Finite Group Databasebeta

← #27425445

Subgroups — #27425445

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810925C211yesmaximal, characteristic, complemented
2C2×#16810681C211yesmaximal, characteristic, complemented
3#20280549C211yesmaximal, characteristic, complemented
4#20277410C211yesmaximal, characteristic
5#20276148C211yesmaximal, characteristic, complemented
6#20276146C211yesmaximal, characteristic, complemented
7#20275894C211yesmaximal, characteristic
8#20275671C211yesmaximal, characteristic, complemented
9#20275470C211yesmaximal, characteristic, complemented
10#20275454C211yesmaximal, characteristic, complemented
11#20275447C211yesmaximal, characteristic, complemented
12#20275409C211yesmaximal, characteristic, complemented
13#20275400C211yesmaximal, characteristic, complemented
14#20274713C211yesmaximal, characteristic, complemented
15C53×#2033C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810925—24no—
20#16810681—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810557—21no—
32#16810555—21no—
33#16810546—21no—
34#16810546—21no—
35#16810526—21no—
36#16810526C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810516—21no—
41#16810503—21no—
42#16810498—21no—
43#16810492C2211yescharacteristic
44#16810483C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133208—22no—
65#13133208—22no—
66D4×Dic53—22no—
67D4×Dic53—22no—
68#13133176—22no—
69#13133176—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53D411yescharacteristic
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53D411yescharacteristic
89C2×C22⋊Dic53D411yescharacteristic
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133150—22no—
94#13133150—22no—
95#13133149—22no—
96#13133146—22no—
97#13133146D414yescomplemented
98#13133138—22no—
99#13133138—22no—
100C22⋊#88322—22no—
101#13133127—22no—
102C8×Dic53—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×D4×C53—21no—
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114#121×C53—22no—
115#121×C53—22no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#107×C53—22no—
122#102×C53—22no—
123#102×C53—22no—
124C22⋊C8×C53—22no—
125#95×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—24no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—81no—
135C2×D4×C53—81no—
136C2×D4×C53#4112yes—
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×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—81no—
154C22×Dic53—81no—
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53C2×D412yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53#4112yes—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—81no—
173C22⋊Dic53—22no—
174C22⋊Dic53—22no—
175C22⋊Dic53—42no—
176C22⋊Dic53—81no—
177C22⋊Dic53—81no—
178C22⋊Dic53—81no—
179C22⋊Dic53—81no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191C4⋊Dic53—81no—
192C4⋊Dic53—81no—
193#11719919—42no—
194#11719919—81no—
195#11719919—81no—
196#11719919—81no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—81no—
200C4×Dic53C2×D412yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C2×D412yes—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53C22×D411yescharacteristic
223D4×C53—22no—
224D4×C53—22no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—84no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
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—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—21no—
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—21no—
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×Dic53—41no—
272C2×Dic53—84no—
273C2×Dic53#12112yes—
274C2×Dic53#12112yes—
275C8×C53—22no—
276C8×C53—41no—
277#88322—22no—
278#88322—41no—
279#24713—531nomaximal, Sylow
280C22×C53—41no—
281C22×C53—42no—
282C22×C53—41no—
283C22×C53—42no—
284C22×C53—41no—
285C22×C53—42no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53C2×#12111yescharacteristic
295C22×C53#52511yescharacteristic
296C22×C53#53611yescharacteristic
297C22×C53#57611yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57811yescharacteristic
301C4×C53—42no—
302C4×C53—41no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53#54711yescharacteristic
306C4×C53#54811yescharacteristic
307C4×C53#55211yescharacteristic
308C4×C53#55611yescharacteristic
309Dic53—22no—
310Dic53—22no—
311Dic53—41no—
312Dic53—41no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317C2×#478—531no—
318C2×#433—531no—
319#2629—531no—
320#2324—531no—
321#2077—531no—
322#2076—531no—
323#2033—531no—
324#2033D5311yescharacteristic
325#1989—531no—
326#1947—531no—
327#1943—531no—
328#1942—531no—
329#1936—531no—
330#1934—531no—
331#1802—531no—
332C2×C53—81no—
333C2×C53—82no—
334C2×C53#241511yescharacteristic
335C2×C53#306111yescharacteristic
336C2×C53#306211yescharacteristic
337C2×C53#312011yescharacteristic
338C2×C53#312211yescharacteristic
339C2×C53#324111yescharacteristic
340C2×C53#324311yescharacteristic
341C2×#125—531no—
342C2×#121C2×D5311yescharacteristic
343C2×C4×D4—531no—
344C2×#117—531no—
345#478—1064no—
346#433—1064no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107C2×D5311yescharacteristic
350C2×#106—531no—
351C2×C4⋊C8—531no—
352C2×C4⋊C8—531no—
353C2×#102—531no—
354C2×#102—531no—
355C2×#102C2×D5311yescharacteristic
356C2×#102C2×D5311yescharacteristic
357C2×C22⋊C8—531no—
358C2×C22⋊C8C2×D5311yescharacteristic
359C2×#97—531no—
360C2×C4×C8—531no—
361#398—1061no—
362#398—1061no—
363#396—1061no—
364#396—1061no—
365#390—531no—
366#390—1061no—
367#389—531no—
368#389—531no—
369#389—531no—
370#388—1061no—
371#386—1061no—
372#385—1061no—
373#384—531no—
374#380—531no—
375#380—531no—
376#380C2×D5311yescharacteristic
377C4×C4⋊C4—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340C2×D5311yescharacteristic
386C53#2471311yescharacteristic, complemented, Sylow
387C22×D4—21no—
388C22×D4C22×D5311yescharacteristic
389C23×C4—1061no—
390C23×C4C22×D5311yescharacteristic
391C22×C8—531no—
392C22×C8—531no—
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395#125—1062no—
396#125—1062no—
397#121—22no—
398#121—22no—
399C4×D4—1062no—
400C4×D4—1062no—
401#117—1062no—
402#117—1062no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—1061no—
431#107—22no—
432#107—1062no—
433#107—1062no—
434#106—1062no—
435C4⋊C8—534no—
436C4⋊C8—1062no—
437#102—22no—
438#102—22no—
439#102—1062no—
440#102—1062no—
441C22⋊C8—22no—
442C22⋊C8—1062no—
443#97—1062no—
444C4×C8—1062no—
445#95—21no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452C24—21no—
453C24D4×D5311yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4#1171994712yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—4241no—
481C22×C4—4241no—
482C22×C4—2122no—
483C22×C4D4×D5311yescharacteristic
484C22×C4C23×D5311yescharacteristic
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8D4×D5312yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—4241no—
513C4⋊C4—4241no—
514C4⋊C4—4241no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C4⋊C4#1171994712yes—
520C22⋊C4—24no—
521C22⋊C4—81no—
522C22⋊C4—1062no—
523C22⋊C4—1062no—
524C22⋊C4—4241no—
525C22⋊C4—4241no—
526C22⋊C4—4241no—
527C22⋊C4—4241no—
528C22⋊C4—2122no—
529C42—532no—
530C42—532no—
531C42—1061no—
532C42—1061no—
533C42—4241no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D5311yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—41no—
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—532no—
571C2×C4—532no—
572C2×C4—1061no—
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—1061no—
582C2×C4—4244no—
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×C4C2×D4×D5311yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4C2×#1171994711yescharacteristic
598C2×C4C2×#1171994711yescharacteristic
599C2×C4#1313326411yescharacteristic
600C2×C4#1313326411yescharacteristic
601C8—22no—
602C8—41no—
603C8—1062no—
604C8—2121no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#121×D5311yescharacteristic
620C22#1681112911yescharacteristic
621C22#1681119511yescharacteristic
622C22#1681142411yescharacteristic
623C22#1681142711yescharacteristic
624C22#1681143011yescharacteristic
625C22#1681143911yescharacteristic
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—1062no—
631C4—1062no—
632C4—4241no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4—2121no—
637C4—2121no—
638C4#1681127311yescharacteristic
639C4#1681127911yescharacteristic
640C4#1681131011yescharacteristic
641C4#1681134211yescharacteristic
642C2—81no—
643C2—82no—
644C2#2027813711yescharacteristic
645C2#2028438311yescharacteristic
646C2#2028439411yescharacteristic
647C2#2028492211yescharacteristic
648C2#2028495211yescharacteristic
649C2#2028596611yescharacteristic
650C2#2028599411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2033C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2033D53
hypocenterC53#24713
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. C2×#16810925C2semidirect
2. C2×#16810681C2semidirect
3. #20280549C2semidirect
5. #20276148C2semidirect
6. #20276146C2semidirect
8. #20275671C2semidirect
9. #20275470C2semidirect
10. #20275454C2semidirect
11. #20275447C2semidirect
12. #20275409C2semidirect
13. #20275400C2semidirect
14. #20274713C2semidirect
97. #13133146D4semidirect
386. C53#24713semidirect

Maximal subgroups

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

subgroupindex
C2×#168109252
C2×#168106812
#202805492
#202774102
#202761482
#202761462
#202758942
#202756712
#202754702
#202754542
#202754472
#202754092
#202754002
#202747132
C53×#20332
#2471353