Finite Group Databasebeta

← #27405894

Subgroups — #27405894

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810902C211yesmaximal, characteristic
2#20277774C211yesmaximal, characteristic, complemented
3#20277616C211yesmaximal, characteristic
4#20275674C211yesmaximal, characteristic, complemented
5#20275472C211yesmaximal, characteristic, complemented
6#20275449C211yesmaximal, characteristic, complemented
7#20275445C211yesmaximal, characteristic, complemented
8#20275289C211yesmaximal, characteristic, complemented
9#20274965C211yesmaximal, characteristic
10#20274963C211yesmaximal, characteristic
11#20274576C211yesmaximal, characteristic, complemented
12C4×#13133137C211yesmaximal, characteristic
13#20274529C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1918C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×C4×Dic106—21no—
18C2×C4×D212C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×C4⋊C4×D53—21no—
22#16810902C418yescomplemented
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33C2×C8×Dic53C2211yescharacteristic
34#16810527—21no—
35#16810522C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810510—21no—
39#16810507—21no—
40#16810497—21no—
41#16810494—21no—
42#16810492C2211yescharacteristic
43#16810482—21no—
44C4⋊C4×Dic53C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C4×#11719919C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×Dic106—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216C2×C414yescomplemented
69C4×Dic106—44no—
70C4×D212—22no—
71C4×D212—41no—
72#13133179—22no—
73#13133179—22no—
74#13133179—41no—
75#13133179C2×C414yes—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919D411yescharacteristic
86C2×#11719919D411yescharacteristic
87C4⋊C4×D53—44no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133151—41no—
96#13133150—41no—
97#13133150C2×C414yescomplemented
98#13133146C2×C414yescomplemented
99#13133139—22no—
100#13133139C2×C414yes—
101#13133137C2×C414yes—
102#13133137Q814yescomplemented
103C4×#88322—41no—
104C8×Dic53—22no—
105C8×Dic53—41no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124D411yescharacteristic
112#13133124D411yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#107×C53C2×C414yescomplemented
123#95×C53—21no—
124#95×C53—21no—
125C23×D53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212C22×C412yes—
144C2×D212C2×Q812yes—
145C2×D212#4112yes—
146C2×C4×D53—24no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—81no—
154C2×C4×D53—81no—
155C2×Dic106—24no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53C22×C412yes—
171C4⋊C4×C53C22×C412yes—
172C4⋊C4×C53C2×Q812yes—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176#11719921—24no—
177#11719921—42no—
178#11719921—42no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53C22×C412yes—
187#11719919—24no—
188#11719919—22no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192#11719919D814yes—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53C22×C412yes—
198C4×Dic53#4112yes—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C22×C412yes—
204C2×#88322C2×Q812yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53C2×D812yes—
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53C4×D412yes—
254C2×Dic53C4×D412yes—
255C2×Dic53C2×D812yes—
256C2×Dic53C2×D812yes—
257C2×Dic53#13512yes—
258D212—22no—
259D212—22no—
260D212—22no—
261D212C4×Q814yescomplemented
262C4×D53—48no—
263C4×D53—84no—
264C4×D53—84no—
265Dic106—48no—
266C8×C53—22no—
267C8×C53—41no—
268#88322—22no—
269#88322—22no—
270#22962—531nomaximal, Sylow
271C22×C53C2×C4×D411yescharacteristic
272C22×C53#54611yescharacteristic
273C22×C53D4×Q811yescharacteristic
274C22×C53C22×D811yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57811yescharacteristic
278C2×D53—24no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—42no—
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—81no—
291C4×C53C2×C4×Q811yescharacteristic
292C4×C53#52011yescharacteristic
293C4×C53#55211yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—22no—
296Dic53—22no—
297Dic53—42no—
298Dic53—41no—
299Dic53—81no—
300Dic53—81no—
301Dic53C4×D814yes—
302C2×#474—531no—
303#2365—531no—
304D4×C4⋊C4—531no—
305#1990—531no—
306#1947—531no—
307#1942—531no—
308#1942—531no—
309#1918—531no—
310#1918D5311yescharacteristic, complemented
311#1862—531no—
312#1862—531no—
313C8×C4⋊C4—531no—
314C4×#102—531no—
315C4×#102—531no—
316C4×#102—531no—
317C2×C53#236011yescharacteristic
318C2×C53C2×C4×D811yescharacteristic
319C2×C53#294711yescharacteristic
320C2×C53#329411yescharacteristic
321C2×C53#330211yescharacteristic
322C2×C53Q8×D811yescharacteristic
323C2×C53#338111yescharacteristic
324D53—44no—
325C2×C4⋊Q8—531no—
326C2×C4×Q8—1061no—
327C2×C4×D4—531no—
328C2×C4×D4—531no—
329C2×C4×D4—531no—
330C22×C4⋊C4—1061no—
331#474—538no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107C2×D5311yescharacteristic
336C2×C4⋊C8—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×#102—531no—
340C2×#102—531no—
341C2×C4×C8—531no—
342C2×C4×C8—531no—
343C2×C4×C8—531no—
344#391—1061no—
345#389—531no—
346#389—531no—
347#389—531no—
348#389C2×D5311yescharacteristic
349#387—1061no—
350#387—1061no—
351#385—1061no—
352#385—1061no—
353#384—531no—
354#380—1061no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4C2×D5311yescharacteristic
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366#340C2×D5311yescharacteristic
367#336—531no—
368#336C2×D5311yescharacteristic
369C53#2296211yescharacteristic, complemented, Sylow
370C22×Q8—1061no—
371C22×D4—531no—
372C23×C4—1061no—
373C23×C4—1061no—
374C23×C4—1061no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8C22×D5311yescharacteristic
378C22×C8C22×D5311yescharacteristic
379C4⋊Q8—534no—
380C4⋊Q8—1062no—
381C4×Q8—2124no—
382C4×D4—534no—
383C4×D4—1062no—
384C4×D4—1062no—
385C4×D4—1062no—
386C4×D4—2121no—
387C4×D4—2121no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—2124no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42C22×D5311yescharacteristic
414#107—534no—
415#107—2121no—
416#107—2121no—
417#107C4×D5314yescomplemented
418C4⋊C8—534no—
419#102—534no—
420#102—534no—
421#102—534no—
422#102—1062no—
423C4×C8—1062no—
424C4×C8—2121no—
425C4×C8—2121no—
426#95—21no—
427#95—21no—
428#95—531no—
429#95—531no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435C24—1061no—
436C2×Q8—1064no—
437C2×Q8—2122no—
438C2×Q8—2122no—
439C2×D4—532no—
440C2×D4—532no—
441C2×D4—532no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1064no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—4241no—
458C22×C4—4241no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4C23×D5311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—532no—
474C2×C8—532no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8C2×C4×D5312yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—534no—
487C4⋊C4—532no—
488C4⋊C4—1062no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4C2×C4×D5312yes—
501C4⋊C4C2×C4×D5312yes—
502C4⋊C4Q8×D5312yes—
503C22⋊C4—1064no—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—532no—
510C42—532no—
511C42—1061no—
512C42—1061no—
513C42—2122no—
514C42—2122no—
515C23—1062no—
516C23—1062no—
517C23—1062no—
518C23—2121no—
519C23—2121no—
520C23—2121no—
521C23—2121no—
522C23C2×D4×D5311yescharacteristic
523Q8—2128no—
524D4—534no—
525D4—1062no—
526D4—1062no—
527D4—1062no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1062no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—4244no—
559C2×C4—4244no—
560C2×C4—2128no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4D8×D5312yes—
570C2×C4C22×C4×D5311yescharacteristic
571C2×C4C2×Q8×D5311yescharacteristic
572C2×C4#41×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#1313326311yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—22no—
577C8—41no—
578C8—1062no—
579C8—1062no—
580C22—1064no—
581C22—2122no—
582C22—2122no—
583C22—2122no—
584C22—2122no—
585C22—2122no—
586C22—2122no—
587C22C4×D4×D5311yescharacteristic
588C22#1681126511yescharacteristic
589C22#1681128611yescharacteristic
590C22C2×D8×D5311yescharacteristic
591C22#135×D5311yescharacteristic
592C22#1681143211yescharacteristic
593C22#1681143911yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—81no—
600C4—534no—
601C4—1062no—
602C4—1062no—
603C4—4241no—
604C4—4241no—
605C4—2122no—
606C4—2121no—
607C4C4×Q8×D5311yescharacteristic
608C4#1681110311yescharacteristic
609C4#1681130911yescharacteristic
610C4#1681132911yescharacteristic
611C2—2124no—
612C2#2027772311yescharacteristic
613C2C4×D8×D5311yescharacteristic
614C2#2028355711yescharacteristic
615C2#2028652611yescharacteristic
616C2#2028663011yescharacteristic
617C2#2028756511yescharacteristic
618C2#2028759311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1918C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1918D53
hypocenterC53#22962
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #20277774C2semidirect
4. #20275674C2semidirect
5. #20275472C2semidirect
6. #20275449C2semidirect
7. #20275445C2semidirect
8. #20275289C2semidirect
11. #20274576C2semidirect
15. C53×#1918C2semidirect
22. #16810902C4semidirect
68. #13133216C2×C4semidirect
97. #13133150C2×C4semidirect
98. #13133146C2×C4semidirect
102. #13133137Q8semidirect
122. #107×C53C2×C4semidirect
261. D212C4×Q8semidirect
310. #1918D53semidirect
369. C53#22962semidirect
417. #107C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109022
#202777742
#202776162
#202756742
#202754722
#202754492
#202754452
#202752892
#202749652
#202749632
#202745762
C4×#131331372
#202745292
C53×#19182
#2296253