Finite Group Databasebeta

← #27387113

Subgroups — #27387113

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810720C211yesmaximal, characteristic
2#20277186C211yesmaximal, characteristic, complemented
3C4×D4×Dic53C211yesmaximal, characteristic
4#20275357C211yesmaximal, characteristic, complemented
5#20275348C211yesmaximal, characteristic, complemented
6#20275304C211yesmaximal, characteristic, complemented
7#20275289C211yesmaximal, characteristic, complemented
8#20275049C211yesmaximal, characteristic
9#20275047C211yesmaximal, characteristic
10#20274964C211yesmaximal, characteristic
11#20274615C211yesmaximal, characteristic, complemented
12#20274544C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14#20274462C211yesmaximal, characteristic, complemented
15C53×#1864C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C22×C4×Dic53—21no—
21#16810720C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810510—21no—
34#16810504—21no—
35C4⋊C4×Dic53—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810468—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923C2×C414yes—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133182—22no—
69#13133182C2×C414yescomplemented
70D4×Dic53—22no—
71D4×Dic53—22no—
72D4×Dic53—22no—
73D4×Dic53C2×C414yes—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—44no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133150C2×C414yescomplemented
95#13133142—22no—
96#13133142C2×C414yescomplemented
97#13133140—22no—
98#13133138—22no—
99#13133127—22no—
100#13133127C2×C414yescomplemented
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53—21no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C4×D4×C53—22no—
115C4×D4×C53—22no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#106×C53—22no—
123#102×C53—22no—
124#102×C53C2×C414yes—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C22×C412yes—
131C2×D4×C53#4112yes—
132C2×D4×C53#4112yes—
133C22×C4×C53—24no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—24no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—24no—
147C22×Dic53—21no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C412yes—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53C22×C412yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53C22×C412yes—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—24no—
176C42×C53—21no—
177C42×C53—21no—
178C22⋊Dic53—24no—
179C22⋊Dic53—24no—
180#11719923—28no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53C22×C412yes—
187C4⋊Dic53#4112yes—
188#11719919—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×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C22×D411yescharacteristic
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—22no—
219D4×C53#11714yes—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—48no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53#13712yes—
236C2×C4×C53C23×C411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—48no—
255C2×Dic53—48no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53C4×D412yes—
266C2×Dic53C4×D412yes—
267C2×Dic53AGL(1,ℤ/8)12yes—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—22no—
273C8×C53—41no—
274#88322—22no—
275#88322—41no—
276#21121—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53#52511yescharacteristic
286C22×C53#52511yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57511yescharacteristic
291C4×C53—22no—
292C4×C53—42no—
293C4×C53—41no—
294C4×C53—42no—
295C4×C53—81no—
296C4×C53C2×#11711yescharacteristic
297C4×C53C4×#4111yescharacteristic
298C4×C53#52911yescharacteristic
299C4×C53#52911yescharacteristic
300Dic53—22no—
301Dic53—22no—
302Dic53—22no—
303Dic53—42no—
304Dic53—41no—
305Dic53—42no—
306Dic53—42no—
307Dic53—81no—
308C2×#440—531no—
309#2295—531no—
310C42×D4—531no—
311#1928—531no—
312#1927—531no—
313#1921—531no—
314#1918—531no—
315#1877—531no—
316#1877—531no—
317#1864D5311yescharacteristic
318#1862—531no—
319#1776—531no—
320#1763—531no—
321#1762—531no—
322#1745—531no—
323C2×C53—44no—
324C2×C53#229011yescharacteristic
325C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C53C4×#13711yescharacteristic
327C2×C53#311011yescharacteristic
328C2×C53#311111yescharacteristic
329C2×C53#311411yescharacteristic
330C2×C53#311611yescharacteristic
331C22×SD16—531no—
332C2×C4×Q8—531no—
333C2×C4×D4—531no—
334C2×C4×D4—531no—
335C2×C4×D4C2×D5311yescharacteristic
336C22×C42—1061no—
337#440—538no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#106C2×D5311yescharacteristic
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#102—531no—
345C2×#102C2×D5311yescharacteristic
346C2×#102C2×D5311yescharacteristic
347C2×#97—531no—
348C2×#97—531no—
349#391—531no—
350#391—531no—
351#389—531no—
352#389C2×D5311yescharacteristic
353#387—1061no—
354#387—1061no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—1061no—
360C4×C22⋊C4—1061no—
361C4×C22⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#376—1061no—
364C43—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373#336—531no—
374#336C2×D5311yescharacteristic
375C53#2112111yescharacteristic, complemented, Sylow
376C22×Q8—531no—
377C22×D4C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4—1061no—
380C23×C4—1061no—
381C2×SD16—534no—
382C2×SD16—1062no—
383C2×SD16—1062no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C4×Q8—534no—
389C4×Q8—1062no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—534no—
393C4×D4—1062no—
394C4×D4—1062no—
395C4×D4—1062no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—2124no—
420C2×C42C22×D5311yescharacteristic
421#107—534no—
422#107—1062no—
423#106—22no—
424#103—534no—
425#103—1062no—
426#103—1062no—
427#102—22no—
428#102—1062no—
429#102C4×D5314yes—
430#97—534no—
431#97—1062no—
432#95—21no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24—21no—
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D5312yes—
453C2×D4#1171994712yes—
454C2×D4#1171994912yes—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—532no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8C2×C4×D5312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—532no—
497C4⋊C4—532no—
498C4⋊C4—1064no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—2122no—
504C4⋊C4C2×C4×D5312yes—
505C4⋊C4#1171994712yes—
506C22⋊C4—24no—
507C22⋊C4—1064no—
508C22⋊C4—1064no—
509C42—21no—
510C42—21no—
511C42—532no—
512C42—532no—
513C42—1062no—
514C42—1064no—
515C42—1062no—
516C42—1061no—
517C42—1061no—
518C42—2121no—
519C42—2121no—
520C42—2122no—
521C42—2122no—
522C42—2122no—
523C42—2122no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D5311yescharacteristic
532Q8—1062no—
533Q8—1062no—
534Q8—2121no—
535Q8—2121no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1313317514yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—48no—
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—1062no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—2128no—
571C2×C4—2128no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4#1313324612yes—
582C2×C4C22×C4×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4C2×#1171994911yescharacteristic
588C8—22no—
589C8—41no—
590C8—1062no—
591C8—2121no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D5311yescharacteristic
600C22#1681112911yescharacteristic
601C22#1681113611yescharacteristic
602C22AGL(1,ℤ/8)×D5311yescharacteristic
603C22C2×#1313324611yescharacteristic
604C22#1681141911yescharacteristic
605C22#1681142011yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—1062no—
612C4—1062no—
613C4—1062no—
614C4—4241no—
615C4—2122no—
616C4—2122no—
617C4—2122no—
618C4—2121no—
619C4C2×#1313317511yescharacteristic
620C4C4×#1171994711yescharacteristic
621C4#1681115011yescharacteristic
622C4#1681115911yescharacteristic
623C2—44no—
624C2#2027715111yescharacteristic
625C2#2028352611yescharacteristic
626C2C4×#1313324611yescharacteristic
627C2#2028480911yescharacteristic
628C2#2028481711yescharacteristic
629C2#2028485811yescharacteristic
630C2#2028487911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1864C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1864D53
hypocenterC53#21121
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #20277186C2semidirect
4. #20275357C2semidirect
5. #20275348C2semidirect
6. #20275304C2semidirect
7. #20275289C2semidirect
11. #20274615C2semidirect
12. #20274544C2semidirect
14. #20274462C2semidirect
21. #16810720C4semidirect
69. #13133182C2×C4semidirect
94. #13133150C2×C4semidirect
96. #13133142C2×C4semidirect
100. #13133127C2×C4semidirect
375. C53#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#168107202
#202771862
C4×D4×Dic532
#202753572
#202753482
#202753042
#202752892
#202750492
#202750472
#202749642
#202746152
#202745442
#202745422
#202744622
C53×#18642
#2112153