Finite Group Databasebeta

← C53×#23876 #27231388

Subgroups — C53×#23876 #27231388

Order 13568.

2012 conjugacy classes of subgroups.

The subgroup lattice has 938 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic, complemented
2C53×#3488C211yesmaximal, characteristic, complemented
3C2×C53×#453C212yesmaximal, complemented
4C53×#3003C211yesmaximal, characteristic, complemented
5C53×#2433C211yesmaximal, characteristic, complemented
6C53×#2006C212yesmaximal, complemented
7C53×#2002C212yesmaximal, complemented
8C53×#1900C211yesmaximal, characteristic, complemented
9C53×#1883C212yesmaximal, complemented
10C53×#1871C211yesmaximal, characteristic, complemented
11C53×#1789C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C53C2211yescharacteristic, complemented
13C22×#41×C53C2211yescharacteristic
14C23×D4×C53C2211yescharacteristic, complemented
15C2×AGL(1,ℤ/8)×C53—21no—
16C2×AGL(1,ℤ/8)×C53—21no—
17C2×AGL(1,ℤ/8)×C53—28no—
18C2×AGL(1,ℤ/8)×C53C2212yes—
19C2×AGL(1,ℤ/8)×C53C2212yescomplemented
20C2×AGL(1,ℤ/8)×C53C2212yescomplemented
21C22×SD16×C53—21no—
22C22×D8×C53—21no—
23C22×M16×C53C2211yescharacteristic, complemented
24C53×#550—21no—
25C53×#548—21no—
26C53×#546—21no—
27C53×#546—22no—
28D42×C53—22no—
29D42×C53—21no—
30C53×#537—21no—
31C53×#537C2211yescharacteristic, complemented
32C53×#535—21no—
33C53×#535C2211yescharacteristic, complemented
34C2×#127×C53C2212yescomplemented
35C53×#525—21no—
36C53×#525C2211yescharacteristic
37C2×#121×C53—21no—
38C2×#121×C53C2211yescharacteristic
39C2×#120×C53—22no—
40C53×#520C2211yescharacteristic, complemented
41C53×#453—28no—
42C53×#450—21no—
43C53×#450—21no—
44C53×#447—21no—
45C53×#447—21no—
46C2×#104×C53C2212yescomplemented
47C2×#104×C53C2212yescomplemented
48C2×#104×C53C2212yescomplemented
49C53×#418C2212yescomplemented
50C2×#100×C53C2212yescomplemented
51C53×#407C2211yescharacteristic, complemented
52C53×#392—22no—
53C53×#390C2212yescomplemented
54C4×C22⋊C4×C53C2211yescharacteristic, complemented
55C53×#343C2212yescomplemented
56C53×#339C2212yescomplemented
57C25×C53—21no—
58H(5,𝔽2)×C53—24no—
59H(5,𝔽2)×C53—42no—
60C2×#41×C53—21no—
61C2×#41×C53—21no—
62C2×#41×C53—21no—
63C2×#41×C53—21no—
64C2×#41×C53—24no—
65C2×#41×C53—42no—
66C2×#41×C53D411yescharacteristic, complemented
67C2×#41×C53D411yescharacteristic, complemented
68C2×#41×C53C2311yescharacteristic
69C2×#41×C53C2311yescharacteristic
70C2×#41×C53C2311yescharacteristic
71C22×Q8×C53D411yescharacteristic, complemented
72C22×D4×C53—24no—
73C22×D4×C53—21no—
74C22×D4×C53—22no—
75C22×D4×C53—22no—
76C22×D4×C53—21no—
77C22×D4×C53—21no—
78C22×D4×C53—24no—
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—21no—
83C22×D4×C53—21no—
84C22×D4×C53—42no—
85C22×D4×C53D412yes—
86C22×D4×C53D411yescharacteristic, complemented
87C22×D4×C53C2311yescharacteristic
88C22×D4×C53C2312yes—
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91AGL(1,ℤ/8)×C53—28no—
92AGL(1,ℤ/8)×C53—28no—
93AGL(1,ℤ/8)×C53—24no—
94AGL(1,ℤ/8)×C53—24no—
95AGL(1,ℤ/8)×C53—24no—
96AGL(1,ℤ/8)×C53—24no—
97AGL(1,ℤ/8)×C53—42no—
98AGL(1,ℤ/8)×C53D418yescomplemented
99C2×SD16×C53—28no—
100C2×SD16×C53—22no—
101C2×SD16×C53—22no—
102C2×SD16×C53—22no—
103C2×D8×C53—28no—
104C2×D8×C53—22no—
105C2×D8×C53—22no—
106C2×D8×C53—22no—
107C2×M16×C53—24no—
108C2×M16×C53—21no—
109C2×M16×C53C2312yes—
110C2×M16×C53C2312yes—
111C22×C8×C53—21no—
112#127×C53—22no—
113#127×C53—21no—
114#127×C53—24no—
115#127×C53—42no—
116#124×C53—22no—
117#124×C53—21no—
118#123×C53—22no—
119#123×C53—21no—
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—21no—
122C22⋊Q8×C53—21no—
123C22⋊Q8×C53D411yescharacteristic, complemented
124C22⋊Q8×C53D411yescharacteristic, complemented
125#121×C53—22no—
126#121×C53—22no—
127#121×C53—22no—
128#121×C53—22no—
129#121×C53—22no—
130#121×C53—21no—
131#121×C53—21no—
132#121×C53—21no—
133#121×C53—21no—
134#121×C53—21no—
135#121×C53—21no—
136#121×C53—21no—
137#121×C53—21no—
138#121×C53—22no—
139#121×C53D411yescharacteristic, complemented
140#121×C53D411yescharacteristic, complemented
141#120×C53—22no—
142#120×C53—22no—
143#120×C53—22no—
144#120×C53—21no—
145#120×C53—22no—
146#120×C53—48no—
147C4×Q8×C53—21no—
148C4×D4×C53—22no—
149C4×D4×C53—22no—
150C4×D4×C53—21no—
151C4×D4×C53—22no—
152C4×D4×C53—21no—
153C4×D4×C53—21no—
154#117×C53—21no—
155#117×C53C2311yescharacteristic
156C2×C4⋊C4×C53—21no—
157C2×C4⋊C4×C53C2311yescharacteristic
158C2×C22⋊C4×C53—22no—
159C2×C22⋊C4×C53—21no—
160C2×C22⋊C4×C53—24no—
161C2×C22⋊C4×C53D412yes—
162C2×C42×C53C2312yes—
163#104×C53—24no—
164#104×C53—24no—
165#104×C53—42no—
166#102×C53—22no—
167#102×C53—22no—
168#100×C53—24no—
169C22⋊C8×C53—21no—
170C22⋊C8×C53—21no—
171#95×C53—21no—
172C24×C53—22no—
173C24×C53—22no—
174C24×C53—21no—
175C24×C53—44no—
176C24×C53—44no—
177C24×C53—41no—
178C24×C53—41no—
179C24×C53—82no—
180C24×C53C2×D411yescharacteristic
181C24×C53C2×D412yes—
182#41×C53—22no—
183#41×C53—22no—
184#41×C53—22no—
185#41×C53—22no—
186#41×C53—22no—
187#41×C53—22no—
188#41×C53—22no—
189#41×C53—44no—
190#41×C53—41no—
191#41×C53—41no—
192#41×C53—44no—
193#41×C53—44no—
194#41×C53—42no—
195#41×C53—42no—
196#41×C53C2×D414yes—
197C2×Q8×C53—21no—
198C2×Q8×C53—22no—
199C2×Q8×C53—41no—
200C2×Q8×C53C2×D411yescharacteristic
201C2×Q8×C53C2×D411yescharacteristic
202C2×Q8×C53C2×D411yescharacteristic
203C2×D4×C53—21no—
204C2×D4×C53—24no—
205C2×D4×C53—24no—
206C2×D4×C53—22no—
207C2×D4×C53—22no—
208C2×D4×C53—22no—
209C2×D4×C53—22no—
210C2×D4×C53—21no—
211C2×D4×C53—21no—
212C2×D4×C53—21no—
213C2×D4×C53—21no—
214C2×D4×C53—21no—
215C2×D4×C53—22no—
216C2×D4×C53—22no—
217C2×D4×C53—22no—
218C2×D4×C53—21no—
219C2×D4×C53—22no—
220C2×D4×C53—21no—
221C2×D4×C53—22no—
222C2×D4×C53—22no—
223C2×D4×C53—22no—
224C2×D4×C53—22no—
225C2×D4×C53—21no—
226C2×D4×C53—21no—
227C2×D4×C53—22no—
228C2×D4×C53—22no—
229C2×D4×C53—22no—
230C2×D4×C53—41no—
231C2×D4×C53—44no—
232C2×D4×C53—44no—
233C2×D4×C53—44no—
234C2×D4×C53—44no—
235C2×D4×C53—42no—
236C2×D4×C53—42no—
237C2×D4×C53—48no—
238C2×D4×C53—44no—
239C2×D4×C53—48no—
240C2×D4×C53—42no—
241C2×D4×C53—42no—
242C2×D4×C53—44no—
243C2×D4×C53—48no—
244C2×D4×C53—41no—
245C2×D4×C53—41no—
246C2×D4×C53—84no—
247C2×D4×C53—84no—
248C2×D4×C53C2×D411yescharacteristic
249C2×D4×C53C2×D411yescharacteristic
250C2×D4×C53C2×D411yescharacteristic
251C2×D4×C53C2×D414yescomplemented
252C22×C4×C53—21no—
253C22×C4×C53—22no—
254C22×C4×C53—22no—
255C22×C4×C53—22no—
256C22×C4×C53—21no—
257C22×C4×C53—22no—
258C22×C4×C53—22no—
259C22×C4×C53—21no—
260C22×C4×C53—21no—
261C22×C4×C53—21no—
262C22×C4×C53—21no—
263C22×C4×C53—44no—
264C22×C4×C53C2×D411yescharacteristic
265C22×C4×C53C2×D411yescharacteristic
266C22×C4×C53C2×D411yescharacteristic
267C22×C4×C53C2×D411yescharacteristic
268C22×C4×C53C2×D411yescharacteristic
269C22×C4×C53C2×D412yes—
270C22×C4×C53C2411yescharacteristic
271SD16×C53—28no—
272SD16×C53—28no—
273SD16×C53—44no—
274D8×C53—28no—
275D8×C53—28no—
276D8×C53—44no—
277M16×C53—24no—
278M16×C53—22no—
279M16×C53—22no—
280M16×C53—22no—
281M16×C53C2×D414yescomplemented
282C2×C8×C53—24no—
283C2×C8×C53—22no—
284C2×C8×C53—22no—
285C4⋊C4×C53—22no—
286C4⋊C4×C53—22no—
287C4⋊C4×C53—21no—
288C4⋊C4×C53—21no—
289C4⋊C4×C53—21no—
290C4⋊C4×C53—21no—
291C4⋊C4×C53C2×D411yescharacteristic
292C4⋊C4×C53C2×D411yescharacteristic
293C22⋊C4×C53—22no—
294C22⋊C4×C53—22no—
295C22⋊C4×C53—24no—
296C22⋊C4×C53—22no—
297C22⋊C4×C53—22no—
298C22⋊C4×C53—22no—
299C22⋊C4×C53—21no—
300C22⋊C4×C53—21no—
301C22⋊C4×C53—21no—
302C22⋊C4×C53—21no—
303C22⋊C4×C53—22no—
304C22⋊C4×C53—22no—
305C22⋊C4×C53—21no—
306C22⋊C4×C53—48no—
307C42×C53—22no—
308C42×C53—21no—
309C42×C53—22no—
310C42×C53—22no—
311C23×C53—21no—
312C23×C53—22no—
313C23×C53—22no—
314C23×C53—21no—
315C23×C53—22no—
316C23×C53—21no—
317C23×C53—24no—
318C23×C53—24no—
319C23×C53—41no—
320C23×C53—42no—
321C23×C53—44no—
322C23×C53—44no—
323C23×C53—44no—
324C23×C53—44no—
325C23×C53—44no—
326C23×C53—42no—
327C23×C53—42no—
328C23×C53—42no—
329C23×C53—42no—
330C23×C53—42no—
331C23×C53—81no—
332C23×C53—82no—
333C23×C53—82no—
334C23×C53—82no—
335C23×C53—82no—
336C23×C53—82no—
337C23×C53—816no—
338C23×C53—82no—
339C23×C53#12011yescharacteristic
340C23×C53#12011yescharacteristic
341C23×C53C22×D411yescharacteristic
342Q8×C53—21no—
343Q8×C53—21no—
344Q8×C53—21no—
345Q8×C53—22no—
346Q8×C53—42no—
347Q8×C53#12012yescomplemented
348D4×C53—24no—
349D4×C53—24no—
350D4×C53—21no—
351D4×C53—21no—
352D4×C53—21no—
353D4×C53—24no—
354D4×C53—22no—
355D4×C53—42no—
356D4×C53—42no—
357D4×C53—44no—
358D4×C53—44no—
359D4×C53—44no—
360D4×C53—42no—
361D4×C53—44no—
362D4×C53—42no—
363D4×C53—42no—
364D4×C53—44no—
365D4×C53—42no—
366D4×C53—41no—
367D4×C53—42no—
368D4×C53—41no—
369D4×C53—42no—
370D4×C53—42no—
371D4×C53—81no—
372D4×C53—82no—
373D4×C53—82no—
374D4×C53—82no—
375D4×C53—82no—
376D4×C53—816no—
377D4×C53#12012yescomplemented
378C2×C4×C53—21no—
379C2×C4×C53—22no—
380C2×C4×C53—21no—
381C2×C4×C53—21no—
382C2×C4×C53—21no—
383C2×C4×C53—21no—
384C2×C4×C53—22no—
385C2×C4×C53—22no—
386C2×C4×C53—24no—
387C2×C4×C53—24no—
388C2×C4×C53—21no—
389C2×C4×C53—21no—
390C2×C4×C53—21no—
391C2×C4×C53—22no—
392C2×C4×C53—21no—
393C2×C4×C53—42no—
394C2×C4×C53—42no—
395C2×C4×C53—42no—
396C2×C4×C53—44no—
397C2×C4×C53—42no—
398C2×C4×C53—42no—
399C2×C4×C53—41no—
400C2×C4×C53—41no—
401C2×C4×C53—42no—
402C2×C4×C53—42no—
403C2×C4×C53—82no—
404C2×C4×C53#12011yescharacteristic
405C2×C4×C53#12011yescharacteristic
406C2×C4×C53C22×D411yescharacteristic
407C2×C4×C53C22×D412yes—
408C2×C4×C53C22×D411yescharacteristic
409C2×C4×C53H(5,𝔽2)12yes—
410C8×C53—24no—
411C8×C53—42no—
412#23876C5311yesmaximal, characteristic, direct product factor, Sylow
413C22×C53—21no—
414C22×C53—21no—
415C22×C53—21no—
416C22×C53—21no—
417C22×C53—21no—
418C22×C53—22no—
419C22×C53—21no—
420C22×C53—21no—
421C22×C53—44no—
422C22×C53—48no—
423C22×C53—41no—
424C22×C53—41no—
425C22×C53—41no—
426C22×C53—41no—
427C22×C53—42no—
428C22×C53—44no—
429C22×C53—42no—
430C22×C53—41no—
431C22×C53—41no—
432C22×C53—41no—
433C22×C53—42no—
434C22×C53—41no—
435C22×C53—84no—
436C22×C53—84no—
437C22×C53—84no—
438C22×C53—88no—
439C22×C53—88no—
440C22×C53—82no—
441C22×C53—82no—
442C22×C53—164no—
443C22×C53#45312yescomplemented
444C22×C53C2×#12011yescharacteristic
445C22×C53D4212yes—
446C4×C53—21no—
447C4×C53—21no—
448C4×C53—22no—
449C4×C53—44no—
450C4×C53—41no—
451C4×C53—41no—
452C4×C53—42no—
453C4×C53—41no—
454C4×C53—41no—
455C4×C53C2×#12011yescharacteristic
456C4×C53#53411yescharacteristic
457C4×C53D4212yes—
458C22×AGL(1,ℤ/8)C2×C5311yescharacteristic, complemented
459#3488C2×C5311yescharacteristic, complemented
460C2×#453C2×C5312yescomplemented
461#3003C2×C5311yescharacteristic, complemented
462#2433C2×C5311yescharacteristic, complemented
463#2006C2×C5312yescomplemented
464#2002C2×C5312yescomplemented
465#1900C2×C5311yescharacteristic, complemented
466#1883C2×C5312yescomplemented
467#1871C2×C5311yescharacteristic, complemented
468#1789C2×C5311yescharacteristic, complemented
469C2×C53—22no—
470C2×C53—22no—
471C2×C53—41no—
472C2×C53—42no—
473C2×C53—81no—
474C2×C53—84no—
475C2×C53—82no—
476C2×C53—81no—
477C2×C53#240311yescharacteristic
478C2×C53C2×#45311yescharacteristic
479C2×C53#302111yescharacteristic
480C2×H(5,𝔽2)C22×C5311yescharacteristic, complemented
481C22×#41C22×C5311yescharacteristic
482C23×D4C22×C5311yescharacteristic, complemented
483C2×AGL(1,ℤ/8)—21no—
484C2×AGL(1,ℤ/8)—21no—
485C2×AGL(1,ℤ/8)—28no—
486C2×AGL(1,ℤ/8)C22×C5312yes—
487C2×AGL(1,ℤ/8)C22×C5312yescomplemented
488C2×AGL(1,ℤ/8)C22×C5312yescomplemented
489C22×SD16—21no—
490C22×D8—21no—
491C22×M16C22×C5311yescharacteristic, complemented
492#550—21no—
493#548—21no—
494#546—21no—
495#546—22no—
496D42—22no—
497D42—21no—
498#537—21no—
499#537C22×C5311yescharacteristic, complemented
500#535—21no—
501#535C22×C5311yescharacteristic, complemented
502C2×#127C22×C5312yescomplemented
503#525—21no—
504#525C22×C5311yescharacteristic
505C2×#121—21no—
506C2×#121C22×C5311yescharacteristic
507C2×#120—22no—
508#520C22×C5311yescharacteristic, complemented
509#453—28no—
510#450—21no—
511#450—21no—
512#447—21no—
513#447—21no—
514C2×#104C22×C5312yescomplemented
515C2×#104C22×C5312yescomplemented
516C2×#104C22×C5312yescomplemented
517#418C22×C5312yescomplemented
518C2×#100C22×C5312yescomplemented
519#407C22×C5311yescharacteristic, complemented
520#392—22no—
521#390C22×C5312yescomplemented
522C4×C22⋊C4C22×C5311yescharacteristic, complemented
523#343C22×C5312yescomplemented
524#339C22×C5312yescomplemented
525C53#2387611yescharacteristic, direct product factor, Sylow
526C25—21no—
527H(5,𝔽2)—24no—
528H(5,𝔽2)—42no—
529C2×#41—21no—
530C2×#41—21no—
531C2×#41—21no—
532C2×#41—21no—
533C2×#41—24no—
534C2×#41—42no—
535C2×#41D4×C5311yescharacteristic, complemented
536C2×#41D4×C5311yescharacteristic, complemented
537C2×#41C23×C5311yescharacteristic
538C2×#41C23×C5311yescharacteristic
539C2×#41C23×C5311yescharacteristic
540C22×Q8D4×C5311yescharacteristic, complemented
541C22×D4—24no—
542C22×D4—21no—
543C22×D4—22no—
544C22×D4—22no—
545C22×D4—21no—
546C22×D4—21no—
547C22×D4—24no—
548C22×D4—21no—
549C22×D4—21no—
550C22×D4—21no—
551C22×D4—21no—
552C22×D4—21no—
553C22×D4—42no—
554C22×D4D4×C5312yes—
555C22×D4D4×C5311yescharacteristic, complemented
556C22×D4C23×C5311yescharacteristic
557C22×D4C23×C5312yes—
558C23×C4—21no—
559C23×C4C23×C5311yescharacteristic
560AGL(1,ℤ/8)—28no—
561AGL(1,ℤ/8)—28no—
562AGL(1,ℤ/8)—24no—
563AGL(1,ℤ/8)—24no—
564AGL(1,ℤ/8)—24no—
565AGL(1,ℤ/8)—24no—
566AGL(1,ℤ/8)—42no—
567AGL(1,ℤ/8)D4×C5318yescomplemented
568C2×SD16—28no—
569C2×SD16—22no—
570C2×SD16—22no—
571C2×SD16—22no—
572C2×D8—28no—
573C2×D8—22no—
574C2×D8—22no—
575C2×D8—22no—
576C2×M16—21no—
577C2×M16—24no—
578C2×M16C23×C5312yes—
579C2×M16C23×C5312yes—
580C22×C8—21no—
581#127—22no—
582#127—21no—
583#127—24no—
584#127—42no—
585#124—22no—
586#124—21no—
587#123—22no—
588#123—21no—
589C22⋊Q8—22no—
590C22⋊Q8—21no—
591C22⋊Q8—21no—
592C22⋊Q8D4×C5311yescharacteristic, complemented
593C22⋊Q8D4×C5311yescharacteristic, complemented
594#121—22no—
595#121—22no—
596#121—22no—
597#121—22no—
598#121—22no—
599#121—21no—
600#121—21no—
601#121—21no—
602#121—21no—
603#121—21no—
604#121—21no—
605#121—22no—
606#121—21no—
607#121—21no—
608#121D4×C5311yescharacteristic, complemented
609#121D4×C5311yescharacteristic, complemented
610#120—22no—
611#120—22no—
612#120—22no—
613#120—21no—
614#120—22no—
615#120—48no—
616C4×Q8—21no—
617C4×D4—22no—
618C4×D4—22no—
619C4×D4—21no—
620C4×D4—22no—
621C4×D4—21no—
622C4×D4—21no—
623#117—21no—
624#117C23×C5311yescharacteristic
625C2×C4⋊C4—21no—
626C2×C4⋊C4C23×C5311yescharacteristic
627C2×C22⋊C4—22no—
628C2×C22⋊C4—21no—
629C2×C22⋊C4—24no—
630C2×C22⋊C4D4×C5312yes—
631C2×C42C23×C5312yes—
632#104—24no—
633#104—24no—
634#104—42no—
635#102—22no—
636#102—22no—
637#100—24no—
638C22⋊C8—21no—
639C22⋊C8—21no—
640#95—21no—
641C24—22no—
642C24—22no—
643C24—21no—
644C24—44no—
645C24—44no—
646C24—41no—
647C24—41no—
648C24—82no—
649C24C2×D4×C5311yescharacteristic
650C24C2×D4×C5312yes—
651#41—22no—
652#41—22no—
653#41—22no—
654#41—22no—
655#41—22no—
656#41—22no—
657#41—22no—
658#41—44no—
659#41—41no—
660#41—41no—
661#41—44no—
662#41—44no—
663#41—42no—
664#41—42no—
665#41C2×D4×C5314yes—
666C2×Q8—21no—
667C2×Q8—22no—
668C2×Q8—41no—
669C2×Q8C2×D4×C5311yescharacteristic
670C2×Q8C2×D4×C5311yescharacteristic
671C2×Q8C2×D4×C5311yescharacteristic
672C2×D4—21no—
673C2×D4—24no—
674C2×D4—24no—
675C2×D4—22no—
676C2×D4—22no—
677C2×D4—22no—
678C2×D4—22no—
679C2×D4—21no—
680C2×D4—21no—
681C2×D4—21no—
682C2×D4—21no—
683C2×D4—21no—
684C2×D4—22no—
685C2×D4—22no—
686C2×D4—22no—
687C2×D4—21no—
688C2×D4—22no—
689C2×D4—21no—
690C2×D4—22no—
691C2×D4—22no—
692C2×D4—22no—
693C2×D4—22no—
694C2×D4—21no—
695C2×D4—21no—
696C2×D4—22no—
697C2×D4—22no—
698C2×D4—22no—
699C2×D4—41no—
700C2×D4—44no—
701C2×D4—44no—
702C2×D4—44no—
703C2×D4—44no—
704C2×D4—42no—
705C2×D4—42no—
706C2×D4—48no—
707C2×D4—44no—
708C2×D4—48no—
709C2×D4—42no—
710C2×D4—42no—
711C2×D4—44no—
712C2×D4—48no—
713C2×D4—41no—
714C2×D4—41no—
715C2×D4—84no—
716C2×D4—84no—
717C2×D4C2×D4×C5311yescharacteristic
718C2×D4C2×D4×C5311yescharacteristic
719C2×D4C2×D4×C5311yescharacteristic
720C2×D4C2×D4×C5314yescomplemented
721C22×C4—21no—
722C22×C4—22no—
723C22×C4—22no—
724C22×C4—22no—
725C22×C4—21no—
726C22×C4—22no—
727C22×C4—22no—
728C22×C4—21no—
729C22×C4—21no—
730C22×C4—21no—
731C22×C4—21no—
732C22×C4—44no—
733C22×C4C2×D4×C5311yescharacteristic
734C22×C4C2×D4×C5311yescharacteristic
735C22×C4C2×D4×C5311yescharacteristic
736C22×C4C2×D4×C5311yescharacteristic
737C22×C4C2×D4×C5311yescharacteristic
738C22×C4C2×D4×C5312yes—
739C22×C4C24×C5311yescharacteristic
740SD16—28no—
741SD16—28no—
742SD16—44no—
743D8—28no—
744D8—28no—
745D8—44no—
746M16—24no—
747M16—22no—
748M16—22no—
749M16—22no—
750M16C2×D4×C5314yescomplemented
751C2×C8—24no—
752C2×C8—22no—
753C2×C8—22no—
754C4⋊C4—22no—
755C4⋊C4—22no—
756C4⋊C4—21no—
757C4⋊C4—21no—
758C4⋊C4—21no—
759C4⋊C4—21no—
760C4⋊C4C2×D4×C5311yescharacteristic
761C4⋊C4C2×D4×C5311yescharacteristic
762C22⋊C4—22no—
763C22⋊C4—22no—
764C22⋊C4—24no—
765C22⋊C4—22no—
766C22⋊C4—22no—
767C22⋊C4—22no—
768C22⋊C4—21no—
769C22⋊C4—21no—
770C22⋊C4—21no—
771C22⋊C4—21no—
772C22⋊C4—22no—
773C22⋊C4—22no—
774C22⋊C4—21no—
775C22⋊C4—48no—
776C42—22no—
777C42—21no—
778C42—22no—
779C42—22no—
780C23—21no—
781C23—22no—
782C23—22no—
783C23—22no—
784C23—21no—
785C23—21no—
786C23—24no—
787C23—24no—
788C23—41no—
789C23—42no—
790C23—44no—
791C23—44no—
792C23—44no—
793C23—44no—
794C23—44no—
795C23—42no—
796C23—42no—
797C23—42no—
798C23—42no—
799C23—42no—
800C23—81no—
801C23—82no—
802C23—82no—
803C23—82no—
804C23—82no—
805C23—82no—
806C23—816no—
807C23—82no—
808C23#120×C5311yescharacteristic
809C23#120×C5311yescharacteristic
810C23C22×D4×C5311yescharacteristic
811Q8—21no—
812Q8—21no—
813Q8—21no—
814Q8—22no—
815Q8—42no—
816Q8#120×C5312yescomplemented
817D4—24no—
818D4—21no—
819D4—24no—
820D4—21no—
821D4—21no—
822D4—22no—
823D4—24no—
824D4—42no—
825D4—42no—
826D4—44no—
827D4—44no—
828D4—44no—
829D4—42no—
830D4—44no—
831D4—42no—
832D4—42no—
833D4—44no—
834D4—42no—
835D4—41no—
836D4—42no—
837D4—41no—
838D4—42no—
839D4—42no—
840D4—81no—
841D4—82no—
842D4—82no—
843D4—82no—
844D4—82no—
845D4—816no—
846D4#120×C5312yescomplemented
847C2×C4—21no—
848C2×C4—22no—
849C2×C4—21no—
850C2×C4—21no—
851C2×C4—21no—
852C2×C4—21no—
853C2×C4—22no—
854C2×C4—22no—
855C2×C4—24no—
856C2×C4—24no—
857C2×C4—21no—
858C2×C4—22no—
859C2×C4—21no—
860C2×C4—21no—
861C2×C4—21no—
862C2×C4—42no—
863C2×C4—42no—
864C2×C4—42no—
865C2×C4—44no—
866C2×C4—42no—
867C2×C4—42no—
868C2×C4—41no—
869C2×C4—41no—
870C2×C4—42no—
871C2×C4—42no—
872C2×C4—82no—
873C2×C4#120×C5311yescharacteristic
874C2×C4#120×C5311yescharacteristic
875C2×C4C22×D4×C5311yescharacteristic
876C2×C4C22×D4×C5312yes—
877C2×C4C22×D4×C5311yescharacteristic
878C2×C4H(5,𝔽2)×C5312yes—
879C8—24no—
880C8—42no—
881C22—21no—
882C22—21no—
883C22—21no—
884C22—21no—
885C22—21no—
886C22—21no—
887C22—21no—
888C22—22no—
889C22—44no—
890C22—48no—
891C22—41no—
892C22—41no—
893C22—42no—
894C22—41no—
895C22—41no—
896C22—41no—
897C22—42no—
898C22—41no—
899C22—41no—
900C22—41no—
901C22—42no—
902C22—44no—
903C22—84no—
904C22—84no—
905C22—84no—
906C22—88no—
907C22—88no—
908C22—82no—
909C22—82no—
910C22—164no—
911C22C53×#45312yescomplemented
912C22C2×#120×C5311yescharacteristic
913C22D42×C5312yes—
914C4—21no—
915C4—21no—
916C4—22no—
917C4—44no—
918C4—42no—
919C4—41no—
920C4—41no—
921C4—41no—
922C4—41no—
923C4C2×#120×C5311yescharacteristic
924C4C53×#53411yescharacteristic
925C4D42×C5312yes—
926C2—22no—
927C2—22no—
928C2—41no—
929C2—42no—
930C2—84no—
931C2—81no—
932C2—82no—
933C2—81no—
934C2C53×#240311yescharacteristic
935C2C2×C53×#45311yescharacteristic
936C2C53×#302111yescharacteristic
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. C22×AGL(1,ℤ/8)×C53C2semidirect
2. C53×#3488C2semidirect
3. C2×C53×#453C2semidirect
4. C53×#3003C2semidirect
5. C53×#2433C2semidirect
6. C53×#2006C2semidirect
7. C53×#2002C2semidirect
8. C53×#1900C2semidirect
9. C53×#1883C2semidirect
10. C53×#1871C2semidirect
11. C53×#1789C2semidirect
12. C2×H(5,𝔽2)×C53C22semidirect
14. C23×D4×C53C22semidirect
19. C2×AGL(1,ℤ/8)×C53C22semidirect
20. C2×AGL(1,ℤ/8)×C53C22semidirect
23. C22×M16×C53C22semidirect
31. C53×#537C22semidirect
33. C53×#535C22semidirect
34. C2×#127×C53C22semidirect
40. C53×#520C22semidirect
46. C2×#104×C53C22semidirect
47. C2×#104×C53C22semidirect
48. C2×#104×C53C22semidirect
49. C53×#418C22semidirect
50. C2×#100×C53C22semidirect
51. C53×#407C22semidirect
53. C53×#390C22semidirect
54. C4×C22⋊C4×C53C22semidirect
55. C53×#343C22semidirect
56. C53×#339C22semidirect
66. C2×#41×C53D4semidirect
67. C2×#41×C53D4semidirect
71. C22×Q8×C53D4semidirect
86. C22×D4×C53D4semidirect
98. AGL(1,ℤ/8)×C53D4semidirect
123. C22⋊Q8×C53D4semidirect
124. C22⋊Q8×C53D4semidirect
139. #121×C53D4semidirect
140. #121×C53D4semidirect
251. C2×D4×C53C2×D4semidirect
281. M16×C53C2×D4semidirect
347. Q8×C53#120semidirect
377. D4×C53#120semidirect
412. #23876C53direct (N × H)
443. C22×C53#453semidirect
458. C22×AGL(1,ℤ/8)C2×C53semidirect
459. #3488C2×C53semidirect
460. C2×#453C2×C53semidirect
461. #3003C2×C53semidirect
462. #2433C2×C53semidirect
463. #2006C2×C53semidirect
464. #2002C2×C53semidirect
465. #1900C2×C53semidirect
466. #1883C2×C53semidirect
467. #1871C2×C53semidirect
468. #1789C2×C53semidirect
480. C2×H(5,𝔽2)C22×C53semidirect
482. C23×D4C22×C53semidirect
487. C2×AGL(1,ℤ/8)C22×C53semidirect
488. C2×AGL(1,ℤ/8)C22×C53semidirect
491. C22×M16C22×C53semidirect
499. #537C22×C53semidirect
501. #535C22×C53semidirect
502. C2×#127C22×C53semidirect
508. #520C22×C53semidirect
514. C2×#104C22×C53semidirect
515. C2×#104C22×C53semidirect
516. C2×#104C22×C53semidirect
517. #418C22×C53semidirect
518. C2×#100C22×C53semidirect
519. #407C22×C53semidirect
521. #390C22×C53semidirect
522. C4×C22⋊C4C22×C53semidirect
523. #343C22×C53semidirect
524. #339C22×C53semidirect
525. C53#23876direct (N × H)
535. C2×#41D4×C53semidirect
536. C2×#41D4×C53semidirect
540. C22×Q8D4×C53semidirect
555. C22×D4D4×C53semidirect
567. AGL(1,ℤ/8)D4×C53semidirect
592. C22⋊Q8D4×C53semidirect
593. C22⋊Q8D4×C53semidirect
608. #121D4×C53semidirect
609. #121D4×C53semidirect
720. C2×D4C2×D4×C53semidirect
750. M16C2×D4×C53semidirect
816. Q8#120×C53semidirect
846. D4#120×C53semidirect
911. C22C53×#453semidirect

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)×C532
C53×#34882
C2×C53×#4532
C53×#30032
C53×#24332
C53×#20062
C53×#20022
C53×#19002
C53×#18832
C53×#18712
C53×#17892
#2387653