Finite Group Databasebeta

← #27429813

Subgroups — #27429813

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810862C211yesmaximal, characteristic
2C2×C4×#11719925C211yesmaximal, characteristic, complemented
3#20278218C211yesmaximal, characteristic
4#20276134C211yesmaximal, characteristic, complemented
5#20276133C211yesmaximal, characteristic, complemented
6#20275911C211yesmaximal, characteristic
7#20275908C211yesmaximal, characteristic
8#20275575C211yesmaximal, characteristic, complemented
9#20275445C211yesmaximal, characteristic, complemented
10#20274983C211yesmaximal, characteristic, complemented
11#20274982C211yesmaximal, characteristic, complemented
12#20274877C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic
14C22⋊C4×#88322C211yesmaximal, characteristic
15C53×#2421C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810862—24no—
21C4×#11719925—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810564—21no—
37#16810551—21no—
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810517—21no—
41#16810505—21no—
42#16810489C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#397—21no—
54C53×#391C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#387—21no—
58C53×#382C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719925—22no—
63C2×#11719925—22no—
64C2×#11719925D414yes—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322—22no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133196—22no—
73#13133196—41no—
74C4×Dic106—22no—
75C4×Dic106—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85#13133150—22no—
86#13133150—22no—
87#13133150—22no—
88#13133148—41no—
89#13133148—41no—
90#13133141—22no—
91#13133141—22no—
92#13133141—22no—
93#13133140—22no—
94#13133140—22no—
95#13133140—22no—
96C22⋊#88322—22no—
97C4×#88322—22no—
98C4×#88322—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53—41no—
109C4×Q8×C53—22no—
110C4×Q8×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
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×C53C2×D412yes—
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×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106C2×D412yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185#11719925—28no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192#11719919—42no—
193#11719919—42no—
194#11719919—42no—
195C4×Dic53—42no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—44no—
205C2×#88322C2×D412yes—
206C2×#88322#4112yes—
207C2×#88322#4112yes—
208C2×#88322#4112yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53#13512yes—
216C23×C53C22×D411yescharacteristic
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—41no—
220Q8×C53—41no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
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×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53#13512yes—
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×Dic53—22no—
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—
270Dic106—22no—
271Dic106—22no—
272Dic106—41no—
273Dic106—41no—
274#88322—22no—
275#88322—41no—
276#88322—41no—
277#88322#12314yes—
278#25033—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53#53211yescharacteristic
287C22×C53D4211yescharacteristic
288C22×C53#54711yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53#57811yescharacteristic
292C22×C53#57811yescharacteristic
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53C2×#12311yescharacteristic
303C4×C53#54211yescharacteristic
304C4×C53#54711yescharacteristic
305C4×C53#55311yescharacteristic
306Dic53—42no—
307Dic53—81no—
308Dic53—81no—
309Dic53—81no—
310C2×#467—531no—
311C2×C4×Q16—531no—
312#2421—531no—
313#2421D5311yescharacteristic
314#2074—531no—
315#2074—531no—
316#2035—531no—
317#2035—531no—
318#1969—531no—
319#1942—531no—
320#1865—531no—
321#1865—531no—
322#1846—531no—
323#1808—531no—
324C8×C22⋊C4—531no—
325C2×C53—42no—
326C2×C53#242211yescharacteristic
327C2×C53#316811yescharacteristic
328C2×C53#317711yescharacteristic
329C2×C53#328511yescharacteristic
330C2×C53#328511yescharacteristic
331C2×C53#333611yescharacteristic
332C2×C53#334211yescharacteristic
333C22×Q16—531no—
334C23×C8—531no—
335C2×C22⋊Q8—531no—
336C2×C22⋊Q8C2×D5311yescharacteristic
337C2×C4×Q8—531no—
338C2×C4×Q8C2×D5311yescharacteristic
339#467—1064no—
340C4×Q16—1064no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#107—531no—
344C2×#106—531no—
345C2×#106—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×C22⋊C8—531no—
353C2×C4×C8—531no—
354C2×C4×C8—531no—
355#400—21no—
356#400—1061no—
357#397—21no—
358#397—1061no—
359#391—531no—
360#391C2×D5311yescharacteristic
361#389—531no—
362#389C2×D5311yescharacteristic
363#388—21no—
364#388—1061no—
365#387—21no—
366#387—1061no—
367#382—531no—
368#382C2×D5311yescharacteristic
369#380—531no—
370#380C2×D5311yescharacteristic
371C4×C22⋊C4C2×D5311yescharacteristic
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#336—531no—
377#336—531no—
378C53#2503311yescharacteristic, complemented, Sylow
379C22×Q8—531no—
380C22×Q8C22×D5311yescharacteristic
381C23×C4C22×D5311yescharacteristic
382C2×Q16—534no—
383C2×Q16—1062no—
384C2×Q16—1062no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—1062no—
390C22×C8—1062no—
391C22×C8—1062no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1062no—
395C22⋊Q8—2121no—
396C4×Q8—22no—
397C4×Q8—41no—
398C4×Q8—1062no—
399C4×Q8—2121no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4C22⋊D5311yescharacteristic
417C2×C22⋊C4C22⋊D5311yescharacteristic
418C2×C42—21no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#107—1062no—
423#107—1062no—
424#107—1062no—
425#106—2121no—
426#106—2121no—
427#103—1062no—
428#103—1062no—
429#103—1062no—
430#103—1062no—
431#103—1062no—
432#103—1062no—
433C22⋊C8—1062no—
434C4×C8—1062no—
435C4×C8—1062no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95C22⋊D5311yescharacteristic
449#95C22⋊D5311yescharacteristic
450C24C2×C22⋊D5311yescharacteristic
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—532no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8D4×D5312yes—
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—21no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C23×D5311yescharacteristic
483Q16—1068no—
484C2×C8—532no—
485C2×C8—532no—
486C2×C8—532no—
487C2×C8—532no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—2124no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23#1313323712yes—
534C23C22×C22⋊D5311yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—1062no—
540Q8—1062no—
541Q8—2121no—
542Q8—2121no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
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—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1062no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4#1313323712yes—
586C2×C4C2×D4×D5311yescharacteristic
587C2×C4#41×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#1313326211yescharacteristic
591C2×C4#1313326411yescharacteristic
592C8—534no—
593C8—1062no—
594C8—2121no—
595C8—2121no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22#1681116911yescharacteristic
604C22D4×C22⋊D5311yescharacteristic
605C22#1681127211yescharacteristic
606C22C2×#1313323711yescharacteristic
607C22C2×#1313323711yescharacteristic
608C22#1681143811yescharacteristic
609C22#1681143811yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4—2122no—
623C4#123×D5311yescharacteristic
624C4#1681124111yescharacteristic
625C4#1681127111yescharacteristic
626C4#1681131811yescharacteristic
627C2—42no—
628C2#2027822611yescharacteristic
629C2#2028527911yescharacteristic
630C2#2028536811yescharacteristic
631C2#2028645211yescharacteristic
632C2#2028645211yescharacteristic
633C2#2028707711yescharacteristic
634C2#2028714111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2421C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2421D53
hypocenterC53#25033
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2421

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719925C2semidirect
4. #20276134C2semidirect
5. #20276133C2semidirect
8. #20275575C2semidirect
9. #20275445C2semidirect
10. #20274983C2semidirect
11. #20274982C2semidirect
12. #20274877C2semidirect
378. C53#25033semidirect

Maximal subgroups

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

subgroupindex
C2×#168108622
C2×C4×#117199252
#202782182
#202761342
#202761332
#202759112
#202759082
#202755752
#202754452
#202749832
#202749822
#202748772
#202747422
C22⋊C4×#883222
C53×#24212
#2503353