Finite Group Databasebeta

← #27456744

Subgroups — #27456744

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279133C211yesmaximal, characteristic, complemented
2#20277735C211yesmaximal, characteristic
3#20276121C211yesmaximal, characteristic, complemented
4#20276101C211yesmaximal, characteristic, complemented
5#20275894C211yesmaximal, characteristic
6#20275470C211yesmaximal, characteristic, complemented
7#20275449C211yesmaximal, characteristic, complemented
8#20275324C211yesmaximal, characteristic, complemented
9#20275289C211yesmaximal, characteristic, complemented
10#20275285C211yesmaximal, characteristic, complemented
11#20274964C211yesmaximal, characteristic, complemented
12#20274905C211yesmaximal, characteristic, complemented
13#20274799C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic, complemented
15C53×#2033C211yesmaximal, characteristic
16C2×#13133210—21no—
17C2×D4×Dic53—21no—
18C2×D4×Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810564—21no—
29#16810551—21no—
30#16810526C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810516—21no—
35#16810510—21no—
36#16810503—21no—
37#16810498—21no—
38#16810492C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133210—44no—
64D4×Dic53—22no—
65D4×Dic53—41no—
66D4×Dic53—44no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—22no—
71C2×C4⋊Dic53—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53D411yescharacteristic
84C2×C22⋊Dic53D411yescharacteristic
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133151—41no—
91#13133150—22no—
92#13133146—22no—
93#13133138—22no—
94#13133138—41no—
95C22⋊#88322—41no—
96C8×Dic53—41no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#121×C53—22no—
111#121×C53—22no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117#107×C53—22no—
118#102×C53—22no—
119#102×C53—22no—
120C22⋊C8×C53—22no—
121#95×C53—21no—
122C24×C53—21no—
123C24×C53C2×D411yescharacteristic
124C2×D4×C53—24no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—42no—
131C2×D4×C53—42no—
132C2×D4×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C22×Dic53—81no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53#4112yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53#4112yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—42no—
172C22⋊Dic53—41no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—42no—
176C22⋊Dic53—81no—
177C22⋊Dic53D814yes—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53#4112yes—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53C2×D812yes—
218C23×C53C22×D411yescharacteristic
219D4×C53—22no—
220D4×C53—22no—
221D4×C53—48no—
222D4×C53—41no—
223D4×C53—41no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
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×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—41no—
258C2×Dic53—41no—
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—84no—
269C2×Dic53—84no—
270C2×Dic53C2×D812yes—
271C2×Dic53C2×D812yes—
272C8×C53—22no—
273C8×C53—41no—
274#88322—41no—
275#88322—41no—
276#27030—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53#52511yescharacteristic
292C22×C53#54611yescharacteristic
293C22×C53#54711yescharacteristic
294C22×C53C22×D811yescharacteristic
295C22×C53#57511yescharacteristic
296C22×C53#57711yescharacteristic
297C22×C53#57811yescharacteristic
298C4×C53—42no—
299C4×C53—41no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53#52911yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53#55311yescharacteristic
305C4×C53#55611yescharacteristic
306Dic53—22no—
307Dic53—22no—
308Dic53—41no—
309Dic53—42no—
310Dic53—41no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314#2496—531no—
315#2361—531no—
316#2071—531no—
317#2067—531no—
318#2033—531no—
319#2033D5311yescharacteristic
320#1947—531no—
321#1942—531no—
322#1923—531no—
323#1918—531no—
324#1918—531no—
325#1862—531no—
326#1852—531no—
327#1823—531no—
328C4×#102—531no—
329C2×C53—42no—
330C2×C53—82no—
331C2×C53#246611yescharacteristic
332C2×C53#310011yescharacteristic
333C2×C53#312011yescharacteristic
334C2×C53#329411yescharacteristic
335C2×C53#331311yescharacteristic
336C2×C53#333411yescharacteristic
337C2×C53#335111yescharacteristic
338C2×#126—1061no—
339C2×#121C2×D5311yescharacteristic
340C2×C4×D4—531no—
341C2×C4×D4—1061no—
342C22×C4⋊C4—531no—
343C2×#107—531no—
344C2×#107—531no—
345C2×#107—531no—
346C2×#107C2×D5311yescharacteristic
347C2×C4⋊C8—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102C2×D5311yescharacteristic
351C2×#102C2×D5311yescharacteristic
352C2×C22⋊C8—531no—
353C2×C22⋊C8C2×D5311yescharacteristic
354C2×C4×C8—531no—
355#400—1061no—
356#397—1061no—
357#390—531no—
358#389—531no—
359#389—531no—
360#389—531no—
361#388—1061no—
362#387—1061no—
363#386—1061no—
364#385—1061no—
365#384—531no—
366#380—531no—
367#380C2×D5311yescharacteristic
368C4×C4⋊C4—531no—
369C4×C4⋊C4—531no—
370C4×C22⋊C4—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340C2×D5311yescharacteristic
379#336—531no—
380#336—531no—
381C53#2703011yescharacteristic, complemented, Sylow
382C22×D4—21no—
383C22×D4C22×D5311yescharacteristic
384C23×C4—1061no—
385C23×C4—1061no—
386C23×C4C22×D5311yescharacteristic
387C22×C8—531no—
388C22×C8—531no—
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391#126—2124no—
392#121—22no—
393#121—22no—
394C4×D4—1062no—
395C4×D4—2121no—
396C4×D4—2124no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—2121no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—1061no—
423C2×C42—1061no—
424#107—22no—
425#107—1062no—
426#107—1062no—
427#107—2121no—
428C4⋊C8—1062no—
429#102—22no—
430#102—22no—
431#102—1062no—
432#102—2121no—
433C22⋊C8—22no—
434C22⋊C8—2121no—
435C4×C8—2121no—
436#95—21no—
437#95—531no—
438#95—531no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24—21no—
446C24D4×D5311yescharacteristic
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4#1171994712yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—4241no—
474C22×C4—4241no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4D4×D5311yescharacteristic
480C22×C4C23×D5311yescharacteristic
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8#1171994712yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—532no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4#1171994712yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—534no—
516C22⋊C4—4241no—
517C22⋊C4—2122no—
518C22⋊C4—2121no—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—1061no—
522C42—1061no—
523C42—1061no—
524C42—1061no—
525C42—2122no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23D8×D5312yes—
541C23C2×D4×D5311yescharacteristic
542D4—22no—
543D4—22no—
544D4—48no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—532no—
564C2×C4—532no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1062no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—4244no—
577C2×C4—4244no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4#41×D5311yescharacteristic
590C2×C4C2×#1171994711yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4#1313326311yescharacteristic
594C2×C4#1313326411yescharacteristic
595C8—22no—
596C8—41no—
597C8—2121no—
598C8—2121no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#1681112911yescharacteristic
614C22#1681126511yescharacteristic
615C22#1681127311yescharacteristic
616C22C2×D8×D5311yescharacteristic
617C22#1681141911yescharacteristic
618C22#1681143211yescharacteristic
619C22#1681143711yescharacteristic
620C4—42no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—1062no—
625C4—1062no—
626C4—4241no—
627C4—4241no—
628C4—4241no—
629C4—2121no—
630C4—2122no—
631C4—2121no—
632C4#1681115411yescharacteristic
633C4#1681121711yescharacteristic
634C4#1681132511yescharacteristic
635C4#1681134211yescharacteristic
636C2—42no—
637C2—82no—
638C2#2027874611yescharacteristic
639C2#2028471011yescharacteristic
640C2#2028493011yescharacteristic
641C2#2028652611yescharacteristic
642C2#2028678411yescharacteristic
643C2#2028705911yescharacteristic
644C2#2028727011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2033C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2033D53
hypocenterC53#27030
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. #20279133C2semidirect
3. #20276121C2semidirect
4. #20276101C2semidirect
6. #20275470C2semidirect
7. #20275449C2semidirect
8. #20275324C2semidirect
9. #20275289C2semidirect
10. #20275285C2semidirect
11. #20274964C2semidirect
12. #20274905C2semidirect
13. #20274799C2semidirect
14. #102×Dic53C2semidirect
381. C53#27030semidirect

Maximal subgroups

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

subgroupindex
#202791332
#202777352
#202761212
#202761012
#202758942
#202754702
#202754492
#202753242
#202752892
#202752852
#202749642
#202749052
#202747992
#102×Dic532
C53×#20332
#2703053