Finite Group Databasebeta

← #27449725

Subgroups — #27449725

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic
2C2×Q16×Dic53C211yesmaximal, characteristic, complemented
3#20278236C211yesmaximal, characteristic
4#20277724C211yesmaximal, characteristic
5#20276251C211yesmaximal, characteristic, complemented
6#20276035C211yesmaximal, characteristic, complemented
7#20275915C211yesmaximal, characteristic
8#20275472C211yesmaximal, characteristic, complemented
9#20275447C211yesmaximal, characteristic, complemented
10#20275359C211yesmaximal, characteristic, complemented
11#20275324C211yesmaximal, characteristic
12#20274971C211yesmaximal, characteristic, complemented
13#20274967C211yesmaximal, characteristic, complemented
14#20274905C211yesmaximal, characteristic
15C53×#2001C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810956—24no—
20Q16×Dic53—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×C22⋊#88322C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810559—21no—
34#16810553—21no—
35#16810530C2211yescharacteristic
36#16810530C2211yescharacteristic
37#16810522—21no—
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810511—21no—
41#16810509—21no—
42#16810504—21no—
43#16810498—21no—
44#16810492C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47C22⋊C4×Dic53—21no—
48C4×C22⋊Dic53C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×Q16×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53—22no—
66Q8×Dic53—41no—
67Q8×Dic53—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—41no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53D411yescharacteristic
83C2×C22⋊Dic53D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133151—41no—
91#13133150—22no—
92#13133150—22no—
93#13133150—22no—
94#13133146—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—22no—
98#13133141D414yes—
99C22⋊#88322—22no—
100C8×Dic53—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124D411yescharacteristic
110#13133124D411yescharacteristic
111C22×Q8×C53C2311yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C23×C4×C53C2311yescharacteristic
114C2×Q16×C53—22no—
115C2×Q16×C53—22no—
116C2×Q16×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22⋊Q8×C53—22no—
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53—41no—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#103×C53—22no—
130#103×C53—41no—
131C22⋊C8×C53—22no—
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53C2×D412yes—
144C2×Q8×C53#4112yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—81no—
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168Q16×C53—28no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53#4112yes—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—42no—
180C4⋊C4×C53#4112yes—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—42no—
183C22⋊Dic53—41no—
184C22⋊Dic53—81no—
185C22⋊Dic53Q1614yescomplemented
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53—42no—
196C4⋊Dic53—81no—
197C4⋊Dic53C2×D412yes—
198#11719919—42no—
199#11719919—81no—
200#11719919—81no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53—81no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322C2×D412yes—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53C2×Q1612yes—
221C23×C53C22×D411yescharacteristic
222Q8×C53—22no—
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53—41no—
227Q8×C53—41no—
228Q8×C53—41no—
229Q8×C53—41no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—22no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—84no—
278C2×Dic53C2×Q1612yes—
279C2×Dic53C2×Q1612yes—
280C8×C53—22no—
281C8×C53—41no—
282#88322—22no—
283#88322—41no—
284#26559—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53#52511yescharacteristic
293C22×C53D4211yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C22×Q1611yescharacteristic
296C22×C53#57511yescharacteristic
297C22×C53#57611yescharacteristic
298C22×C53#57811yescharacteristic
299C4×C53—42no—
300C4×C53—41no—
301C4×C53—42no—
302C4×C53—42no—
303C4×C53—81no—
304C4×C53#52511yescharacteristic
305C4×C53#53911yescharacteristic
306C4×C53#54711yescharacteristic
307C4×C53#55911yescharacteristic
308Dic53—22no—
309Dic53—22no—
310Dic53—41no—
311Dic53—41no—
312Dic53—42no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316C2×#484—531no—
317C2×C4×Q16—531no—
318#2423—531no—
319#2360—531no—
320#2095—531no—
321#2056—531no—
322#2036—531no—
323#2001D5311yescharacteristic
324#1947—531no—
325#1942—531no—
326#1928—531no—
327#1923—531no—
328#1863—531no—
329#1863—531no—
330#1852—531no—
331C2×C53—42no—
332C2×C53#245511yescharacteristic
333C2×C53#310411yescharacteristic
334C2×C53#312411yescharacteristic
335C2×C53#326811yescharacteristic
336C2×C53D4×Q1611yescharacteristic
337C2×C53#333811yescharacteristic
338C2×C53#335211yescharacteristic
339C22×Q16C2×D5311yescharacteristic
340C2×C22⋊Q8C2×D5311yescharacteristic
341C2×C22⋊Q8C2×D5311yescharacteristic
342C2×C4×Q8—531no—
343C2×C4×Q8—531no—
344C22×C4⋊C4—531no—
345#484—1064no—
346C4×Q16—1064no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#107—531no—
351C2×#107—531no—
352C2×C4⋊C8—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103C2×D5311yescharacteristic
358C2×#103C2×D5311yescharacteristic
359C2×C22⋊C8—531no—
360C2×C22⋊C8C2×D5311yescharacteristic
361C2×C4×C8—531no—
362#399—1061no—
363#398—1061no—
364#391—531no—
365#391—531no—
366#389—531no—
367#389—531no—
368#389—1061no—
369#388—1061no—
370#387—1061no—
371#387—1061no—
372#385—1061no—
373#384—531no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—531no—
377C4×C22⋊C4—1061no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340C2×D5311yescharacteristic
382#336—531no—
383#336—531no—
384C53#2655911yescharacteristic, complemented, Sylow
385C22×Q8C22×D5311yescharacteristic
386C22×Q8C22×D5311yescharacteristic
387C23×C4—1061no—
388C23×C4C22×D5311yescharacteristic
389C2×Q16—22no—
390C2×Q16—22no—
391C2×Q16—22no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8C22×D5311yescharacteristic
395C22×C8C22×D5311yescharacteristic
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—41no—
400C4×Q8—1062no—
401C4×Q8—1062no—
402C4×Q8—2121no—
403C4×Q8—2121no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—2121no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—531no—
424C2×C22⋊C4—531no—
425C2×C22⋊C4—1061no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42—1061no—
431#107—1062no—
432#107—1062no—
433#107—1062no—
434#107—1062no—
435#107—2121no—
436C4⋊C8—1062no—
437#103—22no—
438#103—41no—
439#103—534no—
440#103—1062no—
441#103—1062no—
442#103—1062no—
443C22⋊C8—22no—
444C22⋊C8—1062no—
445C4×C8—1062no—
446#95—531no—
447#95—531no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456C24D4×D5311yescharacteristic
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8D4×D5312yes—
468C2×Q8#1171994712yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—4241no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4D4×D5311yescharacteristic
491C22×C4C23×D5311yescharacteristic
492Q16—28no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—532no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8#1171994712yes—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—532no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—4241no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4#1171994712yes—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—534no—
529C22⋊C4—4241no—
530C22⋊C4—2121no—
531C42—1061no—
532C42—1061no—
533C42—1061no—
534C42—1061no—
535C42—4241no—
536C42—2122no—
537C42—2122no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23Q16×D5312yes—
545C23C2×D4×D5311yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
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—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—532no—
574C2×C4—532no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1062no—
586C2×C4—4244no—
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×C4C2×D4×D5311yescharacteristic
599C2×C4C2×#1171994711yescharacteristic
600C2×C4C2×#1171994711yescharacteristic
601C2×C4C2×#1171994711yescharacteristic
602C2×C4#1313326211yescharacteristic
603C2×C4#1313326411yescharacteristic
604C8—22no—
605C8—41no—
606C8—1062no—
607C8—2121no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22#1681112911yescharacteristic
616C22#1681125511yescharacteristic
617C22#1681127311yescharacteristic
618C22C2×Q16×D5311yescharacteristic
619C22#1681142211yescharacteristic
620C22#1681142511yescharacteristic
621C22#1681143911yescharacteristic
622C4—42no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—1062no—
628C4—1062no—
629C4—4241no—
630C4—4241no—
631C4—4241no—
632C4—2121no—
633C4—2122no—
634C4—2121no—
635C4#1681112911yescharacteristic
636C4#1681121811yescharacteristic
637C4#1681127311yescharacteristic
638C4#1681135911yescharacteristic
639C2—42no—
640C2#2027859111yescharacteristic
641C2#2028474911yescharacteristic
642C2#2028497811yescharacteristic
643C2#2028625811yescharacteristic
644C2#2028646011yescharacteristic
645C2#2028710411yescharacteristic
646C2#2028728511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2001C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2001D53
hypocenterC53#26559
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic53C2semidirect
5. #20276251C2semidirect
6. #20276035C2semidirect
8. #20275472C2semidirect
9. #20275447C2semidirect
10. #20275359C2semidirect
12. #20274971C2semidirect
13. #20274967C2semidirect
185. C22⋊Dic53Q16semidirect
384. C53#26559semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×Q16×Dic532
#202782362
#202777242
#202762512
#202760352
#202759152
#202754722
#202754472
#202753592
#202753242
#202749712
#202749672
#202749052
C53×#20012
#2655953