Finite Group Databasebeta

← #27428492

Subgroups — #27428492

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279017C211yesmaximal, characteristic
2C4⋊C4×C22⋊D53C211yesmaximal, characteristic
3#20276280C211yesmaximal, characteristic, complemented
4#20276195C211yesmaximal, characteristic, complemented
5#20276154C211yesmaximal, characteristic, complemented
6#20276150C211yesmaximal, characteristic, complemented
7#20275292C211yesmaximal, characteristic, complemented
8#20275291C211yesmaximal, characteristic, complemented
9#20275007C211yesmaximal, characteristic
10#20274909C211yesmaximal, characteristic, complemented
11#20274908C211yesmaximal, characteristic, complemented
12#20274804C211yesmaximal, characteristic, complemented
13#20274801C211yesmaximal, characteristic, complemented
14#20274740C211yesmaximal, characteristic, complemented
15C2×C53×#340C211yesmaximal, characteristic, complemented
16C2×#13133200—21no—
17C2×C4×C22⋊D53—21no—
18C2×C4×C22⋊D53C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26C2×#13133129C2211yescharacteristic
27#16810554—21no—
28#16810549—21no—
29#16810519C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810518—21no—
32#16810496—21no—
33#16810494—21no—
34#16810492C2211yescharacteristic
35#16810486—21no—
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810480C2211yescharacteristic
39#16810480C2211yescharacteristic
40#16810480C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C22⋊D53D411yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133200—44no—
65C4×C22⋊D53—22no—
66C4×C22⋊D53—22no—
67C4×C22⋊D53—44no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919D411yescharacteristic
78C4⋊C4×D53—22no—
79C4⋊C4×D53—41no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53D411yescharacteristic
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89#13133149—22no—
90#13133149—41no—
91#13133148—22no—
92#13133148—41no—
93C22⋊#88322—22no—
94#13133129—22no—
95#13133129—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C23×C4×C53—21no—
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53—22no—
108C22×C8×C53—22no—
109C22×C8×C53—41no—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—22no—
116C2×C4⋊C4×C53—22no—
117C2×C4⋊C4×C53—41no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C24×C53C2×D411yescharacteristic
123C23×D53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—81no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53C2411yescharacteristic
142C2×C22⋊D53—22no—
143C2×C22⋊D53—22no—
144C2×C22⋊D53SD1614yes—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—44no—
167C2×C8×C53#4112yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53#4112yes—
178C22⋊Dic53—22no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181#11719921—22no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—22no—
194#11719919—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—81no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—42no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53#12312yes—
213C23×C53C2×SD1612yes—
214C23×C53AGL(1,ℤ/8)12yes—
215C23×C53C22×D411yescharacteristic
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53C2×SD1612yes—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—84no—
242C2×C4×C53—84no—
243C2×C4×C53#12312yes—
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C22⋊D53—28no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53C2×SD1612yes—
272C4×D53—41no—
273C4×D53—41no—
274C4×D53—41no—
275C4×D53—41no—
276C4×D53—84no—
277C8×C53—22no—
278C8×C53—41no—
279#88322—41no—
280#88322—41no—
281#24937—531nomaximal, Sylow
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53#48114yes—
289C22×C53C2×#12311yescharacteristic
290C22×C53#54611yescharacteristic
291C22×C53#54711yescharacteristic
292C22×C53C22×SD1611yescharacteristic
293C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C53#57711yescharacteristic
295C22×C53#57811yescharacteristic
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—41no—
300C2×D53—41no—
301C2×D53—41no—
302C2×D53—41no—
303C4×C53—22no—
304C4×C53—42no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53#52911yescharacteristic
309C4×C53#53911yescharacteristic
310C4×C53#55111yescharacteristic
311C4×C53#55211yescharacteristic
312Dic53—22no—
313Dic53—41no—
314Dic53—41no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319#2486—531no—
320D4×C4⋊C4—531no—
321#2100—531no—
322#2084—531no—
323#2077—531no—
324#2077—531no—
325#1919—531no—
326#1919—531no—
327#1869—531no—
328#1853—531no—
329#1853—531no—
330#1824—531no—
331#1823—531no—
332#1807—531no—
333C2×#340D5311yescharacteristic, complemented
334C2×C53—24no—
335C2×C53#241911yescharacteristic
336C2×C53C2×#48111yescharacteristic
337C2×C53#309711yescharacteristic
338C2×C53#329811yescharacteristic
339C2×C53#331011yescharacteristic
340C2×C53#333511yescharacteristic
341C2×C53#334311yescharacteristic
342D53—42no—
343C23×C8C2×D5311yescharacteristic
344C2×#123—1061no—
345C2×C4×D4—531no—
346C2×C4×D4—1061no—
347C22×C4⋊C4—531no—
348C22×C4⋊C4C2×D5311yescharacteristic
349C22×C4⋊C4C2×D5311yescharacteristic
350C2×#106—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×#106—531no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C22⋊C8—531no—
357#398—1061no—
358#397—1061no—
359#389—531no—
360#389—531no—
361#389—1061no—
362#385—1061no—
363#385—1061no—
364#384—531no—
365#381—1061no—
366#380—531no—
367#380—531no—
368#380—531no—
369#380—531no—
370#380—531no—
371C4×C4⋊C4—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340C2×D5311yescharacteristic
381#340C2×D5311yescharacteristic
382#340C2×D5311yescharacteristic
383#340C2×D5311yescharacteristic
384#336—531no—
385#336—531no—
386C53#2493711yescharacteristic, complemented, Sylow
387C22×D4—531no—
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—531no—
391C23×C4—1061no—
392C23×C4C22×D5311yescharacteristic
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—41no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400#123—2124no—
401C4×D4—1062no—
402C4×D4—1062no—
403C4×D4—2124no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—2121no—
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—531no—
427C2×C22⋊C4—531no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C42—531no—
434C2×C42—1061no—
435#106—1062no—
436#106—1062no—
437#106—2121no—
438#106—2121no—
439C22⋊C8—1062no—
440C22⋊C8—1062no—
441C22⋊C8—2121no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450C24—531no—
451C24D4×D5311yescharacteristic
452C2×D4—534no—
453C2×D4—1062no—
454C2×D4—1062no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—4241no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4#1171994512yes—
486C22×C4D4×D5311yescharacteristic
487C22×C4#1171994712yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—44no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8#1171994512yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—1062no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—4241no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4#1171994512yes—
526C22⋊C4—1062no—
527C22⋊C4—1062no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—1061no—
534C42—1061no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—532no—
541C23—1061no—
542C23—1061no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23—1061no—
547C23#1313319912yes—
548C23SD16×D5312yes—
549C23#1313324112yes—
550C23C2×D4×D5311yescharacteristic
551D4—1068no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—532no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—4244no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4#1313319912yes—
599C2×C4C2×#1171994711yescharacteristic
600C2×C4C2×#1171994511yescharacteristic
601C2×C4C2×#1171994511yescharacteristic
602C2×C4C2×#1171994511yescharacteristic
603C2×C4#1313326211yescharacteristic
604C2×C4#1313326511yescharacteristic
605C8—22no—
606C8—41no—
607C8—2121no—
608C8—2121no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—1062no—
616C22—1062no—
617C22—1062no—
618C22—2121no—
619C22—2121no—
620C22—2121no—
621C22—2121no—
622C22#1681094014yes—
623C22C2×#1313319911yescharacteristic
624C22#1681126311yescharacteristic
625C22#1681127411yescharacteristic
626C22C2×SD16×D5311yescharacteristic
627C22C2×#1313324111yescharacteristic
628C22#1681142911yescharacteristic
629C22#1681143611yescharacteristic
630C4—22no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—1062no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—4241no—
640C4—2121no—
641C4—2121no—
642C4#1681115711yescharacteristic
643C4#1681122011yescharacteristic
644C4#1681129911yescharacteristic
645C4#1681130811yescharacteristic
646C2—24no—
647C2—2122no—
648C2#2027819211yescharacteristic
649C2C2×#1681094011yescharacteristic
650C2#2028468011yescharacteristic
651C2#2028657611yescharacteristic
652C2#2028673211yescharacteristic
653C2#2028706811yescharacteristic
654C2#2028715111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC23×C53C22×D4
prime radicalC22×C22⋊D53D4
hypercenterC2×#340D53
hypocenterC53#24937
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #20276280C2semidirect
4. #20276195C2semidirect
5. #20276154C2semidirect
6. #20276150C2semidirect
7. #20275292C2semidirect
8. #20275291C2semidirect
10. #20274909C2semidirect
11. #20274908C2semidirect
12. #20274804C2semidirect
13. #20274801C2semidirect
14. #20274740C2semidirect
15. C2×C53×#340C2semidirect
333. C2×#340D53semidirect
386. C53#24937semidirect

Maximal subgroups

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

subgroupindex
#202790172
C4⋊C4×C22⋊D532
#202762802
#202761952
#202761542
#202761502
#202752922
#202752912
#202750072
#202749092
#202749082
#202748042
#202748012
#202747402
C2×C53×#3402
#2493753