Finite Group Databasebeta

← #27405899

Subgroups — #27405899

Order 13568.

898 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810912C211yesmaximal, characteristic
2#20277801C211yesmaximal, characteristic, complemented
3#20277616C211yesmaximal, characteristic
4#20275674C211yesmaximal, characteristic, complemented
5#20275475C211yesmaximal, characteristic, complemented
6#20275442C211yesmaximal, characteristic, complemented
7#20275295C211yesmaximal, characteristic, complemented
8#20275290C211yesmaximal, characteristic, complemented
9#20274975C211yesmaximal, characteristic
10#20274973C211yesmaximal, characteristic
11C8×#11719919C211yesmaximal, characteristic, complemented
12C4×#13133139C211yesmaximal, characteristic
13#20274529C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1941C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21#16810912C418yescomplemented
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810528—21no—
32#16810527—21no—
33#16810522C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810518—21no—
36#16810497—21no—
37#16810494—21no—
38#16810492C2211yescharacteristic
39#16810482—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719919C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810474—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
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×D212—22no—
70C4×D212—41no—
71#13133179—22no—
72#13133179—22no—
73#13133179—41no—
74#13133179C2×C414yes—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919D411yescharacteristic
84C2×#11719919D411yescharacteristic
85C4⋊C4×D53—44no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133148—41no—
95#13133148C2×C414yescomplemented
96#13133146C2×C414yescomplemented
97#13133139C2×C414yes—
98#13133139Q814yescomplemented
99#13133137—22no—
100#13133137C2×C414yes—
101C8×Dic53—22no—
102C8×Dic53—41no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#106×C53—41no—
120#106×C53C2×C414yescomplemented
121C4×C8×C53—41no—
122#95×C53—21no—
123#95×C53—21no—
124C23×D53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212C22×C412yes—
143C2×D212C2×Q812yes—
144C2×D212#4112yes—
145C2×C4×D53—24no—
146C2×C4×D53—42no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×Dic106—24no—
155C2×Dic106—81no—
156C2×Dic106—81no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C412yes—
162C2×C8×C53C2×Q812yes—
163C4⋊C4×C53—21no—
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×C53—81no—
171C4⋊C4×C53C22×C412yes—
172C4⋊C4×C53C22×C412yes—
173C42×C53—21no—
174C42×C53—21no—
175#11719921—24no—
176#11719921—42no—
177#11719921—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C2×Q812yes—
185#11719919—24no—
186#11719919—22no—
187#11719919—42no—
188#11719919—81no—
189#11719919—81no—
190#11719919SD1614yes—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C4×Dic53—81no—
196C4×Dic53C22×C412yes—
197C4×Dic53#4112yes—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C22×C412yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
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×C53C2×SD1612yes—
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—21no—
242C2×Dic53—22no—
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×Dic53C4×D412yes—
253C2×Dic53C4×D412yes—
254C2×Dic53C2×SD1612yes—
255C2×Dic53C2×SD1612yes—
256C2×Dic53#13512yes—
257D212—22no—
258D212—22no—
259D212—22no—
260D212C4×Q814yes—
261C4×D53—48no—
262C4×D53—84no—
263C4×D53—84no—
264Dic106—84no—
265C8×C53—22no—
266C8×C53—22no—
267#88322—22no—
268#88322—41no—
269#22963—531nomaximal, Sylow
270C22×C53C2×C4×D411yescharacteristic
271C22×C53#54611yescharacteristic
272C22×C53D4×Q811yescharacteristic
273C22×C53C22×SD1611yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53#57611yescharacteristic
276C22×C53#57711yescharacteristic
277C2×D53—24no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—42no—
287C4×C53—41no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53C2×C4×Q811yescharacteristic
291C4×C53#52011yescharacteristic
292C4×C53#55211yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—22no—
295Dic53—22no—
296Dic53—42no—
297Dic53—42no—
298Dic53—41no—
299Dic53—81no—
300Dic53C4×SD1614yes—
301C2×#476—531no—
302#2369—531no—
303D4×C4⋊C4—531no—
304#1990—531no—
305#1948—531no—
306#1941—531no—
307#1941D5311yescharacteristic, complemented
308#1919—531no—
309#1919—531no—
310#1864—531no—
311#1864—531no—
312C8×C4⋊C4—531no—
313C4×#102—531no—
314C4×#102—531no—
315C4×#102—531no—
316C2×C53#236011yescharacteristic
317C2×C53C2×C4×SD1611yescharacteristic
318C2×C53#294611yescharacteristic
319C2×C53#329511yescharacteristic
320C2×C53#330311yescharacteristic
321C2×C53Q8×SD1611yescharacteristic
322C2×C53#338011yescharacteristic
323D53—44no—
324C2×C4⋊Q8—531no—
325C2×C4×D4—531no—
326C2×C4×D4—531no—
327C2×C4×D4—531no—
328C22×C4⋊C4—1061no—
329#476—538no—
330C2×#106—531no—
331C2×#106—531no—
332C2×#106C2×D5311yescharacteristic
333C2×#106C2×D5311yescharacteristic
334C2×C4⋊C8—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×C4×C8—531no—
340C2×C4×C8—531no—
341C2×C4×C8C2×D5311yescharacteristic
342#391—1061no—
343#391—1061no—
344#389—531no—
345#389—531no—
346#389—1061no—
347#389C2×D5311yescharacteristic
348#389C2×D5311yescharacteristic
349#385—1061no—
350#385—1061no—
351#384—531no—
352#380—1061no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366#336—531no—
367#336—531no—
368C53#2296311yescharacteristic, complemented, Sylow
369C22×Q8—1061no—
370C22×D4—531no—
371C23×C4—1061no—
372C23×C4—1061no—
373C23×C4—1061no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C4⋊Q8—534no—
379C4⋊Q8—1062no—
380C4×D4—534no—
381C4×D4—1062no—
382C4×D4—1062no—
383C4×D4—1062no—
384C4×D4—2121no—
385C4×D4—2121no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—2124no—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42C22×D5311yescharacteristic
412#106—41no—
413#106—534no—
414#106—2121no—
415#106C4×D5314yescomplemented
416C4⋊C8—534no—
417#102—534no—
418#102—534no—
419#102—534no—
420#102—1062no—
421C4×C8—41no—
422C4×C8—1062no—
423C4×C8—2121no—
424#95—21no—
425#95—21no—
426#95—531no—
427#95—531no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24—1061no—
434C2×Q8—1064no—
435C2×Q8—4241no—
436C2×Q8—4241no—
437C2×D4—532no—
438C2×D4—532no—
439C2×D4—532no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1064no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—532no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8C2×C4×D5312yes—
477C2×C8Q8×D5312yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—534no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—1062no—
490C4⋊C4—1064no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4C2×C4×D5312yes—
500C4⋊C4C2×C4×D5312yes—
501C22⋊C4—1064no—
502C22⋊C4—2122no—
503C22⋊C4—2122no—
504C42—21no—
505C42—21no—
506C42—532no—
507C42—532no—
508C42—1061no—
509C42—1061no—
510C42—4241no—
511C42—4241no—
512C42—2122no—
513C23—1062no—
514C23—1062no—
515C23—1062no—
516C23—2121no—
517C23—2121no—
518C23—2121no—
519C23—2121no—
520C23C2×D4×D5311yescharacteristic
521Q8—4244no—
522D4—534no—
523D4—1062no—
524D4—1062no—
525D4—1062no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—4244no—
558C2×C4—4244no—
559C2×C4—2128no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4SD16×D5312yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×Q8×D5311yescharacteristic
570C2×C4#41×D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4#1313326211yescharacteristic
573C2×C4#1313326511yescharacteristic
574C8—22no—
575C8—22no—
576C8—1062no—
577C8—2121no—
578C22—1064no—
579C22—2122no—
580C22—2122no—
581C22—2122no—
582C22—2122no—
583C22—2122no—
584C22—2122no—
585C22C4×D4×D5311yescharacteristic
586C22#1681126211yescharacteristic
587C22#1681128811yescharacteristic
588C22C2×SD16×D5311yescharacteristic
589C22#135×D5311yescharacteristic
590C22#1681142511yescharacteristic
591C22#1681142911yescharacteristic
592C4—22no—
593C4—42no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—534no—
599C4—1062no—
600C4—1062no—
601C4—4241no—
602C4—2122no—
603C4—2122no—
604C4—2121no—
605C4C4×Q8×D5311yescharacteristic
606C4#1681110411yescharacteristic
607C4#1681130511yescharacteristic
608C4#1681132711yescharacteristic
609C2—2124no—
610C2#2027771711yescharacteristic
611C2C4×SD16×D5311yescharacteristic
612C2#2028354211yescharacteristic
613C2#2028653411yescharacteristic
614C2#2028663811yescharacteristic
615C2#2028757211yescharacteristic
616C2#2028757911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1941C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1941D53
hypocenterC53#22963
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. #20277801C2semidirect
4. #20275674C2semidirect
5. #20275475C2semidirect
6. #20275442C2semidirect
7. #20275295C2semidirect
8. #20275290C2semidirect
11. C8×#11719919C2semidirect
15. C53×#1941C2semidirect
21. #16810912C4semidirect
68. #13133216C2×C4semidirect
95. #13133148C2×C4semidirect
96. #13133146C2×C4semidirect
98. #13133139Q8semidirect
120. #106×C53C2×C4semidirect
307. #1941D53semidirect
368. C53#22963semidirect
415. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109122
#202778012
#202776162
#202756742
#202754752
#202754422
#202752952
#202752902
#202749752
#202749732
C8×#117199192
C4×#131331392
#202745292
C53×#19412
#2296353