007C2×C4—22no—
1008C2×C4—21no—
1009C2×C4—21no—
1010C2×C4—21no—
1011C2×C4—42no—
1012C2×C4—42no—
1013C2×C4—42no—
1014C2×C4—48no—
1015C2×C4—44no—
1016C2×C4—41no—
1017C2×C4—41no—
1018C2×C4—41no—
1019C2×C4—41no—
1020C2×C4—41no—
1021C2×C4—41no—
1022C2×C4—42no—
1023C2×C4—42no—
1024C2×C4—41no—
1025C2×C4—82no—
1026C2×C4#120×C5311yescharacteristic
1027C2×C4#120×C5311yescharacteristic
1028C2×C4C22×D4×C5311yescharacteristic
1029C2×C4C22×D4×C5311yescharacteristic
1030C2×C4C22×D4×C5311yescharacteristic
1031C2×C4C2×#41×C5311yescharacteristic
1032C2×C4H(5,𝔽2)×C5311yescharacteristic
1033C2×C4H(5,𝔽2)×C5311yescharacteristic
1034C8—22no—
1035C8—41no—
1036C8—41no—
1037C8—41no—
1038C22—21no—
1039C22—21no—
1040C22—21no—
1041C22—21no—
1042C22—21no—
1043C22—21no—
1044C22—21no—
1045C22—22no—
1046C22—42no—
1047C22—42no—
1048C22—41no—
1049C22—41no—
1050C22—42no—
1051C22—41no—
1052C22—41no—
1053C22—41no—
1054C22—42no—
1055C22—41no—
1056C22—41no—
1057C22—41no—
1058C22—42no—
1059C22—84no—
1060C22—82no—
1061C22—82no—
1062C22—82no—
1063C22—82no—
1064C22—81no—
1065C22—81no—
1066C22—81no—
1067C22—164no—
1068C22C53×#45312yescomplemented
1069C22C2×#120×C5311yescharacteristic
1070C22D42×C5311yescharacteristic
1071C22C53×#54611yescharacteristic
1072C4—24no—
1073C4—21no—
1074C4—21no—
1075C4—22no—
1076C4—42no—
1077C4—42no—
1078C4—41no—
1079C4—41no—
1080C4—41no—
1081C4—41no—
1082C4—81no—
1083C4C2×#121×C5311yescharacteristic
1084C4C53×#53411yescharacteristic
1085C4D42×C5311yescharacteristic
1086C4C53×#54611yescharacteristic
1087C2—21no—
1088C2—21no—
1089C2—22no—
1090C2—41no—
1091C2—42no—
1092C2—81no—
1093C2—82no—
1094C2—81no—
1095C2—161no—
1096C2C53×#241011yescharacteristic
1097C2C2×C53×#45311yescharacteristic
1098C2C53×#301811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3488C2semidirect
2. C53×#3055C2semidirect
3. C2×C53×#453C2semidirect
4. C53×#3006C2semidirect
5. C22×#104×C53C2semidirect
6. C53×#2856C2semidirect
7. C53×#2039C2semidirect
8. C53×#2037C2semidirect
9. C53×#2006C2semidirect
10. C53×#2002C2semidirect
11. C53×#1900C2semidirect
12. C53×#1896C2semidirect
13. C53×#1883C2semidirect
14. C53×#1859C2semidirect
15. C53×#1792C2semidirect
19. C2×AGL(1,ℤ/8)×C53C22semidirect
20. C2×AGL(1,ℤ/8)×C53C22semidirect
21. C22×M16×C53C22semidirect
32. C2×#127×C53C22semidirect
38. C2×#121×C53C22semidirect
40. C22×C42×C53C22semidirect
50. C53×#428C22semidirect
54. C2×#104×C53C22semidirect
55. C2×#104×C53C22semidirect
56. C2×#104×C53C22semidirect
57. C2×#104×C53C22semidirect
58. C53×#418C22semidirect
60. C53×#417C22semidirect
61. C53×#417C22semidirect
62. C2×#100×C53C22semidirect
63. C2×#100×C53C22semidirect
64. C53×#407C22semidirect
66. C53×#390C22semidirect
67. C53×#390C22semidirect
68. C53×#384C22semidirect
69. C53×#343C22semidirect
70. C53×#343C22semidirect
71. C53×#339C22semidirect
72. C53×#339C22semidirect
85. C2×#41×C53D4semidirect
86. C2×#41×C53D4semidirect
90. C22×Q8×C53D4semidirect
99. C22×D4×C53D4semidirect
136. C22⋊Q8×C53D4semidirect
137. C22⋊Q8×C53D4semidirect
157. #121×C53D4semidirect
158. #121×C53D4semidirect
194. #104×C53D4semidirect
331. M16×C53C2×D4semidirect
370. C42×C53C2×D4semidirect
403. Q8×C53#121semidirect
438. D4×C53#121semidirect
488. #24464C53direct (N × H)
519. C22×C53#453semidirect
538. #3488C2×C53semidirect
539. #3055C2×C53semidirect
540. C2×#453C2×C53semidirect
541. #3006C2×C53semidirect
542. C22×#104C2×C53semidirect
543. #2856C2×C53semidirect
544. #2039C2×C53semidirect
545. #2037C2×C53semidirect
546. #2006C2×C53semidirect
547. #2002C2×C53semidirect
548. #1900C2×C53semidirect
549. #1896C2×C53semidirect
550. #1883C2×C53semidirect
551. #1859C2×C53semidirect
552. #1792C2×C53semidirect
568. C2×AGL(1,ℤ/8)C22×C53semidirect
569. C2×AGL(1,ℤ/8)C22×C53semidirect
570. C22×M16C22×C53semidirect
581. C2×#127C22×C53semidirect
587. C2×#121C22×C53semidirect
589. C22×C42C22×C53semidirect
599. #428C22×C53semidirect
603. C2×#104C22×C53semidirect
604. C2×#104C22×C53semidirect
605. C2×#104C22×C53semidirect
606. C2×#104C22×C53semidirect
607. #418C22×C53semidirect
609. #417C22×C53semidirect
610. #417C22×C53semidirect
611. C2×#100C22×C53semidirect
612. C2×#100C22×C53semidirect
613. #407C22×C53semidirect
615. #390C22×C53semidirect
616. #390C22×C53semidirect
617. #384C22×C53semidirect
618. #343C22×C53semidirect
619. #343C22×C53semidirect
620. #339C22×C53semidirect
621. #339C22×C53semidirect
622. C53#24464direct (N × H)
635. C2×#41D4×C53semidirect
636. C2×#41D4×C53semidirect
640. C22×Q8D4×C53semidirect
649. C22×D4D4×C53semidirect
686. C22⋊Q8D4×C53semidirect
687. C22⋊Q8D4×C53semidirect
707. #121D4×C53semidirect
708. #121D4×C53semidirect
744. #104D4×C53semidirect
881. M16C2×D4×C53semidirect
920. C42C2×D4×C53semidirect
953. Q8#121×C53semidirect
988. D4#121×C53semidirect
1068. C22C53×#453semidirect

Maximal subgroups

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

subgroupindex
C53×#34882
C53×#30552
C2×C53×#4532
C53×#30062
C22×#104×C532
C53×#28562
C53×#20392
C53×#20372
C53×#20062
C53×#20022
C53×#19002
C53×#18962
C53×#18832
C53×#18592
C53×#17922
#2446453