Finite Group Databasebeta

← C53×#24257 #27231769

Subgroups — C53×#24257 #27231769

Order 13568.

1650 conjugacy classes of subgroups.

The subgroup lattice has 966 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#546C211yesmaximal, characteristic, complemented
2C2×C53×#463C211yesmaximal, characteristic, complemented
3C2×C53×#449C211yesmaximal, characteristic, complemented
4C2×C53×#447C211yesmaximal, characteristic, complemented
5C2×C53×#442C211yesmaximal, characteristic, complemented
6C53×#2537C211yesmaximal, characteristic, complemented
7C53×#2035C211yesmaximal, characteristic, complemented
8C53×#2034C211yesmaximal, characteristic, complemented
9C53×#2000C211yesmaximal, characteristic, complemented
10C53×#1999C211yesmaximal, characteristic, complemented
11C53×#1959C211yesmaximal, characteristic, complemented
12C53×#1955C211yesmaximal, characteristic, complemented
13C53×#1802C211yesmaximal, characteristic, complemented
14C53×#1788C211yesmaximal, characteristic, complemented
15C53×#1786C211yesmaximal, characteristic, complemented
16C22×#41×C53C2211yescharacteristic
17C23×D4×C53C2211yescharacteristic
18C22×SD16×C53C2211yescharacteristic, complemented
19C22×D8×C53C2211yescharacteristic
20C22×D8×C53C2211yescharacteristic
21C53×#546—24no—
22C53×#546—24no—
23C2×#124×C53C2211yescharacteristic
24C2×#123×C53—21no—
25C2×C22⋊Q8×C53C2211yescharacteristic
26C2×#121×C53—21no—
27C2×#121×C53—21no—
28C2×#121×C53C2211yescharacteristic
29C2×#121×C53C2211yescharacteristic
30C2×#121×C53C2211yescharacteristic, complemented
31C2×#120×C53—21no—
32C2×C4×D4×C53C2211yescharacteristic
33C2×C4×D4×C53C2211yescharacteristic
34C2×C4×D4×C53C2211yescharacteristic, complemented
35C22×C22⋊C4×C53—21no—
36C53×#463—24no—
37C53×#449—24no—
38C53×#447—24no—
39C53×#442—24no—
40C2×#106×C53C2211yescharacteristic, complemented
41C2×C4⋊C8×C53C2211yescharacteristic, complemented
42C2×#103×C53C2211yescharacteristic, complemented
43C2×#103×C53C2211yescharacteristic, complemented
44C2×#103×C53C2211yescharacteristic, complemented
45C2×#102×C53C2211yescharacteristic
46C2×#102×C53C2211yescharacteristic
47C2×#102×C53C2211yescharacteristic
48C2×#102×C53C2211yescharacteristic
49C2×#102×C53C2211yescharacteristic
50C2×#102×C53C2211yescharacteristic
51C2×#102×C53C2211yescharacteristic, complemented
52C2×C22⋊C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic, complemented
55C2×C22⋊C8×C53C2211yescharacteristic, complemented
56C2×#97×C53C2211yescharacteristic, complemented
57C53×#394—21no—
58C53×#394—21no—
59C53×#384C2211yescharacteristic, complemented
60C53×#380—21no—
61C53×#380C2211yescharacteristic
62C53×#380C2211yescharacteristic, complemented
63C53×#340C2211yescharacteristic, complemented
64C53×#340C2211yescharacteristic, complemented
65C25×C53—21no—
66C2×#41×C53—22no—
67C2×#41×C53—22no—
68C2×#41×C53—22no—
69C2×#41×C53—22no—
70C2×#41×C53—41no—
71C2×#41×C53—41no—
72C22×Q8×C53C2311yescharacteristic
73C22×D4×C53—21no—
74C22×D4×C53—21no—
75C22×D4×C53—22no—
76C22×D4×C53—22no—
77C22×D4×C53—22no—
78C22×D4×C53—22no—
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—41no—
83C22×D4×C53—41no—
84C22×D4×C53D411yescharacteristic
85C22×D4×C53C2311yescharacteristic
86C22×D4×C53C2311yescharacteristic
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53—21no—
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C2×SD16×C53—22no—
94C2×SD16×C53—22no—
95C2×SD16×C53—41no—
96C2×D8×C53—22no—
97C2×D8×C53—22no—
98C2×D8×C53—22no—
99C2×D8×C53—22no—
100C2×D8×C53—41no—
101C2×D8×C53D414yescomplemented
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106#124×C53—22no—
107#124×C53D414yescomplemented
108#123×C53—28no—
109C22⋊Q8×C53—22no—
110C22⋊Q8×C53—22no—
111#121×C53—28no—
112#121×C53—22no—
113#121×C53—22no—
114#121×C53—22no—
115#121×C53—22no—
116#121×C53—41no—
117#121×C53—41no—
118#121×C53—44no—
119#120×C53—28no—
120C4×D4×C53—22no—
121C4×D4×C53—22no—
122C4×D4×C53—22no—
123C4×D4×C53—22no—
124C4×D4×C53—22no—
125C4×D4×C53D414yescomplemented
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53—21no—
129C2×C4⋊C4×C53D411yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C4⋊C4×C53C2311yescharacteristic
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C22⋊C4×C53—28no—
135C2×C22⋊C4×C53—21no—
136C2×C22⋊C4×C53—21no—
137C2×C22⋊C4×C53—21no—
138C2×C22⋊C4×C53—21no—
139C2×C22⋊C4×C53—21no—
140C2×C22⋊C4×C53—21no—
141C2×C22⋊C4×C53—21no—
142C2×C22⋊C4×C53—21no—
143C2×C22⋊C4×C53—21no—
144C2×C22⋊C4×C53—21no—
145C2×C22⋊C4×C53D411yescharacteristic
146C2×C22⋊C4×C53D411yescharacteristic
147C2×C42×C53C2311yescharacteristic
148#106×C53—22no—
149C4⋊C8×C53—22no—
150#103×C53—22no—
151#103×C53—22no—
152#103×C53—41no—
153#102×C53—22no—
154#102×C53—22no—
155#102×C53—22no—
156#102×C53—22no—
157#102×C53—22no—
158#102×C53—22no—
159#102×C53—41no—
160C22⋊C8×C53—22no—
161C22⋊C8×C53—22no—
162C22⋊C8×C53—41no—
163C22⋊C8×C53—41no—
164#97×C53—22no—
165#95×C53—21no—
166#95×C53—21no—
167C24×C53—21no—
168C24×C53—24no—
169C24×C53—21no—
170C24×C53—21no—
171C24×C53—42no—
172C24×C53—42no—
173C24×C53—42no—
174C24×C53—42no—
175C24×C53—81no—
176C24×C53—81no—
177C24×C53C2×D411yescharacteristic
178C24×C53C2×D411yescharacteristic
179#41×C53—44no—
180#41×C53—44no—
181#41×C53—44no—
182#41×C53—44no—
183C2×Q8×C53—21no—
184C2×Q8×C53—21no—
185C2×Q8×C53—21no—
186C2×Q8×C53—21no—
187C2×Q8×C53—21no—
188C2×Q8×C53C2×D412yes—
189C2×D4×C53—24no—
190C2×D4×C53—24no—
191C2×D4×C53—22no—
192C2×D4×C53—24no—
193C2×D4×C53—22no—
194C2×D4×C53—21no—
195C2×D4×C53—21no—
196C2×D4×C53—21no—
197C2×D4×C53—21no—
198C2×D4×C53—21no—
199C2×D4×C53—21no—
200C2×D4×C53—21no—
201C2×D4×C53—21no—
202C2×D4×C53—21no—
203C2×D4×C53—21no—
204C2×D4×C53—21no—
205C2×D4×C53—21no—
206C2×D4×C53—21no—
207C2×D4×C53—41no—
208C2×D4×C53—42no—
209C2×D4×C53—42no—
210C2×D4×C53—44no—
211C2×D4×C53—44no—
212C2×D4×C53—44no—
213C2×D4×C53—42no—
214C2×D4×C53—42no—
215C2×D4×C53—44no—
216C2×D4×C53—42no—
217C2×D4×C53—42no—
218C2×D4×C53—42no—
219C2×D4×C53—42no—
220C2×D4×C53—81no—
221C2×D4×C53—81no—
222C2×D4×C53—81no—
223C2×D4×C53—81no—
224C2×D4×C53C2×D412yes—
225C2×D4×C53C2×D412yes—
226C2×D4×C53C2×D412yes—
227C2×D4×C53C2×D412yes—
228C2×D4×C53#4112yes—
229C22×C4×C53—24no—
230C22×C4×C53—21no—
231C22×C4×C53—21no—
232C22×C4×C53—21no—
233C22×C4×C53—21no—
234C22×C4×C53—21no—
235C22×C4×C53—21no—
236C22×C4×C53—21no—
237C22×C4×C53—21no—
238C22×C4×C53—21no—
239C22×C4×C53—21no—
240C22×C4×C53—21no—
241C22×C4×C53—21no—
242C22×C4×C53—21no—
243C22×C4×C53—21no—
244C22×C4×C53—24no—
245C22×C4×C53—21no—
246C22×C4×C53—21no—
247C22×C4×C53—21no—
248C22×C4×C53—42no—
249C22×C4×C53—42no—
250C22×C4×C53—42no—
251C22×C4×C53—42no—
252C22×C4×C53—42no—
253C22×C4×C53—81no—
254C22×C4×C53—81no—
255C22×C4×C53C2×D411yescharacteristic
256C22×C4×C53C2×D411yescharacteristic
257C22×C4×C53C2×D411yescharacteristic
258C22×C4×C53C2×D411yescharacteristic
259C22×C4×C53C2411yescharacteristic
260SD16×C53—44no—
261D8×C53—28no—
262D8×C53—44no—
263C2×C8×C53—21no—
264C2×C8×C53—21no—
265C2×C8×C53—21no—
266C2×C8×C53—21no—
267C2×C8×C53—21no—
268C2×C8×C53—21no—
269C2×C8×C53—21no—
270C2×C8×C53—21no—
271C2×C8×C53—21no—
272C2×C8×C53—21no—
273C2×C8×C53—21no—
274C2×C8×C53C2×D412yes—
275C4⋊C4×C53—24no—
276C4⋊C4×C53—22no—
277C4⋊C4×C53—21no—
278C4⋊C4×C53—21no—
279C4⋊C4×C53—21no—
280C4⋊C4×C53—21no—
281C4⋊C4×C53—21no—
282C4⋊C4×C53—21no—
283C4⋊C4×C53—21no—
284C4⋊C4×C53—42no—
285C4⋊C4×C53—81no—
286C4⋊C4×C53C2×D412yes—
287C22⋊C4×C53—24no—
288C22⋊C4×C53—24no—
289C22⋊C4×C53—22no—
290C22⋊C4×C53—24no—
291C22⋊C4×C53—24no—
292C22⋊C4×C53—24no—
293C22⋊C4×C53—22no—
294C22⋊C4×C53—42no—
295C22⋊C4×C53—42no—
296C22⋊C4×C53—42no—
297C22⋊C4×C53—42no—
298C22⋊C4×C53—42no—
299C42×C53C2×D412yes—
300C42×C53C2×D412yes—
301C23×C53—22no—
302C23×C53—21no—
303C23×C53—21no—
304C23×C53—22no—
305C23×C53—21no—
306C23×C53—22no—
307C23×C53—21no—
308C23×C53—21no—
309C23×C53—21no—
310C23×C53—21no—
311C23×C53—22no—
312C23×C53—21no—
313C23×C53—22no—
314C23×C53—21no—
315C23×C53—21no—
316C23×C53—21no—
317C23×C53—21no—
318C23×C53—21no—
319C23×C53—48no—
320C23×C53—48no—
321C23×C53—41no—
322C23×C53—41no—
323C23×C53—41no—
324C23×C53—41no—
325C23×C53—41no—
326C23×C53—41no—
327C23×C53—41no—
328C23×C53—41no—
329C23×C53—41no—
330C23×C53—41no—
331C23×C53—41no—
332C23×C53—41no—
333C23×C53—41no—
334C23×C53—41no—
335C23×C53—41no—
336C23×C53—41no—
337C23×C53—84no—
338C23×C53—84no—
339C23×C53—84no—
340C23×C53—84no—
341C23×C53—84no—
342C23×C53AGL(1,ℤ/8)12yes—
343C23×C53C22×D411yescharacteristic
344Q8×C53—22no—
345Q8×C53—22no—
346Q8×C53—41no—
347Q8×C53—41no—
348D4×C53—22no—
349D4×C53—22no—
350D4×C53—22no—
351D4×C53—22no—
352D4×C53—22no—
353D4×C53—22no—
354D4×C53—22no—
355D4×C53—44no—
356D4×C53—48no—
357D4×C53—48no—
358D4×C53—41no—
359D4×C53—41no—
360D4×C53—41no—
361D4×C53—41no—
362D4×C53—84no—
363D4×C53—84no—
364D4×C53—84no—
365D4×C53#12114yescomplemented
366C2×C4×C53—21no—
367C2×C4×C53—21no—
368C2×C4×C53—21no—
369C2×C4×C53—21no—
370C2×C4×C53—21no—
371C2×C4×C53—22no—
372C2×C4×C53—22no—
373C2×C4×C53—21no—
374C2×C4×C53—21no—
375C2×C4×C53—21no—
376C2×C4×C53—21no—
377C2×C4×C53—21no—
378C2×C4×C53—48no—
379C2×C4×C53—41no—
380C2×C4×C53—41no—
381C2×C4×C53—41no—
382C2×C4×C53—41no—
383C2×C4×C53—41no—
384C2×C4×C53—41no—
385C2×C4×C53—41no—
386C2×C4×C53—41no—
387C2×C4×C53—41no—
388C2×C4×C53—41no—
389C2×C4×C53—41no—
390C2×C4×C53—41no—
391C2×C4×C53—41no—
392C2×C4×C53—41no—
393C2×C4×C53—41no—
394C2×C4×C53—41no—
395C2×C4×C53—41no—
396C2×C4×C53—41no—
397C2×C4×C53—41no—
398C2×C4×C53—41no—
399C2×C4×C53—41no—
400C2×C4×C53—84no—
401C2×C4×C53—84no—
402C2×C4×C53#12012yes—
403C2×C4×C53#12012yes—
404C2×C4×C53C22×D411yescharacteristic
405C2×C4×C53C22×D411yescharacteristic
406C2×C4×C53C22×D411yescharacteristic
407C2×C4×C53C2×#4111yescharacteristic
408C2×C4×C53H(5,𝔽2)11yescharacteristic
409C2×C4×C53H(5,𝔽2)11yescharacteristic
410C8×C53—22no—
411C8×C53—41no—
412C8×C53—41no—
413C8×C53—41no—
414#24257C5311yesmaximal, characteristic, direct product factor, Sylow
415C22×C53—24no—
416C22×C53—22no—
417C22×C53—22no—
418C22×C53—22no—
419C22×C53—41no—
420C22×C53—42no—
421C22×C53—42no—
422C22×C53—41no—
423C22×C53—42no—
424C22×C53—42no—
425C22×C53—41no—
426C22×C53—42no—
427C22×C53—41no—
428C22×C53—42no—
429C22×C53—41no—
430C22×C53—42no—
431C22×C53—42no—
432C22×C53—41no—
433C22×C53—42no—
434C22×C53—41no—
435C22×C53—42no—
436C22×C53—41no—
437C22×C53—42no—
438C22×C53—41no—
439C22×C53—42no—
440C22×C53—41no—
441C22×C53—41no—
442C22×C53—816no—
443C22×C53—81no—
444C22×C53—81no—
445C22×C53—81no—
446C22×C53—81no—
447C22×C53—81no—
448C22×C53—81no—
449C22×C53—81no—
450C22×C53—81no—
451C22×C53C2×#12011yescharacteristic
452C22×C53D4211yescharacteristic
453C22×C53#54611yescharacteristic
454C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
455C22×C53#57511yescharacteristic
456C22×C53#57611yescharacteristic
457C22×C53#57711yescharacteristic
458C4×C53—22no—
459C4×C53—22no—
460C4×C53—41no—
461C4×C53—41no—
462C4×C53—42no—
463C4×C53—42no—
464C4×C53—81no—
465C4×C53—81no—
466C4×C53—81no—
467C4×C53C2×#12111yescharacteristic
468C4×C53#53511yescharacteristic
469C4×C53D4211yescharacteristic
470C4×C53#54611yescharacteristic
471C2×#546C2×C5311yescharacteristic, complemented
472C2×#463C2×C5311yescharacteristic, complemented
473C2×#449C2×C5311yescharacteristic, complemented
474C2×#447C2×C5311yescharacteristic, complemented
475C2×#442C2×C5311yescharacteristic, complemented
476#2537C2×C5311yescharacteristic, complemented
477#2035C2×C5311yescharacteristic, complemented
478#2034C2×C5311yescharacteristic, complemented
479#2000C2×C5311yescharacteristic, complemented
480#1999C2×C5311yescharacteristic, complemented
481#1959C2×C5311yescharacteristic, complemented
482#1955C2×C5311yescharacteristic, complemented
483#1802C2×C5311yescharacteristic, complemented
484#1788C2×C5311yescharacteristic, complemented
485#1786C2×C5311yescharacteristic, complemented
486C2×C53—42no—
487C2×C53—44no—
488C2×C53—81no—
489C2×C53—82no—
490C2×C53—82no—
491C2×C53#240811yescharacteristic
492C2×C53#300611yescharacteristic
493C2×C53#301211yescharacteristic
494C2×C53#326411yescharacteristic
495C2×C53#326711yescharacteristic
496C2×C53#330611yescharacteristic
497C2×C53#331911yescharacteristic
498C22×#41C22×C5311yescharacteristic
499C23×D4C22×C5311yescharacteristic
500C22×SD16C22×C5311yescharacteristic, complemented
501C22×D8C22×C5311yescharacteristic
502C22×D8C22×C5311yescharacteristic
503#546—24no—
504#546—24no—
505C2×#124C22×C5311yescharacteristic
506C2×#123—21no—
507C2×C22⋊Q8C22×C5311yescharacteristic
508C2×#121—21no—
509C2×#121—21no—
510C2×#121C22×C5311yescharacteristic
511C2×#121C22×C5311yescharacteristic
512C2×#121C22×C5311yescharacteristic, complemented
513C2×#120—21no—
514C2×C4×D4C22×C5311yescharacteristic
515C2×C4×D4C22×C5311yescharacteristic
516C2×C4×D4C22×C5311yescharacteristic, complemented
517C22×C22⋊C4—21no—
518#463—24no—
519#449—24no—
520#447—24no—
521#442—24no—
522C2×#106C22×C5311yescharacteristic, complemented
523C2×C4⋊C8C22×C5311yescharacteristic, complemented
524C2×#103C22×C5311yescharacteristic, complemented
525C2×#103C22×C5311yescharacteristic, complemented
526C2×#103C22×C5311yescharacteristic, complemented
527C2×#102C22×C5311yescharacteristic
528C2×#102C22×C5311yescharacteristic
529C2×#102C22×C5311yescharacteristic
530C2×#102C22×C5311yescharacteristic
531C2×#102C22×C5311yescharacteristic
532C2×#102C22×C5311yescharacteristic
533C2×#102C22×C5311yescharacteristic, complemented
534C2×C22⋊C8C22×C5311yescharacteristic
535C2×C22⋊C8C22×C5311yescharacteristic
536C2×C22⋊C8C22×C5311yescharacteristic, complemented
537C2×C22⋊C8C22×C5311yescharacteristic, complemented
538C2×#97C22×C5311yescharacteristic, complemented
539#394—21no—
540#394—21no—
541#384C22×C5311yescharacteristic, complemented
542#380—21no—
543#380C22×C5311yescharacteristic
544#380C22×C5311yescharacteristic, complemented
545#340C22×C5311yescharacteristic, complemented
546#340C22×C5311yescharacteristic, complemented
547C53#2425711yescharacteristic, direct product factor, Sylow
548C25—21no—
549C2×#41—22no—
550C2×#41—22no—
551C2×#41—22no—
552C2×#41—22no—
553C2×#41—41no—
554C2×#41—41no—
555C22×Q8C23×C5311yescharacteristic
556C22×D4—21no—
557C22×D4—21no—
558C22×D4—22no—
559C22×D4—22no—
560C22×D4—22no—
561C22×D4—22no—
562C22×D4—21no—
563C22×D4—21no—
564C22×D4—21no—
565C22×D4—41no—
566C22×D4—41no—
567C22×D4D4×C5311yescharacteristic
568C22×D4C23×C5311yescharacteristic
569C22×D4C23×C5311yescharacteristic
570C22×D4C23×C5311yescharacteristic
571C23×C4—21no—
572C23×C4—21no—
573C23×C4—21no—
574C23×C4C23×C5311yescharacteristic
575C23×C4C23×C5311yescharacteristic
576C2×SD16—22no—
577C2×SD16—22no—
578C2×SD16—41no—
579C2×D8—22no—
580C2×D8—22no—
581C2×D8—22no—
582C2×D8—22no—
583C2×D8—41no—
584C2×D8D4×C5314yescomplemented
585C22×C8C23×C5311yescharacteristic
586C22×C8C23×C5311yescharacteristic
587C22×C8C23×C5311yescharacteristic
588C22×C8C23×C5311yescharacteristic
589#124—22no—
590#124D4×C5314yescomplemented
591#123—28no—
592C22⋊Q8—22no—
593C22⋊Q8—22no—
594#121—28no—
595#121—22no—
596#121—22no—
597#121—22no—
598#121—22no—
599#121—41no—
600#121—41no—
601#121—44no—
602#120—28no—
603C4×D4—22no—
604C4×D4—22no—
605C4×D4—22no—
606C4×D4—22no—
607C4×D4—22no—
608C4×D4D4×C5314yescomplemented
609C2×C4⋊C4—21no—
610C2×C4⋊C4—21no—
611C2×C4⋊C4—21no—
612C2×C4⋊C4D4×C5311yescharacteristic
613C2×C4⋊C4C23×C5311yescharacteristic
614C2×C4⋊C4C23×C5311yescharacteristic
615C2×C4⋊C4C23×C5311yescharacteristic
616C2×C4⋊C4C23×C5311yescharacteristic
617C2×C22⋊C4—28no—
618C2×C22⋊C4—21no—
619C2×C22⋊C4—21no—
620C2×C22⋊C4—21no—
621C2×C22⋊C4—21no—
622C2×C22⋊C4—21no—
623C2×C22⋊C4—21no—
624C2×C22⋊C4—21no—
625C2×C22⋊C4—21no—
626C2×C22⋊C4—21no—
627C2×C22⋊C4—21no—
628C2×C22⋊C4D4×C5311yescharacteristic
629C2×C22⋊C4D4×C5311yescharacteristic
630C2×C42C23×C5311yescharacteristic
631#106—22no—
632C4⋊C8—22no—
633#103—22no—
634#103—22no—
635#103—41no—
636#102—22no—
637#102—22no—
638#102—22no—
639#102—22no—
640#102—22no—
641#102—22no—
642#102—41no—
643C22⋊C8—22no—
644C22⋊C8—22no—
645C22⋊C8—41no—
646C22⋊C8—41no—
647#97—22no—
648#95—21no—
649#95—21no—
650C24—21no—
651C24—24no—
652C24—21no—
653C24—21no—
654C24—42no—
655C24—42no—
656C24—42no—
657C24—42no—
658C24—81no—
659C24—81no—
660C24C2×D4×C5311yescharacteristic
661C24C2×D4×C5311yescharacteristic
662#41—44no—
663#41—44no—
664#41—44no—
665#41—44no—
666C2×Q8—21no—
667C2×Q8—21no—
668C2×Q8—21no—
669C2×Q8—21no—
670C2×Q8—21no—
671C2×Q8C2×D4×C5312yes—
672C2×D4—24no—
673C2×D4—24no—
674C2×D4—22no—
675C2×D4—24no—
676C2×D4—22no—
677C2×D4—21no—
678C2×D4—21no—
679C2×D4—21no—
680C2×D4—21no—
681C2×D4—21no—
682C2×D4—21no—
683C2×D4—21no—
684C2×D4—21no—
685C2×D4—21no—
686C2×D4—21no—
687C2×D4—21no—
688C2×D4—21no—
689C2×D4—21no—
690C2×D4—41no—
691C2×D4—42no—
692C2×D4—42no—
693C2×D4—44no—
694C2×D4—44no—
695C2×D4—44no—
696C2×D4—42no—
697C2×D4—42no—
698C2×D4—44no—
699C2×D4—42no—
700C2×D4—42no—
701C2×D4—42no—
702C2×D4—42no—
703C2×D4—81no—
704C2×D4—81no—
705C2×D4—81no—
706C2×D4—81no—
707C2×D4C2×D4×C5312yes—
708C2×D4C2×D4×C5312yes—
709C2×D4C2×D4×C5312yes—
710C2×D4C2×D4×C5312yes—
711C2×D4#41×C5312yes—
712C22×C4—24no—
713C22×C4—21no—
714C22×C4—21no—
715C22×C4—21no—
716C22×C4—21no—
717C22×C4—21no—
718C22×C4—21no—
719C22×C4—21no—
720C22×C4—21no—
721C22×C4—21no—
722C22×C4—21no—
723C22×C4—21no—
724C22×C4—21no—
725C22×C4—21no—
726C22×C4—21no—
727C22×C4—24no—
728C22×C4—21no—
729C22×C4—21no—
730C22×C4—21no—
731C22×C4—42no—
732C22×C4—42no—
733C22×C4—42no—
734C22×C4—42no—
735C22×C4—42no—
736C22×C4—81no—
737C22×C4—81no—
738C22×C4C2×D4×C5311yescharacteristic
739C22×C4C2×D4×C5311yescharacteristic
740C22×C4C2×D4×C5311yescharacteristic
741C22×C4C2×D4×C5311yescharacteristic
742C22×C4C24×C5311yescharacteristic
743SD16—44no—
744D8—28no—
745D8—44no—
746C2×C8—21no—
747C2×C8—21no—
748C2×C8—21no—
749C2×C8—21no—
750C2×C8—21no—
751C2×C8—21no—
752C2×C8—21no—
753C2×C8—21no—
754C2×C8—21no—
755C2×C8—21no—
756C2×C8—21no—
757C2×C8C2×D4×C5312yes—
758C4⋊C4—24no—
759C4⋊C4—22no—
760C4⋊C4—21no—
761C4⋊C4—21no—
762C4⋊C4—21no—
763C4⋊C4—21no—
764C4⋊C4—21no—
765C4⋊C4—21no—
766C4⋊C4—21no—
767C4⋊C4—42no—
768C4⋊C4—81no—
769C4⋊C4C2×D4×C5312yes—
770C22⋊C4—24no—
771C22⋊C4—24no—
772C22⋊C4—22no—
773C22⋊C4—24no—
774C22⋊C4—24no—
775C22⋊C4—24no—
776C22⋊C4—22no—
777C22⋊C4—42no—
778C22⋊C4—42no—
779C22⋊C4—42no—
780C22⋊C4—42no—
781C22⋊C4—42no—
782C42C2×D4×C5312yes—
783C42C2×D4×C5312yes—
784C23—21no—
785C23—22no—
786C23—21no—
787C23—22no—
788C23—21no—
789C23—22no—
790C23—21no—
791C23—21no—
792C23—21no—
793C23—22no—
794C23—21no—
795C23—22no—
796C23—21no—
797C23—21no—
798C23—21no—
799C23—21no—
800C23—21no—
801C23—21no—
802C23—48no—
803C23—48no—
804C23—41no—
805C23—41no—
806C23—41no—
807C23—41no—
808C23—41no—
809C23—41no—
810C23—41no—
811C23—41no—
812C23—41no—
813C23—41no—
814C23—41no—
815C23—41no—
816C23—41no—
817C23—41no—
818C23—41no—
819C23—41no—
820C23—84no—
821C23—84no—
822C23—84no—
823C23—84no—
824C23—84no—
825C23AGL(1,ℤ/8)×C5312yes—
826C23C22×D4×C5311yescharacteristic
827Q8—22no—
828Q8—22no—
829Q8—41no—
830Q8—41no—
831D4—22no—
832D4—22no—
833D4—22no—
834D4—22no—
835D4—22no—
836D4—22no—
837D4—22no—
838D4—44no—
839D4—48no—
840D4—48no—
841D4—41no—
842D4—41no—
843D4—41no—
844D4—41no—
845D4—84no—
846D4—84no—
847D4—84no—
848D4#121×C5314yescomplemented
849C2×C4—21no—
850C2×C4—21no—
851C2×C4—21no—
852C2×C4—21no—
853C2×C4—22no—
854C2×C4—21no—
855C2×C4—22no—
856C2×C4—21no—
857C2×C4—21no—
858C2×C4—21no—
859C2×C4—21no—
860C2×C4—21no—
861C2×C4—48no—
862C2×C4—41no—
863C2×C4—41no—
864C2×C4—41no—
865C2×C4—41no—
866C2×C4—41no—
867C2×C4—41no—
868C2×C4—41no—
869C2×C4—41no—
870C2×C4—41no—
871C2×C4—41no—
872C2×C4—41no—
873C2×C4—41no—
874C2×C4—41no—
875C2×C4—41no—
876C2×C4—41no—
877C2×C4—41no—
878C2×C4—41no—
879C2×C4—41no—
880C2×C4—41no—
881C2×C4—41no—
882C2×C4—41no—
883C2×C4—84no—
884C2×C4—84no—
885C2×C4#120×C5312yes—
886C2×C4#120×C5312yes—
887C2×C4C22×D4×C5311yescharacteristic
888C2×C4C22×D4×C5311yescharacteristic
889C2×C4C22×D4×C5311yescharacteristic
890C2×C4C2×#41×C5311yescharacteristic
891C2×C4H(5,𝔽2)×C5311yescharacteristic
892C2×C4H(5,𝔽2)×C5311yescharacteristic
893C8—22no—
894C8—41no—
895C8—41no—
896C8—41no—
897C22—22no—
898C22—22no—
899C22—22no—
900C22—24no—
901C22—42no—
902C22—42no—
903C22—41no—
904C22—42no—
905C22—42no—
906C22—41no—
907C22—42no—
908C22—41no—
909C22—42no—
910C22—41no—
911C22—41no—
912C22—42no—
913C22—41no—
914C22—42no—
915C22—41no—
916C22—42no—
917C22—41no—
918C22—42no—
919C22—41no—
920C22—41no—
921C22—41no—
922C22—42no—
923C22—42no—
924C22—816no—
925C22—81no—
926C22—81no—
927C22—81no—
928C22—81no—
929C22—81no—
930C22—81no—
931C22—81no—
932C22—81no—
933C22C2×#120×C5311yescharacteristic
934C22D42×C5311yescharacteristic
935C22C53×#54611yescharacteristic
936C22C2×AGL(1,ℤ/8)×C5311yescharacteristic
937C22C53×#57511yescharacteristic
938C22C53×#57611yescharacteristic
939C22C53×#57711yescharacteristic
940C4—22no—
941C4—22no—
942C4—42no—
943C4—41no—
944C4—41no—
945C4—42no—
946C4—81no—
947C4—81no—
948C4—81no—
949C4C2×#121×C5311yescharacteristic
950C4C53×#53511yescharacteristic
951C4D42×C5311yescharacteristic
952C4C53×#54611yescharacteristic
953C2—42no—
954C2—44no—
955C2—81no—
956C2—82no—
957C2—82no—
958C2C53×#240811yescharacteristic
959C2C53×#300611yescharacteristic
960C2C53×#301211yescharacteristic
961C2C53×#326411yescharacteristic
962C2C53×#326711yescharacteristic
963C2C53×#330611yescharacteristic
964C2C53×#331911yescharacteristic
subgroupG/H
center, socleC23×C53C22×D4
commutator, Frattini, cosocleC22×C4C24×C53
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C4×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#546C2semidirect
2. C2×C53×#463C2semidirect
3. C2×C53×#449C2semidirect
4. C2×C53×#447C2semidirect
5. C2×C53×#442C2semidirect
6. C53×#2537C2semidirect
7. C53×#2035C2semidirect
8. C53×#2034C2semidirect
9. C53×#2000C2semidirect
10. C53×#1999C2semidirect
11. C53×#1959C2semidirect
12. C53×#1955C2semidirect
13. C53×#1802C2semidirect
14. C53×#1788C2semidirect
15. C53×#1786C2semidirect
18. C22×SD16×C53C22semidirect
30. C2×#121×C53C22semidirect
34. C2×C4×D4×C53C22semidirect
40. C2×#106×C53C22semidirect
41. C2×C4⋊C8×C53C22semidirect
42. C2×#103×C53C22semidirect
43. C2×#103×C53C22semidirect
44. C2×#103×C53C22semidirect
51. C2×#102×C53C22semidirect
54. C2×C22⋊C8×C53C22semidirect
55. C2×C22⋊C8×C53C22semidirect
56. C2×#97×C53C22semidirect
59. C53×#384C22semidirect
62. C53×#380C22semidirect
63. C53×#340C22semidirect
64. C53×#340C22semidirect
101. C2×D8×C53D4semidirect
107. #124×C53D4semidirect
125. C4×D4×C53D4semidirect
365. D4×C53#121semidirect
414. #24257C53direct (N × H)
471. C2×#546C2×C53semidirect
472. C2×#463C2×C53semidirect
473. C2×#449C2×C53semidirect
474. C2×#447C2×C53semidirect
475. C2×#442C2×C53semidirect
476. #2537C2×C53semidirect
477. #2035C2×C53semidirect
478. #2034C2×C53semidirect
479. #2000C2×C53semidirect
480. #1999C2×C53semidirect
481. #1959C2×C53semidirect
482. #1955C2×C53semidirect
483. #1802C2×C53semidirect
484. #1788C2×C53semidirect
485. #1786C2×C53semidirect
500. C22×SD16C22×C53semidirect
512. C2×#121C22×C53semidirect
516. C2×C4×D4C22×C53semidirect
522. C2×#106C22×C53semidirect
523. C2×C4⋊C8C22×C53semidirect
524. C2×#103C22×C53semidirect
525. C2×#103C22×C53semidirect
526. C2×#103C22×C53semidirect
533. C2×#102C22×C53semidirect
536. C2×C22⋊C8C22×C53semidirect
537. C2×C22⋊C8C22×C53semidirect
538. C2×#97C22×C53semidirect
541. #384C22×C53semidirect
544. #380C22×C53semidirect
545. #340C22×C53semidirect
546. #340C22×C53semidirect
547. C53#24257direct (N × H)
584. C2×D8D4×C53semidirect
590. #124D4×C53semidirect
608. C4×D4D4×C53semidirect
848. D4#121×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5462
C2×C53×#4632
C2×C53×#4492
C2×C53×#4472
C2×C53×#4422
C53×#25372
C53×#20352
C53×#20342
C53×#20002
C53×#19992
C53×#19592
C53×#19552
C53×#18022
C53×#17882
C53×#17862
#2425753