Finite Group Databasebeta

← #27484933

Subgroups — #27484933

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic53C211yesmaximal, characteristic
2#20277199C211yesmaximal, characteristic, complemented
3#20276215C211yesmaximal, characteristic, complemented
4#20276210C211yesmaximal, characteristic, complemented
5#20276099C211yesmaximal, characteristic, complemented
6#20276085C211yesmaximal, characteristic, complemented
7#20275303C211yesmaximal, characteristic, complemented
8#20275289C211yesmaximal, characteristic, complemented
9#20275038C211yesmaximal, characteristic
10#20274978C211yesmaximal, characteristic, complemented
11#20274964C211yesmaximal, characteristic, complemented
12#20274916C211yesmaximal, characteristic, complemented
13#20274914C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic, complemented
15C53×#1875C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810526C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810516—21no—
36#16810503—21no—
37#16810498—21no—
38#16810498—21no—
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C4⋊Dic53—21no—
42C4×C4⋊Dic53—21no—
43C22⋊C4×Dic53—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64D4×Dic53—44no—
65D4×Dic53—41no—
66D4×Dic53—41no—
67#13133176—41no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—41no—
86C2×C4×Dic53—41no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151—41no—
92#13133150—22no—
93#13133149—41no—
94#13133148—22no—
95#13133146—41no—
96#13133146—41no—
97#13133138—41no—
98#13133138—41no—
99C22⋊#88322—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C22×D4×C53—21no—
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#121×C53—22no—
113#121×C53Q814yes—
114#117×C53—22no—
115#117×C53—22no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C42×C53—21no—
122#102×C53—22no—
123#102×C53—22no—
124C22⋊C8×C53—22no—
125C24×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×D4×C53—24no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—42no—
134C2×D4×C53—42no—
135C2×D4×C53C2×Q812yes—
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×C53—21no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×Q812yes—
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—41no—
154C22×Dic53—41no—
155C22×Dic53—41no—
156C22×Dic53—41no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C2×Q812yes—
174C4⋊C4×C53#4112yes—
175C4⋊C4×C53#4112yes—
176C22⋊C4×C53—24no—
177C22⋊C4×C53—24no—
178C42×C53—24no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—41no—
202C4×Dic53—41no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—22no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53C22⋊Q812yes—
225C23×C53C22×D411yescharacteristic
226D4×C53—22no—
227D4×C53—22no—
228D4×C53—48no—
229D4×C53—41no—
230D4×C53—41no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
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×C53C22⋊Q812yes—
247C2×C4×C53C22×Q811yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—44no—
264C2×Dic53—44no—
265C2×Dic53—41no—
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×Dic53#13512yes—
279C2×Dic53AGL(1,ℤ/8)12yes—
280C8×C53—41no—
281C8×C53—41no—
282#88322—41no—
283#88322—41no—
284#29236—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—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53C2×C22⋊Q811yescharacteristic
300C22×C53#52511yescharacteristic
301C22×C53#53411yescharacteristic
302C22×C53C2×#13511yescharacteristic
303C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C53#57511yescharacteristic
305C22×C53#57511yescharacteristic
306C4×C53—22no—
307C4×C53—42no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53#55311yescharacteristic
311C4×C53#55311yescharacteristic
312C4×C53#55411yescharacteristic
313C4×C53#55411yescharacteristic
314Dic53—22no—
315Dic53—22no—
316Dic53—41no—
317Dic53—41no—
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322C4×#121—531no—
323#2296—531no—
324#2088—531no—
325#2087—531no—
326#2067—531no—
327#2065—531no—
328#1920—531no—
329#1918—531no—
330#1875—531no—
331#1875D5311yescharacteristic
332#1864—531no—
333#1862—531no—
334#1854—531no—
335#1854—531no—
336#1797—531no—
337C2×C53—42no—
338C2×C53—82no—
339C2×C53#257911yescharacteristic
340C2×C53#307611yescharacteristic
341C2×C53#307711yescharacteristic
342C2×C53#310611yescharacteristic
343C2×C53#311111yescharacteristic
344C2×C53#319111yescharacteristic
345C2×C53#319411yescharacteristic
346C2×#121C2×D5311yescharacteristic
347C2×C4×D4—531no—
348C2×C4×D4—1061no—
349C2×#117—531no—
350C2×#117C2×D5311yescharacteristic
351C22×C42—531no—
352C2×#107—531no—
353C2×#107—531no—
354C2×#106—531no—
355C2×#106—531no—
356C2×C4⋊C8—531no—
357C2×C4⋊C8—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×#102C2×D5311yescharacteristic
361C2×#102C2×D5311yescharacteristic
362C2×C22⋊C8—531no—
363C2×C22⋊C8C2×D5311yescharacteristic
364#390—531no—
365#389—531no—
366#389—531no—
367#389—531no—
368#389—531no—
369#389—531no—
370#388—1061no—
371#386—1061no—
372#385—1061no—
373#385—1061no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—1061no—
377C4×C4⋊C4—1061no—
378C4×C22⋊C4—1061no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340C2×D5311yescharacteristic
386#340C2×D5311yescharacteristic
387#336—531no—
388#336—531no—
389C53#2923611yescharacteristic, complemented, Sylow
390C22×D4—21no—
391C22×D4C22×D5311yescharacteristic
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—1061no—
395C23×C4C22×D5311yescharacteristic
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400#121—22no—
401#121Dic10614yes—
402C4×D4—2121no—
403C4×D4—2121no—
404C4×D4—2124no—
405#117—22no—
406#117—22no—
407#117—2121no—
408#117—2121no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—21no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—531no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42—2121no—
436C2×C42—2121no—
437#107—1062no—
438#107—2121no—
439#106—1062no—
440#106—2121no—
441C4⋊C8—2121no—
442C4⋊C8—2121no—
443#102—22no—
444#102—22no—
445#102—2121no—
446#102—2121no—
447C22⋊C8—22no—
448C22⋊C8—1062no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456C24—21no—
457C24D4×D5311yescharacteristic
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×Dic10612yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—2122no—
485C22×C4—2121no—
486C22×C4—2121no—
487C22×C4—2121no—
488C22×C4—2121no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4C2×Dic10612yes—
492C22×C4D4×D5311yescharacteristic
493C22×C4#1171994712yes—
494C22×C4C23×D5311yescharacteristic
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—2122no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4C2×Dic10612yes—
524C4⋊C4#1171994712yes—
525C4⋊C4#1171994712yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—24no—
534C42—1061no—
535C42—1061no—
536C42—1061no—
537C42—1061no—
538C42—2121no—
539C42—2121no—
540C42—2122no—
541C42—2122no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#1313319412yes—
556C23C2×D4×D5311yescharacteristic
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—4244no—
590C2×C4—2124no—
591C2×C4—2124no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4#1313319412yes—
605C2×C4C22×Dic10611yescharacteristic
606C2×C4C2×#1171994711yescharacteristic
607C2×C4C2×#1171994711yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4#1313326311yescharacteristic
610C2×C4#1313326311yescharacteristic
611C8—41no—
612C8—41no—
613C8—2121no—
614C8—2121no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#1313319411yescharacteristic
630C22#1681112911yescharacteristic
631C22#1681118411yescharacteristic
632C22#135×D5311yescharacteristic
633C22AGL(1,ℤ/8)×D5311yescharacteristic
634C22#1681141911yescharacteristic
635C22#1681142011yescharacteristic
636C4—22no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—1062no—
641C4—1062no—
642C4—4241no—
643C4—4241no—
644C4—4241no—
645C4—4241no—
646C4—2121no—
647C4—2121no—
648C4#1681132511yescharacteristic
649C4#1681132511yescharacteristic
650C4#1681133011yescharacteristic
651C4#1681133011yescharacteristic
652C2—42no—
653C2—82no—
654C2#2028005911yescharacteristic
655C2#2028449911yescharacteristic
656C2#2028450911yescharacteristic
657C2#2028476811yescharacteristic
658C2#2028481711yescharacteristic
659C2#2028550611yescharacteristic
660C2#2028553311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1875C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1875D53
hypocenterC53#29236
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. #20277199C2semidirect
3. #20276215C2semidirect
4. #20276210C2semidirect
5. #20276099C2semidirect
6. #20276085C2semidirect
7. #20275303C2semidirect
8. #20275289C2semidirect
10. #20274978C2semidirect
11. #20274964C2semidirect
12. #20274916C2semidirect
13. #20274914C2semidirect
14. #20274693C2semidirect
389. C53#29236semidirect

Maximal subgroups

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

subgroupindex
#121×Dic532
#202771992
#202762152
#202762102
#202760992
#202760852
#202753032
#202752892
#202750382
#202749782
#202749642
#202749162
#202749142
#202746932
C53×#18752
#2923653