Finite Group Databasebeta

← #27387158

Subgroups — #27387158

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810726C211yesmaximal, characteristic
2#20277173C211yesmaximal, characteristic
3C4×Q8×Dic53C211yesmaximal, characteristic, complemented
4#20275343C211yesmaximal, characteristic
5#20275334C211yesmaximal, characteristic
6#20275304C211yesmaximal, characteristic, complemented
7#20275289C211yesmaximal, characteristic, complemented
8#20275070C211yesmaximal, characteristic
9#20275068C211yesmaximal, characteristic
10#20274985C211yesmaximal, characteristic, complemented
11#20274614C211yesmaximal, characteristic, complemented
12#20274547C211yesmaximal, characteristic, complemented
13#20274539C211yesmaximal, characteristic
14#20274462C211yesmaximal, characteristic, complemented
15C53×#1863C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133175—21no—
21#16810726C418yescomplemented
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30#16810525C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810515—21no—
34#16810513—21no—
35C4⋊C4×Dic53—21no—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53—21no—
41C4×C4⋊Dic53C2211yescharacteristic
42#16810470—21no—
43C42×Dic53—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C2×#11719924C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69Q8×Dic53—22no—
70Q8×Dic53C2×C414yes—
71#13133179—22no—
72#13133179C2×C414yes—
73#13133175—44no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133150C2×C414yes—
98#13133149—22no—
99#13133141—22no—
100#13133139—22no—
101#13133139C2×C414yes—
102#13133137—22no—
103#13133127—22no—
104#13133127C2×C414yes—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×Q8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×Q8×C53—22no—
114C4×Q8×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#107×C53—22no—
123#103×C53—22no—
124#103×C53C2×C414yes—
125#95×C53—21no—
126C23×D53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C22×C412yes—
131C2×Q8×C53#4112yes—
132C2×Q8×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212C22×C412yes—
153C2×C4×D53—24no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C412yes—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53C22×C412yes—
169C4⋊C4×C53#4112yes—
170C42×C53—24no—
171C42×C53—21no—
172C42×C53—21no—
173#11719924—28no—
174#11719921—24no—
175#11719921—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179C4⋊Dic53—24no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—24no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53C22×C412yes—
187C4⋊Dic53#4112yes—
188C4×Dic53—24no—
189C4×Dic53—22no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—24no—
193C4×Dic53—22no—
194C4×Dic53—41no—
195C4×Dic53—41no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53C22×C412yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C412yes—
208C23×C53C22×D411yescharacteristic
209C22×D53—22no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53#11714yescomplemented
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53#13712yes—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53C4×D412yes—
263C2×Dic53C4×D412yes—
264C2×Dic53AGL(1,ℤ/8)12yes—
265D212—22no—
266D212—22no—
267D212—41no—
268D212—41no—
269C4×D53—48no—
270C8×C53—22no—
271C8×C53—41no—
272#88322—22no—
273#88322—41no—
274#21124—531nomaximal, Sylow
275C22×C53C2×C4×D411yescharacteristic
276C22×C53#52511yescharacteristic
277C22×C53#52511yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53C2×#13711yescharacteristic
280C22×C53#57511yescharacteristic
281C22×C53#57511yescharacteristic
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—81no—
286C2×D53—81no—
287C2×D53—81no—
288C2×D53—81no—
289C4×C53—22no—
290C4×C53—24no—
291C4×C53—41no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—81no—
295C4×C53C2×#11711yescharacteristic
296C4×C53C4×#4111yescharacteristic
297C4×C53#52911yescharacteristic
298C4×C53#52911yescharacteristic
299Dic53—22no—
300Dic53—22no—
301Dic53—22no—
302Dic53—41no—
303Dic53—42no—
304Dic53—42no—
305Dic53—81no—
306C2×#440—531no—
307#2293—531no—
308C42×Q8—531no—
309#1926—531no—
310#1925—531no—
311#1921—531no—
312#1918—531no—
313#1880—531no—
314#1880—531no—
315#1865—531no—
316#1863D5311yescharacteristic, complemented
317#1776—531no—
318#1763—531no—
319#1762—531no—
320#1745—531no—
321C2×C53#229011yescharacteristic
322C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C53C4×#13711yescharacteristic
324C2×C53#311111yescharacteristic
325C2×C53#311211yescharacteristic
326C2×C53#311511yescharacteristic
327C2×C53#311711yescharacteristic
328D53—82no—
329C22×SD16—531no—
330C2×C4×Q8—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4—531no—
334C2×#117—1061no—
335#440—538no—
336C2×#107—531no—
337C2×#107C2×D5311yescharacteristic
338C2×#106—531no—
339C2×#103—531no—
340C2×#103C2×D5311yescharacteristic
341C2×#103C2×D5311yescharacteristic
342C2×#102—531no—
343C2×#102—531no—
344C2×#102—531no—
345C2×#97—531no—
346C2×#97—531no—
347#390—531no—
348#390—531no—
349#389—531no—
350#389C2×D5311yescharacteristic
351#388—1061no—
352#388—1061no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C4⋊C4—1061no—
359C4×C4⋊C4—1061no—
360C4×C22⋊C4—1061no—
361C43—531no—
362C43—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371#336—531no—
372#336C2×D5311yescharacteristic
373C53#2112411yescharacteristic, complemented, Sylow
374C22×Q8C22×D5311yescharacteristic
375C22×D4—531no—
376C23×C4—1061no—
377C2×SD16—534no—
378C2×SD16—1062no—
379C2×SD16—1062no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—534no—
387C4×Q8—1062no—
388C4×Q8—1062no—
389C4×Q8—1062no—
390C4×D4—534no—
391C4×D4—1062no—
392#117—2124no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C42—21no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423#107—22no—
424#107—534no—
425#106—1062no—
426#103—22no—
427#103—1062no—
428#103C4×D5314yes—
429#102—534no—
430#102—1062no—
431#102—1062no—
432#97—534no—
433#97—1062no—
434#95—21no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—1061no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D5312yes—
445C2×Q8#1171994712yes—
446C2×Q8#1171994912yes—
447C2×D4—532no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—1064no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470SD16—1068no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8C2×C4×D5312yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—532no—
490C4⋊C4—532no—
491C4⋊C4—1064no—
492C4⋊C4—1064no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4C2×C4×D5312yes—
499C4⋊C4#1171994712yes—
500C22⋊C4—1064no—
501C22⋊C4—2122no—
502C22⋊C4—2122no—
503C22⋊C4—2122no—
504C22⋊C4—2122no—
505C42—24no—
506C42—21no—
507C42—21no—
508C42—532no—
509C42—532no—
510C42—1062no—
511C42—1064no—
512C42—1064no—
513C42—1062no—
514C42—1061no—
515C42—1061no—
516C42—2121no—
517C42—2121no—
518C42—2122no—
519C42—2122no—
520C42—2122no—
521C42—2122no—
522C23—1062no—
523C23—2121no—
524C23—2121no—
525C23—2121no—
526C23—2121no—
527C23—2121no—
528C23—2121no—
529C23C2×D4×D5311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1313317514yescomplemented
534D4—1062no—
535D4—1062no—
536D4—2121no—
537D4—2121no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—2128no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4#1313324912yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4C2×#1171994911yescharacteristic
584C8—22no—
585C8—41no—
586C8—1062no—
587C8—2121no—
588C22—4241no—
589C22—4241no—
590C22—4241no—
591C22—4241no—
592C22—2122no—
593C22—2122no—
594C22—2122no—
595C22C4×D4×D5311yescharacteristic
596C22#1681112911yescharacteristic
597C22#1681113611yescharacteristic
598C22AGL(1,ℤ/8)×D5311yescharacteristic
599C22C2×#1313324911yescharacteristic
600C22#1681142011yescharacteristic
601C22#1681142211yescharacteristic
602C4—24no—
603C4—22no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—81no—
608C4—1062no—
609C4—1062no—
610C4—1062no—
611C4—4241no—
612C4—2122no—
613C4—2121no—
614C4—2122no—
615C4C2×#1313317511yescharacteristic
616C4C4×#1171994711yescharacteristic
617C4#1681115011yescharacteristic
618C4#1681115911yescharacteristic
619C2—4242no—
620C2#2027715111yescharacteristic
621C2#2028352011yescharacteristic
622C2C4×#1313324911yescharacteristic
623C2#2028483111yescharacteristic
624C2#2028483911yescharacteristic
625C2#2028485911yescharacteristic
626C2#2028488711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1863C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1863D53
hypocenterC53#21124
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. C4×Q8×Dic53C2semidirect
6. #20275304C2semidirect
7. #20275289C2semidirect
10. #20274985C2semidirect
11. #20274614C2semidirect
12. #20274547C2semidirect
14. #20274462C2semidirect
15. C53×#1863C2semidirect
21. #16810726C4semidirect
219. Q8×C53#117semidirect
316. #1863D53semidirect
373. C53#21124semidirect
533. Q8#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×#168107262
#202771732
C4×Q8×Dic532
#202753432
#202753342
#202753042
#202752892
#202750702
#202750682
#202749852
#202746142
#202745472
#202745392
#202744622
C53×#18632
#2112453