Finite Group Databasebeta

← C53×#33551 #27241063

Subgroups — C53×#33551 #27241063

Order 13568.

1326 conjugacy classes of subgroups.

The subgroup lattice has 1056 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3532C211yesmaximal, characteristic, complemented
2C53×#3026C211yesmaximal, characteristic, complemented
3C2×C53×B(4,𝔽2)C211yesmaximal, characteristic, complemented
4C53×#3019C211yesmaximal, characteristic, complemented
5C53×#2200C212yesmaximal, complemented
6C53×#2199C212yesmaximal, complemented
7C53×#2197C212yesmaximal, complemented
8C53×#2196C212yesmaximal, complemented
9C53×#2127C211yesmaximal, characteristic, complemented
10C53×#2121C211yesmaximal, characteristic, complemented
11C2×C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C53C2211yescharacteristic, complemented
13C53×#575—21no—
14C53×H(3,𝔽4)C2211yescharacteristic, complemented
15C53×#560—21no—
16C53×#560C2211yescharacteristic, complemented
17C53×#551—21no—
18C53×#551C2211yescharacteristic, complemented
19C53×#546—21no—
20C53×#546—21no—
21C53×#546—21no—
22D42×C53—21no—
23D42×C53C2211yescharacteristic, complemented
24C53×#538—21no—
25C53×#538—21no—
26C53×#538—21no—
27C53×#535—21no—
28C53×#535C2211yescharacteristic
29C53×#534—21no—
30C53×#534—21no—
31C53×#534C2211yescharacteristic
32C2×#120×C53—21no—
33C2×#120×C53—21no—
34C2×#120×C53C2211yescharacteristic
35C53×#458—21no—
36C53×#458C2212yescomplemented
37C53×B(4,𝔽2)—21no—
38C53×B(4,𝔽2)—21no—
39C53×B(4,𝔽2)—21no—
40C53×B(4,𝔽2)C2212yes—
41C53×B(4,𝔽2)C2212yescomplemented
42C53×B(4,𝔽2)C2212yescomplemented
43C53×#455—21no—
44C53×#455C2212yescomplemented
45C53×#453—21no—
46C53×#453C2212yescomplemented
47C53×#421—21no—
48C2×#100×C53C2211yescharacteristic, complemented
49C2×#99×C53—21no—
50C2×#99×C53—21no—
51C2×#99×C53C2211yescharacteristic, complemented
52C2×#99×C53C2211yescharacteristic, complemented
53C53×#354C2212yescomplemented
54C53×#353C2212yescomplemented
55C53×#352C2212yescomplemented
56C53×AΣL(1,𝔽16)C2212yescomplemented
57C53×AΣL(1,𝔽16)C2212yescomplemented
58C53×AΣL(1,𝔽16)C2212yescomplemented
59C25×C53D411yescharacteristic, complemented
60H(5,𝔽2)×C53—21no—
61H(5,𝔽2)×C53—21no—
62H(5,𝔽2)×C53—21no—
63H(5,𝔽2)×C53—42no—
64H(5,𝔽2)×C53C2312yes—
65C2×#41×C53—21no—
66C2×#41×C53—21no—
67C2×#41×C53—41no—
68C2×#41×C53D411yescharacteristic, complemented
69C22×D4×C53—21no—
70C22×D4×C53—21no—
71C22×D4×C53—21no—
72C22×D4×C53—21no—
73C22×D4×C53—21no—
74C22×D4×C53—21no—
75C22×D4×C53—21no—
76C22×D4×C53—41no—
77C22×D4×C53—41no—
78C22×D4×C53D411yescharacteristic, complemented
79C22×D4×C53C2311yescharacteristic
80#137×C53—22no—
81#137×C53—41no—
82AGL(1,ℤ/8)×C53—22no—
83AGL(1,ℤ/8)×C53—41no—
84#135×C53—42no—
85C2×M16×C53—21no—
86#127×C53—21no—
87#127×C53D411yescharacteristic, complemented
88#127×C53C2311yescharacteristic
89#126×C53—21no—
90#126×C53—21no—
91#126×C53—21no—
92#126×C53—21no—
93#124×C53—21no—
94#124×C53—21no—
95#124×C53—21no—
96#124×C53—21no—
97#124×C53—21no—
98#124×C53D411yescharacteristic, complemented
99#124×C53C2311yescharacteristic
100#123×C53—21no—
101#123×C53—21no—
102#123×C53—21no—
103#123×C53—21no—
104#123×C53—21no—
105#123×C53—21no—
106#123×C53—21no—
107#123×C53—21no—
108#123×C53—22no—
109#123×C53—41no—
110#123×C53D411yescharacteristic, complemented
111#123×C53C2311yescharacteristic
112C22⋊Q8×C53—21no—
113C22⋊Q8×C53—21no—
114C22⋊Q8×C53—21no—
115#121×C53—21no—
116#121×C53—21no—
117#121×C53—21no—
118#121×C53—21no—
119#121×C53—21no—
120#121×C53—21no—
121#121×C53—21no—
122#121×C53—21no—
123#121×C53—21no—
124#121×C53—21no—
125#121×C53—21no—
126#121×C53—21no—
127#121×C53—21no—
128#121×C53—21no—
129#121×C53—21no—
130#121×C53—42no—
131#120×C53—21no—
132#120×C53—21no—
133#120×C53—21no—
134#120×C53—21no—
135#120×C53—21no—
136#120×C53—21no—
137#120×C53—21no—
138#120×C53—21no—
139#120×C53—21no—
140#120×C53—21no—
141#120×C53—21no—
142#120×C53—21no—
143#120×C53—21no—
144#120×C53—21no—
145#120×C53—22no—
146#120×C53—22no—
147#120×C53—22no—
148#120×C53—42no—
149#120×C53—41no—
150#120×C53D411yescharacteristic, complemented
151#120×C53D411yescharacteristic, complemented
152#120×C53D411yescharacteristic, complemented
153#120×C53C2311yescharacteristic
154#120×C53C2311yescharacteristic
155#120×C53C2311yescharacteristic
156C4×D4×C53—21no—
157C4×D4×C53—21no—
158C4×D4×C53—21no—
159C4×D4×C53—21no—
160C4×D4×C53—21no—
161C4×D4×C53—21no—
162#117×C53—21no—
163#117×C53D411yescharacteristic, complemented
164C2×C22⋊C4×C53—21no—
165C2×C22⋊C4×C53—21no—
166C2×C22⋊C4×C53—21no—
167C2×C22⋊C4×C53—21no—
168C2×C22⋊C4×C53—21no—
169C2×C22⋊C4×C53—21no—
170C2×C22⋊C4×C53—21no—
171C2×C22⋊C4×C53—41no—
172C2×C22⋊C4×C53D411yescharacteristic, complemented
173C2×C22⋊C4×C53D411yescharacteristic, complemented
174#104×C53—22no—
175#104×C53—22no—
176#100×C53—21no—
177#100×C53C2312yes—
178#99×C53—22no—
179#99×C53—22no—
180#99×C53—22no—
181#99×C53—22no—
182#99×C53—21no—
183#99×C53—21no—
184#99×C53C2312yes—
185#99×C53C2312yes—
186C24×C53—21no—
187C24×C53—21no—
188C24×C53—21no—
189C24×C53—21no—
190C24×C53—21no—
191C24×C53—41no—
192C24×C53—41no—
193C24×C53—82no—
194C24×C53C2×D411yescharacteristic
195C24×C53C2×D411yescharacteristic
196C24×C53C2×D411yescharacteristic
197C24×C53C2×D411yescharacteristic
198C24×C53C2×D411yescharacteristic, complemented
199#41×C53—22no—
200#41×C53—22no—
201#41×C53—42no—
202#41×C53—41no—
203#41×C53—42no—
204#41×C53—42no—
205#41×C53—41no—
206#41×C53—41no—
207#41×C53—42no—
208#41×C53—41no—
209#41×C53—82no—
210C2×Q8×C53—21no—
211C2×Q8×C53—21no—
212C2×Q8×C53C2×D411yescharacteristic, complemented
213C2×D4×C53—21no—
214C2×D4×C53—21no—
215C2×D4×C53—21no—
216C2×D4×C53—22no—
217C2×D4×C53—22no—
218C2×D4×C53—21no—
219C2×D4×C53—21no—
220C2×D4×C53—21no—
221C2×D4×C53—22no—
222C2×D4×C53—22no—
223C2×D4×C53—21no—
224C2×D4×C53—21no—
225C2×D4×C53—21no—
226C2×D4×C53—21no—
227C2×D4×C53—21no—
228C2×D4×C53—21no—
229C2×D4×C53—21no—
230C2×D4×C53—21no—
231C2×D4×C53—21no—
232C2×D4×C53—21no—
233C2×D4×C53—21no—
234C2×D4×C53—21no—
235C2×D4×C53—21no—
236C2×D4×C53—21no—
237C2×D4×C53—21no—
238C2×D4×C53—41no—
239C2×D4×C53—41no—
240C2×D4×C53—42no—
241C2×D4×C53—42no—
242C2×D4×C53—41no—
243C2×D4×C53—41no—
244C2×D4×C53—41no—
245C2×D4×C53—41no—
246C2×D4×C53—42no—
247C2×D4×C53—42no—
248C2×D4×C53—41no—
249C2×D4×C53—41no—
250C2×D4×C53—41no—
251C2×D4×C53—41no—
252C2×D4×C53—41no—
253C2×D4×C53—42no—
254C2×D4×C53—42no—
255C2×D4×C53—42no—
256C2×D4×C53—82no—
257C2×D4×C53—82no—
258C2×D4×C53—82no—
259C2×D4×C53—82no—
260C2×D4×C53—81no—
261C2×D4×C53—81no—
262C2×D4×C53—84no—
263C2×D4×C53C2×D411yescharacteristic
264C2×D4×C53C2×D411yescharacteristic
265C2×D4×C53C2×D411yescharacteristic
266C2×D4×C53C2×D411yescharacteristic, complemented
267C2×D4×C53C2411yescharacteristic
268C22×C4×C53—21no—
269C22×C4×C53—21no—
270C22×C4×C53—21no—
271C22×C4×C53—21no—
272C22×C4×C53—21no—
273C22×C4×C53—21no—
274C22×C4×C53—21no—
275C22×C4×C53—21no—
276C22×C4×C53—21no—
277C22×C4×C53—21no—
278C22×C4×C53—41no—
279C22×C4×C53—41no—
280C22×C4×C53—41no—
281C22×C4×C53C2×D411yescharacteristic
282C22×C4×C53C2×D411yescharacteristic, complemented
283Q16×C53—42no—
284SD16×C53—42no—
285SD16×C53—42no—
286D8×C53—42no—
287M16×C53—22no—
288M16×C53—41no—
289C2×C8×C53—41no—
290C4⋊C4×C53—21no—
291C4⋊C4×C53—21no—
292C4⋊C4×C53—21no—
293C4⋊C4×C53—21no—
294C4⋊C4×C53—21no—
295C4⋊C4×C53—21no—
296C4⋊C4×C53—21no—
297C4⋊C4×C53—21no—
298C4⋊C4×C53—21no—
299C4⋊C4×C53—42no—
300C22⋊C4×C53—21no—
301C22⋊C4×C53—21no—
302C22⋊C4×C53—21no—
303C22⋊C4×C53—21no—
304C22⋊C4×C53—21no—
305C22⋊C4×C53—21no—
306C22⋊C4×C53—21no—
307C22⋊C4×C53—21no—
308C22⋊C4×C53—21no—
309C22⋊C4×C53—21no—
310C22⋊C4×C53—21no—
311C22⋊C4×C53—21no—
312C22⋊C4×C53—21no—
313C22⋊C4×C53—21no—
314C22⋊C4×C53—21no—
315C22⋊C4×C53—21no—
316C22⋊C4×C53—21no—
317C22⋊C4×C53—21no—
318C22⋊C4×C53—21no—
319C22⋊C4×C53—21no—
320C22⋊C4×C53—21no—
321C22⋊C4×C53—21no—
322C22⋊C4×C53—21no—
323C22⋊C4×C53—21no—
324C22⋊C4×C53—21no—
325C22⋊C4×C53—22no—
326C22⋊C4×C53—22no—
327C22⋊C4×C53—42no—
328C22⋊C4×C53—42no—
329C22⋊C4×C53—42no—
330C22⋊C4×C53—41no—
331C22⋊C4×C53—41no—
332C22⋊C4×C53—42no—
333C22⋊C4×C53C2×D411yescharacteristic
334C22⋊C4×C53C2×D411yescharacteristic
335C22⋊C4×C53C2×D411yescharacteristic, complemented
336C22⋊C4×C53C2×D411yescharacteristic, complemented
337C42×C53—21no—
338C42×C53—21no—
339C42×C53C2×D411yescharacteristic, complemented
340C42×C53C2×D411yescharacteristic, complemented
341C23×C53—21no—
342C23×C53—21no—
343C23×C53—21no—
344C23×C53—21no—
345C23×C53—21no—
346C23×C53—22no—
347C23×C53—22no—
348C23×C53—21no—
349C23×C53—21no—
350C23×C53—21no—
351C23×C53—41no—
352C23×C53—41no—
353C23×C53—42no—
354C23×C53—42no—
355C23×C53—42no—
356C23×C53—41no—
357C23×C53—41no—
358C23×C53—42no—
359C23×C53—41no—
360C23×C53—44no—
361C23×C53—41no—
362C23×C53—41no—
363C23×C53—41no—
364C23×C53—81no—
365C23×C53—81no—
366C23×C53—81no—
367C23×C53—81no—
368C23×C53—81no—
369C23×C53—81no—
370C23×C53—81no—
371C23×C53—81no—
372C23×C53—81no—
373C23×C53—82no—
374C23×C53—82no—
375C23×C53—82no—
376C23×C53—82no—
377C23×C53—161no—
378C23×C53#12011yescharacteristic
379C23×C53#12011yescharacteristic
380C23×C53#12011yescharacteristic
381C23×C53C22×D411yescharacteristic
382C23×C53C22×D411yescharacteristic
383Q8×C53—22no—
384Q8×C53—41no—
385Q8×C53—41no—
386Q8×C53—81no—
387D4×C53—22no—
388D4×C53—41no—
389D4×C53—41no—
390D4×C53—41no—
391D4×C53—41no—
392D4×C53—41no—
393D4×C53—41no—
394D4×C53—42no—
395D4×C53—42no—
396D4×C53—42no—
397D4×C53—42no—
398D4×C53—42no—
399D4×C53—42no—
400D4×C53—42no—
401D4×C53—42no—
402D4×C53—42no—
403D4×C53—42no—
404D4×C53—41no—
405D4×C53—41no—
406D4×C53—41no—
407D4×C53—41no—
408D4×C53—81no—
409D4×C53—81no—
410D4×C53—81no—
411D4×C53—81no—
412D4×C53—81no—
413D4×C53—81no—
414D4×C53—81no—
415D4×C53—81no—
416D4×C53—81no—
417D4×C53—81no—
418D4×C53—81no—
419D4×C53—81no—
420D4×C53—81no—
421D4×C53—81no—
422D4×C53—84no—
423D4×C53—84no—
424D4×C53—162no—
425C2×C4×C53—21no—
426C2×C4×C53—21no—
427C2×C4×C53—22no—
428C2×C4×C53—22no—
429C2×C4×C53—21no—
430C2×C4×C53—21no—
431C2×C4×C53—21no—
432C2×C4×C53—21no—
433C2×C4×C53—21no—
434C2×C4×C53—21no—
435C2×C4×C53—21no—
436C2×C4×C53—21no—
437C2×C4×C53—42no—
438C2×C4×C53—42no—
439C2×C4×C53—41no—
440C2×C4×C53—41no—
441C2×C4×C53—42no—
442C2×C4×C53—41no—
443C2×C4×C53—41no—
444C2×C4×C53—42no—
445C2×C4×C53—42no—
446C2×C4×C53—81no—
447C2×C4×C53—81no—
448C2×C4×C53—81no—
449C2×C4×C53—81no—
450C2×C4×C53—81no—
451C2×C4×C53—81no—
452C2×C4×C53—81no—
453C2×C4×C53—82no—
454C2×C4×C53—161no—
455C2×C4×C53#12011yescharacteristic
456C2×C4×C53C22×D411yescharacteristic
457C8×C53—42no—
458#33551C5311yesmaximal, characteristic, direct product factor, Sylow
459C22×C53—22no—
460C22×C53—21no—
461C22×C53—21no—
462C22×C53—21no—
463C22×C53—21no—
464C22×C53—41no—
465C22×C53—41no—
466C22×C53—41no—
467C22×C53—41no—
468C22×C53—41no—
469C22×C53—41no—
470C22×C53—42no—
471C22×C53—41no—
472C22×C53—41no—
473C22×C53—41no—
474C22×C53—41no—
475C22×C53—41no—
476C22×C53—41no—
477C22×C53—42no—
478C22×C53—41no—
479C22×C53—42no—
480C22×C53—42no—
481C22×C53—41no—
482C22×C53—41no—
483C22×C53—41no—
484C22×C53—81no—
485C22×C53—81no—
486C22×C53—81no—
487C22×C53—82no—
488C22×C53—82no—
489C22×C53—82no—
490C22×C53—81no—
491C22×C53—84no—
492C22×C53—82no—
493C22×C53—82no—
494C22×C53—82no—
495C22×C53—81no—
496C22×C53—162no—
497C22×C53—161no—
498C22×C53—161no—
499C22×C53B(4,𝔽2)11yescharacteristic
500C22×C53B(4,𝔽2)11yescharacteristic
501C22×C53C2×#12011yescharacteristic
502C4×C53—21no—
503C4×C53—21no—
504C4×C53—41no—
505C4×C53—41no—
506C4×C53—41no—
507C4×C53—41no—
508C4×C53—41no—
509C4×C53—41no—
510C4×C53—42no—
511C4×C53—41no—
512C4×C53—41no—
513C4×C53—82no—
514C4×C53—82no—
515#3532C2×C5311yescharacteristic, complemented
516#3026C2×C5311yescharacteristic, complemented
517C2×B(4,𝔽2)C2×C5311yescharacteristic, complemented
518#3019C2×C5311yescharacteristic, complemented
519#2200C2×C5312yescomplemented
520#2199C2×C5312yescomplemented
521#2197C2×C5312yescomplemented
522#2196C2×C5312yescomplemented
523#2127C2×C5311yescharacteristic, complemented
524#2121C2×C5311yescharacteristic, complemented
525C2×AΣL(1,𝔽16)C2×C5311yescharacteristic, complemented
526C2×C53—21no—
527C2×C53—21no—
528C2×C53—21no—
529C2×C53—41no—
530C2×C53—41no—
531C2×C53—42no—
532C2×C53—41no—
533C2×C53—41no—
534C2×C53—81no—
535C2×C53—81no—
536C2×C53—81no—
537C2×C53—82no—
538C2×C53C2×B(4,𝔽2)11yescharacteristic
539C2×H(5,𝔽2)C22×C5311yescharacteristic, complemented
540#575—21no—
541H(3,𝔽4)C22×C5311yescharacteristic, complemented
542#560—21no—
543#560C22×C5311yescharacteristic, complemented
544#551—21no—
545#551C22×C5311yescharacteristic, complemented
546#546—21no—
547#546—21no—
548#546—21no—
549D42—21no—
550D42C22×C5311yescharacteristic, complemented
551#538—21no—
552#538—21no—
553#538—21no—
554#535—21no—
555#535C22×C5311yescharacteristic
556#534—21no—
557#534—21no—
558#534C22×C5311yescharacteristic
559C2×#120—21no—
560C2×#120—21no—
561C2×#120C22×C5311yescharacteristic
562#458—21no—
563#458C22×C5312yescomplemented
564B(4,𝔽2)—21no—
565B(4,𝔽2)—21no—
566B(4,𝔽2)—21no—
567B(4,𝔽2)C22×C5312yes—
568B(4,𝔽2)C22×C5312yescomplemented
569B(4,𝔽2)C22×C5312yescomplemented
570#455—21no—
571#455C22×C5312yescomplemented
572#453—21no—
573#453C22×C5312yescomplemented
574#421—21no—
575C2×#100C22×C5311yescharacteristic, complemented
576C2×#99—21no—
577C2×#99—21no—
578C2×#99C22×C5311yescharacteristic, complemented
579C2×#99C22×C5311yescharacteristic, complemented
580#354C22×C5312yescomplemented
581#353C22×C5312yescomplemented
582#352C22×C5312yescomplemented
583AΣL(1,𝔽16)C22×C5312yescomplemented
584AΣL(1,𝔽16)C22×C5312yescomplemented
585AΣL(1,𝔽16)C22×C5312yescomplemented
586C53#3355111yescharacteristic, direct product factor, Sylow
587C25D4×C5311yescharacteristic, complemented
588H(5,𝔽2)—21no—
589H(5,𝔽2)—21no—
590H(5,𝔽2)—21no—
591H(5,𝔽2)—42no—
592H(5,𝔽2)C23×C5312yes—
593C2×#41—21no—
594C2×#41—21no—
595C2×#41—41no—
596C2×#41D4×C5311yescharacteristic, complemented
597C22×D4—21no—
598C22×D4—21no—
599C22×D4—21no—
600C22×D4—21no—
601C22×D4—21no—
602C22×D4—21no—
603C22×D4—21no—
604C22×D4—41no—
605C22×D4—41no—
606C22×D4D4×C5311yescharacteristic, complemented
607C22×D4C23×C5311yescharacteristic
608#137—22no—
609#137—41no—
610AGL(1,ℤ/8)—22no—
611AGL(1,ℤ/8)—41no—
612#135—42no—
613C2×M16—21no—
614#127—21no—
615#127D4×C5311yescharacteristic, complemented
616#127C23×C5311yescharacteristic
617#126—21no—
618#126—21no—
619#126—21no—
620#126—21no—
621#124—21no—
622#124—21no—
623#124—21no—
624#124—21no—
625#124—21no—
626#124D4×C5311yescharacteristic, complemented
627#124C23×C5311yescharacteristic
628#123—21no—
629#123—21no—
630#123—21no—
631#123—21no—
632#123—21no—
633#123—21no—
634#123—21no—
635#123—21no—
636#123—22no—
637#123—41no—
638#123D4×C5311yescharacteristic, complemented
639#123C23×C5311yescharacteristic
640C22⋊Q8—21no—
641C22⋊Q8—21no—
642C22⋊Q8—21no—
643#121—21no—
644#121—21no—
645#121—21no—
646#121—21no—
647#121—21no—
648#121—21no—
649#121—21no—
650#121—21no—
651#121—21no—
652#121—21no—
653#121—21no—
654#121—21no—
655#121—21no—
656#121—21no—
657#121—21no—
658#121—42no—
659#120—21no—
660#120—21no—
661#120—21no—
662#120—21no—
663#120—21no—
664#120—21no—
665#120—21no—
666#120—21no—
667#120—21no—
668#120—21no—
669#120—21no—
670#120—21no—
671#120—21no—
672#120—21no—
673#120—22no—
674#120—22no—
675#120—22no—
676#120—42no—
677#120—41no—
678#120D4×C5311yescharacteristic, complemented
679#120D4×C5311yescharacteristic, complemented
680#120D4×C5311yescharacteristic, complemented
681#120C23×C5311yescharacteristic
682#120C23×C5311yescharacteristic
683#120C23×C5311yescharacteristic
684C4×D4—21no—
685C4×D4—21no—
686C4×D4—21no—
687C4×D4—21no—
688C4×D4—21no—
689C4×D4—21no—
690#117—21no—
691#117D4×C5311yescharacteristic, complemented
692C2×C22⋊C4—21no—
693C2×C22⋊C4—21no—
694C2×C22⋊C4—21no—
695C2×C22⋊C4—21no—
696C2×C22⋊C4—21no—
697C2×C22⋊C4—21no—
698C2×C22⋊C4—21no—
699C2×C22⋊C4—41no—
700C2×C22⋊C4D4×C5311yescharacteristic, complemented
701C2×C22⋊C4D4×C5311yescharacteristic, complemented
702#104—22no—
703#104—22no—
704#100—21no—
705#100C23×C5312yes—
706#99—21no—
707#99—21no—
708#99—22no—
709#99—22no—
710#99—22no—
711#99—22no—
712#99C23×C5312yes—
713#99C23×C5312yes—
714C24—21no—
715C24—21no—
716C24—21no—
717C24—21no—
718C24—21no—
719C24—41no—
720C24—41no—
721C24—82no—
722C24C2×D4×C5311yescharacteristic
723C24C2×D4×C5311yescharacteristic
724C24C2×D4×C5311yescharacteristic
725C24C2×D4×C5311yescharacteristic
726C24C2×D4×C5311yescharacteristic, complemented
727#41—22no—
728#41—22no—
729#41—42no—
730#41—41no—
731#41—42no—
732#41—42no—
733#41—41no—
734#41—41no—
735#41—42no—
736#41—41no—
737#41—82no—
738C2×Q8—21no—
739C2×Q8—21no—
740C2×Q8C2×D4×C5311yescharacteristic, complemented
741C2×D4—21no—
742C2×D4—21no—
743C2×D4—21no—
744C2×D4—22no—
745C2×D4—22no—
746C2×D4—21no—
747C2×D4—21no—
748C2×D4—21no—
749C2×D4—22no—
750C2×D4—22no—
751C2×D4—21no—
752C2×D4—21no—
753C2×D4—21no—
754C2×D4—21no—
755C2×D4—21no—
756C2×D4—21no—
757C2×D4—21no—
758C2×D4—21no—
759C2×D4—21no—
760C2×D4—21no—
761C2×D4—21no—
762C2×D4—21no—
763C2×D4—21no—
764C2×D4—21no—
765C2×D4—21no—
766C2×D4—41no—
767C2×D4—41no—
768C2×D4—42no—
769C2×D4—42no—
770C2×D4—41no—
771C2×D4—41no—
772C2×D4—41no—
773C2×D4—41no—
774C2×D4—42no—
775C2×D4—42no—
776C2×D4—41no—
777C2×D4—41no—
778C2×D4—41no—
779C2×D4—41no—
780C2×D4—41no—
781C2×D4—42no—
782C2×D4—42no—
783C2×D4—42no—
784C2×D4—82no—
785C2×D4—82no—
786C2×D4—82no—
787C2×D4—82no—
788C2×D4—81no—
789C2×D4—81no—
790C2×D4—84no—
791C2×D4C2×D4×C5311yescharacteristic
792C2×D4C2×D4×C5311yescharacteristic
793C2×D4C2×D4×C5311yescharacteristic
794C2×D4C2×D4×C5311yescharacteristic, complemented
795C2×D4C24×C5311yescharacteristic
796C22×C4—21no—
797C22×C4—21no—
798C22×C4—21no—
799C22×C4—21no—
800C22×C4—21no—
801C22×C4—21no—
802C22×C4—21no—
803C22×C4—21no—
804C22×C4—21no—
805C22×C4—21no—
806C22×C4—41no—
807C22×C4—41no—
808C22×C4—41no—
809C22×C4C2×D4×C5311yescharacteristic
810C22×C4C2×D4×C5311yescharacteristic, complemented
811Q16—42no—
812SD16—42no—
813SD16—42no—
814D8—42no—
815M16—22no—
816M16—41no—
817C2×C8—41no—
818C4⋊C4—21no—
819C4⋊C4—21no—
820C4⋊C4—21no—
821C4⋊C4—21no—
822C4⋊C4—21no—
823C4⋊C4—21no—
824C4⋊C4—21no—
825C4⋊C4—21no—
826C4⋊C4—21no—
827C4⋊C4—42no—
828C22⋊C4—21no—
829C22⋊C4—21no—
830C22⋊C4—21no—
831C22⋊C4—21no—
832C22⋊C4—21no—
833C22⋊C4—21no—
834C22⋊C4—21no—
835C22⋊C4—21no—
836C22⋊C4—21no—
837C22⋊C4—21no—
838C22⋊C4—21no—
839C22⋊C4—21no—
840C22⋊C4—21no—
841C22⋊C4—21no—
842C22⋊C4—21no—
843C22⋊C4—21no—
844C22⋊C4—21no—
845C22⋊C4—21no—
846C22⋊C4—21no—
847C22⋊C4—21no—
848C22⋊C4—21no—
849C22⋊C4—21no—
850C22⋊C4—21no—
851C22⋊C4—21no—
852C22⋊C4—21no—
853C22⋊C4—22no—
854C22⋊C4—22no—
855C22⋊C4—42no—
856C22⋊C4—42no—
857C22⋊C4—42no—
858C22⋊C4—41no—
859C22⋊C4—41no—
860C22⋊C4—42no—
861C22⋊C4C2×D4×C5311yescharacteristic
862C22⋊C4C2×D4×C5311yescharacteristic
863C22⋊C4C2×D4×C5311yescharacteristic, complemented
864C22⋊C4C2×D4×C5311yescharacteristic, complemented
865C42—21no—
866C42—21no—
867C42C2×D4×C5311yescharacteristic, complemented
868C42C2×D4×C5311yescharacteristic, complemented
869C23—21no—
870C23—21no—
871C23—21no—
872C23—21no—
873C23—21no—
874C23—22no—
875C23—22no—
876C23—21no—
877C23—21no—
878C23—21no—
879C23—41no—
880C23—41no—
881C23—42no—
882C23—42no—
883C23—42no—
884C23—41no—
885C23—41no—
886C23—42no—
887C23—41no—
888C23—44no—
889C23—41no—
890C23—41no—
891C23—41no—
892C23—81no—
893C23—81no—
894C23—81no—
895C23—81no—
896C23—81no—
897C23—81no—
898C23—81no—
899C23—81no—
900C23—81no—
901C23—82no—
902C23—82no—
903C23—82no—
904C23—82no—
905C23—161no—
906C23#120×C5311yescharacteristic
907C23#120×C5311yescharacteristic
908C23#120×C5311yescharacteristic
909C23C22×D4×C5311yescharacteristic
910C23C22×D4×C5311yescharacteristic
911Q8—22no—
912Q8—41no—
913Q8—41no—
914Q8—81no—
915D4—22no—
916D4—41no—
917D4—41no—
918D4—41no—
919D4—41no—
920D4—41no—
921D4—41no—
922D4—42no—
923D4—42no—
924D4—42no—
925D4—42no—
926D4—42no—
927D4—42no—
928D4—42no—
929D4—42no—
930D4—42no—
931D4—42no—
932D4—41no—
933D4—41no—
934D4—41no—
935D4—41no—
936D4—81no—
937D4—81no—
938D4—81no—
939D4—81no—
940D4—81no—
941D4—81no—
942D4—81no—
943D4—81no—
944D4—81no—
945D4—81no—
946D4—81no—
947D4—81no—
948D4—81no—
949D4—81no—
950D4—84no—
951D4—84no—
952D4—162no—
953C2×C4—21no—
954C2×C4—21no—
955C2×C4—22no—
956C2×C4—22no—
957C2×C4—21no—
958C2×C4—21no—
959C2×C4—21no—
960C2×C4—21no—
961C2×C4—21no—
962C2×C4—21no—
963C2×C4—21no—
964C2×C4—21no—
965C2×C4—42no—
966C2×C4—42no—
967C2×C4—41no—
968C2×C4—41no—
969C2×C4—42no—
970C2×C4—41no—
971C2×C4—41no—
972C2×C4—42no—
973C2×C4—42no—
974C2×C4—81no—
975C2×C4—81no—
976C2×C4—81no—
977C2×C4—81no—
978C2×C4—81no—
979C2×C4—81no—
980C2×C4—81no—
981C2×C4—82no—
982C2×C4—161no—
983C2×C4#120×C5311yescharacteristic
984C2×C4C22×D4×C5311yescharacteristic
985C8—42no—
986C22—22no—
987C22—21no—
988C22—21no—
989C22—21no—
990C22—21no—
991C22—41no—
992C22—41no—
993C22—41no—
994C22—41no—
995C22—41no—
996C22—41no—
997C22—42no—
998C22—41no—
999C22—41no—
1000C22—41no—
1001C22—41no—
1002C22—41no—
1003C22—41no—
1004C22—42no—
1005C22—41no—
1006C22—42no—
1007C22—42no—
1008C22—41no—
1009C22—41no—
1010C22—41no—
1011C22—81no—
1012C22—81no—
1013C22—81no—
1014C22—82no—
1015C22—82no—
1016C22—82no—
1017C22—81no—
1018C22—84no—
1019C22—82no—
1020C22—82no—
1021C22—82no—
1022C22—81no—
1023C22—162no—
1024C22—161no—
1025C22—161no—
1026C22C53×B(4,𝔽2)11yescharacteristic
1027C22C53×B(4,𝔽2)11yescharacteristic
1028C22C2×#120×C5311yescharacteristic
1029C4—21no—
1030C4—21no—
1031C4—41no—
1032C4—41no—
1033C4—41no—
1034C4—41no—
1035C4—41no—
1036C4—41no—
1037C4—42no—
1038C4—41no—
1039C4—41no—
1040C4—82no—
1041C4—82no—
1042C2—21no—
1043C2—21no—
1044C2—21no—
1045C2—41no—
1046C2—41no—
1047C2—42no—
1048C2—41no—
1049C2—41no—
1050C2—81no—
1051C2—81no—
1052C2—81no—
1053C2—82no—
1054C2C2×C53×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C53C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C53
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3532C2semidirect
2. C53×#3026C2semidirect
3. C2×C53×B(4,𝔽2)C2semidirect
4. C53×#3019C2semidirect
5. C53×#2200C2semidirect
6. C53×#2199C2semidirect
7. C53×#2197C2semidirect
8. C53×#2196C2semidirect
9. C53×#2127C2semidirect
10. C53×#2121C2semidirect
11. C2×C53×AΣL(1,𝔽16)C2semidirect
12. C2×H(5,𝔽2)×C53C22semidirect
14. C53×H(3,𝔽4)C22semidirect
16. C53×#560C22semidirect
18. C53×#551C22semidirect
23. D42×C53C22semidirect
36. C53×#458C22semidirect
41. C53×B(4,𝔽2)C22semidirect
42. C53×B(4,𝔽2)C22semidirect
44. C53×#455C22semidirect
46. C53×#453C22semidirect
48. C2×#100×C53C22semidirect
51. C2×#99×C53C22semidirect
52. C2×#99×C53C22semidirect
53. C53×#354C22semidirect
54. C53×#353C22semidirect
55. C53×#352C22semidirect
56. C53×AΣL(1,𝔽16)C22semidirect
57. C53×AΣL(1,𝔽16)C22semidirect
58. C53×AΣL(1,𝔽16)C22semidirect
59. C25×C53D4semidirect
68. C2×#41×C53D4semidirect
78. C22×D4×C53D4semidirect
87. #127×C53D4semidirect
98. #124×C53D4semidirect
110. #123×C53D4semidirect
150. #120×C53D4semidirect
151. #120×C53D4semidirect
152. #120×C53D4semidirect
163. #117×C53D4semidirect
172. C2×C22⋊C4×C53D4semidirect
173. C2×C22⋊C4×C53D4semidirect
198. C24×C53C2×D4semidirect
212. C2×Q8×C53C2×D4semidirect
266. C2×D4×C53C2×D4semidirect
282. C22×C4×C53C2×D4semidirect
335. C22⋊C4×C53C2×D4semidirect
336. C22⋊C4×C53C2×D4semidirect
339. C42×C53C2×D4semidirect
340. C42×C53C2×D4semidirect
458. #33551C53direct (N × H)
515. #3532C2×C53semidirect
516. #3026C2×C53semidirect
517. C2×B(4,𝔽2)C2×C53semidirect
518. #3019C2×C53semidirect
519. #2200C2×C53semidirect
520. #2199C2×C53semidirect
521. #2197C2×C53semidirect
522. #2196C2×C53semidirect
523. #2127C2×C53semidirect
524. #2121C2×C53semidirect
525. C2×AΣL(1,𝔽16)C2×C53semidirect
539. C2×H(5,𝔽2)C22×C53semidirect
541. H(3,𝔽4)C22×C53semidirect
543. #560C22×C53semidirect
545. #551C22×C53semidirect
550. D42C22×C53semidirect
563. #458C22×C53semidirect
568. B(4,𝔽2)C22×C53semidirect
569. B(4,𝔽2)C22×C53semidirect
571. #455C22×C53semidirect
573. #453C22×C53semidirect
575. C2×#100C22×C53semidirect
578. C2×#99C22×C53semidirect
579. C2×#99C22×C53semidirect
580. #354C22×C53semidirect
581. #353C22×C53semidirect
582. #352C22×C53semidirect
583. AΣL(1,𝔽16)C22×C53semidirect
584. AΣL(1,𝔽16)C22×C53semidirect
585. AΣL(1,𝔽16)C22×C53semidirect
586. C53#33551direct (N × H)
587. C25D4×C53semidirect
596. C2×#41D4×C53semidirect
606. C22×D4D4×C53semidirect
615. #127D4×C53semidirect
626. #124D4×C53semidirect
638. #123D4×C53semidirect
678. #120D4×C53semidirect
679. #120D4×C53semidirect
680. #120D4×C53semidirect
691. #117D4×C53semidirect
700. C2×C22⋊C4D4×C53semidirect
701. C2×C22⋊C4D4×C53semidirect
726. C24C2×D4×C53semidirect
740. C2×Q8C2×D4×C53semidirect
794. C2×D4C2×D4×C53semidirect
810. C22×C4C2×D4×C53semidirect
863. C22⋊C4C2×D4×C53semidirect
864. C22⋊C4C2×D4×C53semidirect
867. C42C2×D4×C53semidirect
868. C42C2×D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#35322
C53×#30262
C2×C53×B(4,𝔽2)2
C53×#30192
C53×#22002
C53×#21992
C53×#21972
C53×#21962
C53×#21272
C53×#21212
C2×C53×AΣL(1,𝔽16)2
#3355153