Finite Group Databasebeta

← C53×#24142 #27231654

Subgroups — C53×#24142 #27231654

Order 13568.

1292 conjugacy classes of subgroups.

The subgroup lattice has 880 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3505C211yesmaximal, characteristic, complemented
2C53×#2626C211yesmaximal, characteristic, complemented
3C53×#2617C211yesmaximal, characteristic, complemented
4C53×#2321C211yesmaximal, characteristic, complemented
5C53×#2044C212yesmaximal, complemented
6C53×#2021C212yesmaximal, complemented
7C53×#1996C211yesmaximal, characteristic, complemented
8C53×#1964C211yesmaximal, characteristic, complemented
9C53×#1889C212yesmaximal, complemented
10C53×#1867C212yesmaximal, complemented
11C53×#1799C211yesmaximal, characteristic, complemented
12C53×#553—22no—
13C53×#553—21no—
14C53×#552—21no—
15C53×#551—22no—
16C53×#548—21no—
17C53×#546—21no—
18C53×#539—21no—
19C53×#538—21no—
20C53×#534C2212yes—
21C53×#529—21no—
22C53×#525—21no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×#121×C53C2211yescharacteristic
25C2×#120×C53C2211yescharacteristic
26C2×#120×C53C2212yes—
27C53×#518C2212yes—
28C2×C4×D4×C53C2211yescharacteristic
29C22×C22⋊C4×C53C2211yescharacteristic
30C2×#99×C53C2212yes—
31C2×#99×C53C2212yes—
32C2×#99×C53C2212yes—
33C53×#399—21no—
34C53×#397—21no—
35C53×#396—22no—
36C53×#394—22no—
37C53×#393—21no—
38C53×#392—21no—
39C53×#390—21no—
40C53×#388—21no—
41C53×#388—21no—
42C53×#386—21no—
43C53×#386C2211yescharacteristic
44C53×#386C2212yes—
45C53×#385C2212yes—
46C53×#385C2211yescharacteristic
47C53×#380C2211yescharacteristic
48C53×#380C2211yescharacteristic
49C53×#379C2212yes—
50C53×#379C2211yescharacteristic
51C4×C22⋊C4×C53C2211yescharacteristic
52C53×#342C2212yes—
53C53×#342C2212yes—
54C53×#342C2212yes—
55C25×C53C2311yescharacteristic
56C2×#41×C53—21no—
57C22×Q8×C53D411yescharacteristic, complemented
58C22×D4×C53—21no—
59C22×D4×C53—22no—
60C22×D4×C53D411yescharacteristic, complemented
61C22×D4×C53C2311yescharacteristic
62C22×D4×C53C2312yes—
63C23×C4×C53D411yescharacteristic, complemented
64C23×C4×C53C2311yescharacteristic
65C23×C4×C53C2311yescharacteristic
66#126×C53—22no—
67#126×C53—22no—
68#126×C53—22no—
69#125×C53—21no—
70#125×C53—22no—
71#124×C53—21no—
72#124×C53—22no—
73#123×C53—22no—
74#123×C53—22no—
75#123×C53—22no—
76#123×C53—22no—
77#123×C53—22no—
78C22⋊Q8×C53—21no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53—21no—
84#121×C53—21no—
85#121×C53—22no—
86#121×C53—21no—
87#121×C53—21no—
88#121×C53—21no—
89#121×C53—21no—
90#121×C53—22no—
91#121×C53—22no—
92#120×C53—22no—
93#120×C53—22no—
94#120×C53—22no—
95#120×C53—22no—
96#120×C53—24no—
97#120×C53—42no—
98C4×Q8×C53—21no—
99C4×D4×C53—22no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53—22no—
106C4×D4×C53—22no—
107C4×D4×C53—22no—
108C4×D4×C53—22no—
109#117×C53—21no—
110#117×C53—22no—
111#117×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—22no—
116C2×C4⋊C4×C53D411yescharacteristic, complemented
117C2×C4⋊C4×C53D411yescharacteristic, complemented
118C2×C4⋊C4×C53D412yescomplemented
119C2×C4⋊C4×C53D411yescharacteristic, complemented
120C2×C22⋊C4×C53—22no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—22no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53—21no—
126C2×C22⋊C4×C53—21no—
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129C2×C22⋊C4×C53—21no—
130C2×C22⋊C4×C53—22no—
131C2×C22⋊C4×C53—21no—
132C2×C22⋊C4×C53—21no—
133C2×C22⋊C4×C53—22no—
134C2×C22⋊C4×C53—22no—
135C2×C22⋊C4×C53D411yescharacteristic, complemented
136C2×C22⋊C4×C53D412yescomplemented
137C2×C22⋊C4×C53C2312yes—
138C2×C22⋊C4×C53C2311yescharacteristic
139C2×C22⋊C4×C53C2312yes—
140C2×C22⋊C4×C53C2311yescharacteristic
141C2×C22⋊C4×C53C2311yescharacteristic
142C2×C22⋊C4×C53C2312yes—
143C2×C42×C53—21no—
144C2×C42×C53D411yescharacteristic, complemented
145#99×C53—24no—
146#99×C53—42no—
147#99×C53—42no—
148#95×C53—21no—
149#95×C53—21no—
150#95×C53—21no—
151#95×C53—21no—
152#95×C53—22no—
153#95×C53—21no—
154#95×C53—22no—
155C24×C53—21no—
156C24×C53—21no—
157C24×C53—21no—
158C24×C53—21no—
159C24×C53—21no—
160C24×C53—21no—
161C24×C53—81no—
162C24×C53—81no—
163C24×C53C2×D412yes—
164C24×C53C2×D411yescharacteristic
165C24×C53#4112yescomplemented
166C24×C53C2411yescharacteristic
167#41×C53—82no—
168C2×Q8×C53—21no—
169C2×Q8×C53—21no—
170C2×Q8×C53—81no—
171C2×D4×C53—21no—
172C2×D4×C53—22no—
173C2×D4×C53—21no—
174C2×D4×C53—22no—
175C2×D4×C53—22no—
176C2×D4×C53—21no—
177C2×D4×C53—21no—
178C2×D4×C53—22no—
179C2×D4×C53—21no—
180C2×D4×C53—22no—
181C2×D4×C53—21no—
182C2×D4×C53—22no—
183C2×D4×C53—22no—
184C2×D4×C53—22no—
185C2×D4×C53—24no—
186C2×D4×C53—21no—
187C2×D4×C53—21no—
188C2×D4×C53—44no—
189C2×D4×C53—81no—
190C2×D4×C53—81no—
191C2×D4×C53—81no—
192C2×D4×C53—82no—
193C2×D4×C53—82no—
194C22×C4×C53—21no—
195C22×C4×C53—22no—
196C22×C4×C53—22no—
197C22×C4×C53—21no—
198C22×C4×C53—21no—
199C22×C4×C53—21no—
200C22×C4×C53—21no—
201C22×C4×C53—21no—
202C22×C4×C53—22no—
203C22×C4×C53—22no—
204C22×C4×C53—21no—
205C22×C4×C53—81no—
206C22×C4×C53—81no—
207C22×C4×C53—81no—
208C22×C4×C53C2×D411yescharacteristic
209C22×C4×C53C2×D412yes—
210C22×C4×C53C2×D411yescharacteristic
211C22×C4×C53C2×D412yes—
212C22×C4×C53C2×D411yescharacteristic
213C22×C4×C53C2×D412yes—
214C22×C4×C53C2×D411yescharacteristic
215C22×C4×C53C2×D411yescharacteristic
216C22×C4×C53C2×D411yescharacteristic
217C22×C4×C53C2×D411yescharacteristic
218C22×C4×C53C2×D411yescharacteristic
219C22×C4×C53C2×D411yescharacteristic
220C4⋊C4×C53—22no—
221C4⋊C4×C53—22no—
222C4⋊C4×C53—22no—
223C4⋊C4×C53—22no—
224C4⋊C4×C53—21no—
225C4⋊C4×C53—22no—
226C4⋊C4×C53—21no—
227C4⋊C4×C53—21no—
228C4⋊C4×C53—21no—
229C4⋊C4×C53—21no—
230C4⋊C4×C53—22no—
231C4⋊C4×C53—21no—
232C4⋊C4×C53—81no—
233C4⋊C4×C53—81no—
234C4⋊C4×C53—81no—
235C4⋊C4×C53—82no—
236C22⋊C4×C53—22no—
237C22⋊C4×C53—22no—
238C22⋊C4×C53—24no—
239C22⋊C4×C53—22no—
240C22⋊C4×C53—22no—
241C22⋊C4×C53—22no—
242C22⋊C4×C53—22no—
243C22⋊C4×C53—21no—
244C22⋊C4×C53—22no—
245C22⋊C4×C53—21no—
246C22⋊C4×C53—22no—
247C22⋊C4×C53—21no—
248C22⋊C4×C53—22no—
249C22⋊C4×C53—21no—
250C22⋊C4×C53—21no—
251C22⋊C4×C53—22no—
252C22⋊C4×C53—22no—
253C22⋊C4×C53—22no—
254C22⋊C4×C53—21no—
255C22⋊C4×C53—22no—
256C22⋊C4×C53—22no—
257C22⋊C4×C53—21no—
258C22⋊C4×C53—44no—
259C22⋊C4×C53—41no—
260C22⋊C4×C53—41no—
261C22⋊C4×C53—41no—
262C22⋊C4×C53—41no—
263C22⋊C4×C53—81no—
264C22⋊C4×C53—82no—
265C22⋊C4×C53—81no—
266C22⋊C4×C53—82no—
267C22⋊C4×C53—82no—
268C22⋊C4×C53—81no—
269C22⋊C4×C53#4112yescomplemented
270C42×C53—22no—
271C42×C53—22no—
272C42×C53—21no—
273C42×C53—21no—
274C42×C53—81no—
275C23×C53—21no—
276C23×C53—22no—
277C23×C53—21no—
278C23×C53—22no—
279C23×C53—22no—
280C23×C53—22no—
281C23×C53—21no—
282C23×C53—22no—
283C23×C53—22no—
284C23×C53—41no—
285C23×C53—41no—
286C23×C53—41no—
287C23×C53—41no—
288C23×C53—41no—
289C23×C53—42no—
290C23×C53—41no—
291C23×C53—42no—
292C23×C53—42no—
293C23×C53—41no—
294C23×C53—42no—
295C23×C53—41no—
296C23×C53—42no—
297C23×C53—42no—
298C23×C53—41no—
299C23×C53—41no—
300C23×C53—41no—
301C23×C53—41no—
302C23×C53—82no—
303C23×C53—82no—
304C23×C53—82no—
305C23×C53—82no—
306C23×C53C22×D411yescharacteristic
307C23×C53C22×D411yescharacteristic
308C23×C53C22×D411yescharacteristic
309C23×C53C2×#4111yescharacteristic
310C23×C53C2×#4111yescharacteristic
311C23×C53H(5,𝔽2)11yescharacteristic
312C23×C532−1+411yescharacteristic
313Q8×C53—42no—
314Q8×C53—81no—
315D4×C53—42no—
316D4×C53—42no—
317D4×C53—42no—
318D4×C53—41no—
319D4×C53—41no—
320D4×C53—41no—
321D4×C53—42no—
322D4×C53—42no—
323D4×C53—41no—
324D4×C53—81no—
325D4×C53—82no—
326D4×C53—84no—
327D4×C53—82no—
328D4×C53—88no—
329C2×C4×C53—21no—
330C2×C4×C53—21no—
331C2×C4×C53—22no—
332C2×C4×C53—22no—
333C2×C4×C53—21no—
334C2×C4×C53—22no—
335C2×C4×C53—22no—
336C2×C4×C53—21no—
337C2×C4×C53—21no—
338C2×C4×C53—22no—
339C2×C4×C53—22no—
340C2×C4×C53—41no—
341C2×C4×C53—41no—
342C2×C4×C53—41no—
343C2×C4×C53—41no—
344C2×C4×C53—41no—
345C2×C4×C53—41no—
346C2×C4×C53—42no—
347C2×C4×C53—41no—
348C2×C4×C53—41no—
349C2×C4×C53—41no—
350C2×C4×C53—42no—
351C2×C4×C53—41no—
352C2×C4×C53—42no—
353C2×C4×C53—41no—
354C2×C4×C53—42no—
355C2×C4×C53—42no—
356C2×C4×C53—41no—
357C2×C4×C53—42no—
358C2×C4×C53—42no—
359C2×C4×C53—41no—
360C2×C4×C53—41no—
361C2×C4×C53—42no—
362C2×C4×C53—42no—
363C2×C4×C53—41no—
364C2×C4×C53—41no—
365C2×C4×C53—42no—
366C2×C4×C53—42no—
367C2×C4×C53—81no—
368C2×C4×C53—82no—
369C2×C4×C53—82no—
370C2×C4×C53#12011yescharacteristic
371C2×C4×C53#12011yescharacteristic
372C2×C4×C53#12011yescharacteristic
373C2×C4×C53#12011yescharacteristic
374#24142C5311yesmaximal, characteristic, direct product factor, Sylow
375C22×C53—21no—
376C22×C53—21no—
377C22×C53—21no—
378C22×C53—21no—
379C22×C53—21no—
380C22×C53—21no—
381C22×C53—41no—
382C22×C53—41no—
383C22×C53—41no—
384C22×C53—41no—
385C22×C53—42no—
386C22×C53—42no—
387C22×C53—44no—
388C22×C53—42no—
389C22×C53—44no—
390C22×C53—41no—
391C22×C53—42no—
392C22×C53—42no—
393C22×C53—41no—
394C22×C53—42no—
395C22×C53—42no—
396C22×C53—41no—
397C22×C53—82no—
398C22×C53—81no—
399C22×C53—82no—
400C22×C53—82no—
401C22×C53—81no—
402C22×C53—81no—
403C22×C53—82no—
404C22×C53—81no—
405C22×C53—81no—
406C22×C53C2×#12011yescharacteristic
407C22×C53#52511yescharacteristic
408C22×C53#53711yescharacteristic
409C22×C53#54611yescharacteristic
410C22×C53#54711yescharacteristic
411C22×C53#54811yescharacteristic
412C22×C53#55011yescharacteristic
413C4×C53—21no—
414C4×C53—21no—
415C4×C53—21no—
416C4×C53—21no—
417C4×C53—42no—
418C4×C53—41no—
419C4×C53—42no—
420C4×C53—41no—
421C4×C53—42no—
422C4×C53—41no—
423C4×C53—42no—
424C4×C53—82no—
425C4×C53—81no—
426C4×C53—82no—
427C4×C53—81no—
428#3505C2×C5311yescharacteristic, complemented
429#2626C2×C5311yescharacteristic, complemented
430#2617C2×C5311yescharacteristic, complemented
431#2321C2×C5311yescharacteristic, complemented
432#2044C2×C5312yescomplemented
433#2021C2×C5312yescomplemented
434#1996C2×C5311yescharacteristic, complemented
435#1964C2×C5311yescharacteristic, complemented
436#1889C2×C5312yescomplemented
437#1867C2×C5312yescomplemented
438#1799C2×C5311yescharacteristic, complemented
439C2×C53—21no—
440C2×C53—21no—
441C2×C53—21no—
442C2×C53—21no—
443C2×C53—21no—
444C2×C53—21no—
445C2×C53—82no—
446C2×C53—81no—
447C2×C53—82no—
448C2×C53#240711yescharacteristic
449C2×C53#302611yescharacteristic
450C2×C53#302711yescharacteristic
451#553—22no—
452#553—21no—
453#552—21no—
454#551—22no—
455#548—21no—
456#546—21no—
457#539—21no—
458#538—21no—
459#534C22×C5312yes—
460#529—21no—
461#525—21no—
462C2×C22⋊Q8C22×C5311yescharacteristic
463C2×#121C22×C5311yescharacteristic
464C2×#120C22×C5311yescharacteristic
465C2×#120C22×C5312yes—
466#518C22×C5312yes—
467C2×C4×D4C22×C5311yescharacteristic
468C22×C22⋊C4C22×C5311yescharacteristic
469C2×#99C22×C5312yes—
470C2×#99C22×C5312yes—
471C2×#99C22×C5312yes—
472#399—21no—
473#397—21no—
474#396—22no—
475#394—22no—
476#393—21no—
477#392—21no—
478#390—21no—
479#388—21no—
480#388—21no—
481#386—21no—
482#386C22×C5311yescharacteristic
483#386C22×C5312yes—
484#385C22×C5311yescharacteristic
485#385C22×C5312yes—
486#380C22×C5311yescharacteristic
487#380C22×C5311yescharacteristic
488#379C22×C5311yescharacteristic
489#379C22×C5312yes—
490C4×C22⋊C4C22×C5311yescharacteristic
491#342C22×C5312yes—
492#342C22×C5312yes—
493#342C22×C5312yes—
494C53#2414211yescharacteristic, direct product factor, Sylow
495C25C23×C5311yescharacteristic
496C2×#41—21no—
497C22×Q8D4×C5311yescharacteristic, complemented
498C22×D4—21no—
499C22×D4—22no—
500C22×D4D4×C5311yescharacteristic, complemented
501C22×D4C23×C5311yescharacteristic
502C22×D4C23×C5312yes—
503C23×C4D4×C5311yescharacteristic, complemented
504C23×C4C23×C5311yescharacteristic
505C23×C4C23×C5311yescharacteristic
506#126—22no—
507#126—22no—
508#126—22no—
509#125—21no—
510#125—22no—
511#124—21no—
512#124—22no—
513#123—22no—
514#123—22no—
515#123—22no—
516#123—22no—
517#123—22no—
518C22⋊Q8—21no—
519C22⋊Q8—22no—
520C22⋊Q8—21no—
521C22⋊Q8—21no—
522C22⋊Q8—21no—
523C22⋊Q8—21no—
524#121—21no—
525#121—22no—
526#121—21no—
527#121—21no—
528#121—21no—
529#121—21no—
530#121—22no—
531#121—22no—
532#120—22no—
533#120—22no—
534#120—22no—
535#120—22no—
536#120—24no—
537#120—42no—
538C4×Q8—21no—
539C4×D4—22no—
540C4×D4—21no—
541C4×D4—21no—
542C4×D4—21no—
543C4×D4—21no—
544C4×D4—21no—
545C4×D4—22no—
546C4×D4—22no—
547C4×D4—22no—
548C4×D4—22no—
549#117—21no—
550#117—22no—
551#117—22no—
552C2×C4⋊C4—21no—
553C2×C4⋊C4—21no—
554C2×C4⋊C4—21no—
555C2×C4⋊C4—22no—
556C2×C4⋊C4D4×C5311yescharacteristic, complemented
557C2×C4⋊C4D4×C5311yescharacteristic, complemented
558C2×C4⋊C4D4×C5312yescomplemented
559C2×C4⋊C4D4×C5311yescharacteristic, complemented
560C2×C22⋊C4—22no—
561C2×C22⋊C4—21no—
562C2×C22⋊C4—21no—
563C2×C22⋊C4—21no—
564C2×C22⋊C4—21no—
565C2×C22⋊C4—21no—
566C2×C22⋊C4—21no—
567C2×C22⋊C4—21no—
568C2×C22⋊C4—22no—
569C2×C22⋊C4—21no—
570C2×C22⋊C4—22no—
571C2×C22⋊C4—21no—
572C2×C22⋊C4—21no—
573C2×C22⋊C4—22no—
574C2×C22⋊C4—22no—
575C2×C22⋊C4D4×C5311yescharacteristic, complemented
576C2×C22⋊C4D4×C5312yescomplemented
577C2×C22⋊C4C23×C5312yes—
578C2×C22⋊C4C23×C5311yescharacteristic
579C2×C22⋊C4C23×C5312yes—
580C2×C22⋊C4C23×C5311yescharacteristic
581C2×C22⋊C4C23×C5312yes—
582C2×C22⋊C4C23×C5311yescharacteristic
583C2×C42—21no—
584C2×C42D4×C5311yescharacteristic, complemented
585#99—24no—
586#99—42no—
587#99—42no—
588#95—21no—
589#95—21no—
590#95—21no—
591#95—21no—
592#95—22no—
593#95—21no—
594#95—22no—
595C24—21no—
596C24—21no—
597C24—21no—
598C24—21no—
599C24—21no—
600C24—21no—
601C24—81no—
602C24—81no—
603C24C2×D4×C5312yes—
604C24C2×D4×C5311yescharacteristic
605C24#41×C5312yescomplemented
606C24C24×C5311yescharacteristic
607#41—82no—
608C2×Q8—21no—
609C2×Q8—21no—
610C2×Q8—81no—
611C2×D4—21no—
612C2×D4—22no—
613C2×D4—21no—
614C2×D4—22no—
615C2×D4—22no—
616C2×D4—21no—
617C2×D4—21no—
618C2×D4—22no—
619C2×D4—21no—
620C2×D4—22no—
621C2×D4—21no—
622C2×D4—22no—
623C2×D4—22no—
624C2×D4—22no—
625C2×D4—24no—
626C2×D4—21no—
627C2×D4—21no—
628C2×D4—44no—
629C2×D4—81no—
630C2×D4—81no—
631C2×D4—81no—
632C2×D4—82no—
633C2×D4—82no—
634C22×C4—21no—
635C22×C4—22no—
636C22×C4—22no—
637C22×C4—21no—
638C22×C4—21no—
639C22×C4—21no—
640C22×C4—21no—
641C22×C4—21no—
642C22×C4—22no—
643C22×C4—22no—
644C22×C4—21no—
645C22×C4—81no—
646C22×C4—81no—
647C22×C4—81no—
648C22×C4C2×D4×C5311yescharacteristic
649C22×C4C2×D4×C5312yes—
650C22×C4C2×D4×C5311yescharacteristic
651C22×C4C2×D4×C5312yes—
652C22×C4C2×D4×C5311yescharacteristic
653C22×C4C2×D4×C5312yes—
654C22×C4C2×D4×C5311yescharacteristic
655C22×C4C2×D4×C5311yescharacteristic
656C22×C4C2×D4×C5311yescharacteristic
657C22×C4C2×D4×C5311yescharacteristic
658C22×C4C2×D4×C5311yescharacteristic
659C22×C4C2×D4×C5311yescharacteristic
660C4⋊C4—22no—
661C4⋊C4—22no—
662C4⋊C4—22no—
663C4⋊C4—22no—
664C4⋊C4—21no—
665C4⋊C4—22no—
666C4⋊C4—21no—
667C4⋊C4—21no—
668C4⋊C4—21no—
669C4⋊C4—21no—
670C4⋊C4—22no—
671C4⋊C4—21no—
672C4⋊C4—81no—
673C4⋊C4—81no—
674C4⋊C4—81no—
675C4⋊C4—82no—
676C22⋊C4—22no—
677C22⋊C4—22no—
678C22⋊C4—24no—
679C22⋊C4—22no—
680C22⋊C4—22no—
681C22⋊C4—22no—
682C22⋊C4—22no—
683C22⋊C4—21no—
684C22⋊C4—22no—
685C22⋊C4—21no—
686C22⋊C4—22no—
687C22⋊C4—21no—
688C22⋊C4—22no—
689C22⋊C4—21no—
690C22⋊C4—21no—
691C22⋊C4—22no—
692C22⋊C4—22no—
693C22⋊C4—22no—
694C22⋊C4—21no—
695C22⋊C4—22no—
696C22⋊C4—22no—
697C22⋊C4—21no—
698C22⋊C4—44no—
699C22⋊C4—41no—
700C22⋊C4—41no—
701C22⋊C4—41no—
702C22⋊C4—41no—
703C22⋊C4—81no—
704C22⋊C4—82no—
705C22⋊C4—81no—
706C22⋊C4—82no—
707C22⋊C4—81no—
708C22⋊C4—82no—
709C22⋊C4#41×C5312yescomplemented
710C42—22no—
711C42—22no—
712C42—21no—
713C42—21no—
714C42—81no—
715C23—21no—
716C23—22no—
717C23—22no—
718C23—21no—
719C23—22no—
720C23—22no—
721C23—21no—
722C23—22no—
723C23—22no—
724C23—41no—
725C23—41no—
726C23—41no—
727C23—41no—
728C23—42no—
729C23—41no—
730C23—42no—
731C23—41no—
732C23—41no—
733C23—42no—
734C23—41no—
735C23—42no—
736C23—42no—
737C23—42no—
738C23—41no—
739C23—41no—
740C23—41no—
741C23—41no—
742C23—82no—
743C23—82no—
744C23—82no—
745C23—82no—
746C23C22×D4×C5311yescharacteristic
747C23C22×D4×C5311yescharacteristic
748C23C22×D4×C5311yescharacteristic
749C23C2×#41×C5311yescharacteristic
750C23C2×#41×C5311yescharacteristic
751C23H(5,𝔽2)×C5311yescharacteristic
752C232−1+4×C5311yescharacteristic
753Q8—42no—
754Q8—81no—
755D4—42no—
756D4—42no—
757D4—42no—
758D4—41no—
759D4—41no—
760D4—41no—
761D4—42no—
762D4—42no—
763D4—41no—
764D4—81no—
765D4—82no—
766D4—84no—
767D4—82no—
768D4—88no—
769C2×C4—21no—
770C2×C4—22no—
771C2×C4—21no—
772C2×C4—21no—
773C2×C4—22no—
774C2×C4—22no—
775C2×C4—21no—
776C2×C4—22no—
777C2×C4—21no—
778C2×C4—22no—
779C2×C4—22no—
780C2×C4—41no—
781C2×C4—41no—
782C2×C4—41no—
783C2×C4—41no—
784C2×C4—41no—
785C2×C4—41no—
786C2×C4—42no—
787C2×C4—41no—
788C2×C4—41no—
789C2×C4—41no—
790C2×C4—42no—
791C2×C4—41no—
792C2×C4—42no—
793C2×C4—41no—
794C2×C4—41no—
795C2×C4—42no—
796C2×C4—42no—
797C2×C4—41no—
798C2×C4—42no—
799C2×C4—42no—
800C2×C4—42no—
801C2×C4—41no—
802C2×C4—42no—
803C2×C4—41no—
804C2×C4—41no—
805C2×C4—42no—
806C2×C4—42no—
807C2×C4—81no—
808C2×C4—82no—
809C2×C4—82no—
810C2×C4#120×C5311yescharacteristic
811C2×C4#120×C5311yescharacteristic
812C2×C4#120×C5311yescharacteristic
813C2×C4#120×C5311yescharacteristic
814C22—21no—
815C22—21no—
816C22—21no—
817C22—21no—
818C22—21no—
819C22—21no—
820C22—41no—
821C22—41no—
822C22—41no—
823C22—41no—
824C22—42no—
825C22—44no—
826C22—42no—
827C22—42no—
828C22—44no—
829C22—41no—
830C22—42no—
831C22—42no—
832C22—41no—
833C22—41no—
834C22—42no—
835C22—42no—
836C22—82no—
837C22—81no—
838C22—82no—
839C22—82no—
840C22—81no—
841C22—81no—
842C22—82no—
843C22—81no—
844C22—81no—
845C22C2×#120×C5311yescharacteristic
846C22C53×#52511yescharacteristic
847C22C53×#53711yescharacteristic
848C22C53×#54611yescharacteristic
849C22C53×#54711yescharacteristic
850C22C53×#54811yescharacteristic
851C22C53×#55011yescharacteristic
852C4—21no—
853C4—21no—
854C4—21no—
855C4—21no—
856C4—42no—
857C4—41no—
858C4—42no—
859C4—41no—
860C4—42no—
861C4—41no—
862C4—42no—
863C4—82no—
864C4—82no—
865C4—81no—
866C4—81no—
867C2—21no—
868C2—21no—
869C2—21no—
870C2—21no—
871C2—21no—
872C2—21no—
873C2—82no—
874C2—81no—
875C2—82no—
876C2C53×#240711yescharacteristic
877C2C53×#302611yescharacteristic
878C2C53×#302711yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC24C24×C53
LCSG ▷ C24 ▷ 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×#3505C2semidirect
2. C53×#2626C2semidirect
3. C53×#2617C2semidirect
4. C53×#2321C2semidirect
5. C53×#2044C2semidirect
6. C53×#2021C2semidirect
7. C53×#1996C2semidirect
8. C53×#1964C2semidirect
9. C53×#1889C2semidirect
10. C53×#1867C2semidirect
11. C53×#1799C2semidirect
57. C22×Q8×C53D4semidirect
60. C22×D4×C53D4semidirect
63. C23×C4×C53D4semidirect
116. C2×C4⋊C4×C53D4semidirect
117. C2×C4⋊C4×C53D4semidirect
118. C2×C4⋊C4×C53D4semidirect
119. C2×C4⋊C4×C53D4semidirect
135. C2×C22⋊C4×C53D4semidirect
136. C2×C22⋊C4×C53D4semidirect
144. C2×C42×C53D4semidirect
165. C24×C53#41semidirect
269. C22⋊C4×C53#41semidirect
374. #24142C53direct (N × H)
428. #3505C2×C53semidirect
429. #2626C2×C53semidirect
430. #2617C2×C53semidirect
431. #2321C2×C53semidirect
432. #2044C2×C53semidirect
433. #2021C2×C53semidirect
434. #1996C2×C53semidirect
435. #1964C2×C53semidirect
436. #1889C2×C53semidirect
437. #1867C2×C53semidirect
438. #1799C2×C53semidirect
494. C53#24142direct (N × H)
497. C22×Q8D4×C53semidirect
500. C22×D4D4×C53semidirect
503. C23×C4D4×C53semidirect
556. C2×C4⋊C4D4×C53semidirect
557. C2×C4⋊C4D4×C53semidirect
558. C2×C4⋊C4D4×C53semidirect
559. C2×C4⋊C4D4×C53semidirect
575. C2×C22⋊C4D4×C53semidirect
576. C2×C22⋊C4D4×C53semidirect
584. C2×C42D4×C53semidirect
605. C24#41×C53semidirect
709. C22⋊C4#41×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#35052
C53×#26262
C53×#26172
C53×#23212
C53×#20442
C53×#20212
C53×#19962
C53×#19642
C53×#18892
C53×#18672
C53×#17992
#2414253