Finite Group Databasebeta

← #27391314

Subgroups — #27391314

Order 13568.

960 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810948C211yesmaximal, characteristic
2#20277221C211yesmaximal, characteristic
3#117×Dic53C211yesmaximal, characteristic, complemented
4#20275341C211yesmaximal, characteristic, complemented
5#20275334C211yesmaximal, characteristic, complemented
6#20275292C211yesmaximal, characteristic, complemented
7#20275285C211yesmaximal, characteristic, complemented
8#20275037C211yesmaximal, characteristic
9#20274807C211yesmaximal, characteristic, complemented
10#20274615C211yesmaximal, characteristic, complemented
11#20274614C211yesmaximal, characteristic, complemented
12#20274539C211yesmaximal, characteristic, complemented
13#20274539C211yesmaximal, characteristic, complemented
14#20274479C211yesmaximal, characteristic, complemented
15C53×#1854C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×#13133179C2211yescharacteristic
18C2×#13133177—21no—
19C2×#13133176C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21#16810948C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810512—21no—
34#16810500—21no—
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42#16810470—21no—
43#16810469—21no—
44C42×Dic53—21no—
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C23×Dic53D411yescharacteristic
63C23×Dic53D411yescharacteristic
64C22×C4×D53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133191—22no—
68#13133191C2×C414yes—
69#13133179—22no—
70#13133179—22no—
71#13133177—44no—
72#13133176—22no—
73#13133176C2×C414yescomplemented
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—22no—
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×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133151C2×C414yescomplemented
96#13133149C2×C414yescomplemented
97#13133139C2×C414yescomplemented
98#13133139C2×C414yescomplemented
99#13133137—22no—
100#13133137—22no—
101C22⋊#88322—22no—
102#13133127—22no—
103#13133127—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113#117×C53—22no—
114#117×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C42×C53—21no—
122#107×C53—22no—
123#106×C53—22no—
124C22⋊C8×C53C2×C414yescomplemented
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—24no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—41no—
150C22×Dic53—41no—
151C22×Dic53—41no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212C22×C412yes—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53C22×C412yes—
170C2×C8×C53C22×C412yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53#4112yes—
178C4⋊C4×C53#4112yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—42no—
181C42×C53—24no—
182C22⋊Dic53—24no—
183#11719921—24no—
184#11719921—42no—
185#11719921—42no—
186#11719921—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C412yes—
191C4⋊Dic53C22×C412yes—
192C4⋊Dic53C22×C412yes—
193#11719919—42no—
194C4×Dic53—24no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—22no—
198C4×Dic53—22no—
199C4×Dic53—42no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53#4112yes—
205C4×Dic53#4112yes—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53C4×D412yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—22no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
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×C53C4×D412yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C22⋊D53—48no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—44no—
266C2×Dic53—44no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53#12312yes—
274C2×Dic53#12312yes—
275C2×Dic53AGL(1,ℤ/8)12yes—
276D212—22no—
277D212—22no—
278D212—41no—
279D212—41no—
280C4×D53—84no—
281C8×C53—22no—
282C8×C53—22no—
283#88322—41no—
284#88322—41no—
285#21582—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×C4×D411yescharacteristic
294C22×C53C2×#12311yescharacteristic
295C22×C53#52511yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53#57511yescharacteristic
298C22×C53#57511yescharacteristic
299C22×C53#57511yescharacteristic
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53C4×#4111yescharacteristic
313C4×C53C4×#4111yescharacteristic
314C4×C53#52911yescharacteristic
315C4×C53#52911yescharacteristic
316Dic53—22no—
317Dic53—22no—
318Dic53—22no—
319Dic53—41no—
320Dic53—42no—
321Dic53—42no—
322Dic53—42no—
323C2×#482—531no—
324C4×#121—531no—
325C4×#117—531no—
326#1926—531no—
327#1925—531no—
328#1919—531no—
329#1918—531no—
330#1875—531no—
331#1854D5311yescharacteristic, complemented
332#1825—531no—
333#1776—531no—
334#1776—531no—
335#1762—531no—
336#1762—531no—
337#1749—531no—
338C2×C53—42no—
339C2×C53C4×#12311yescharacteristic
340C2×C53#294311yescharacteristic
341C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
342C2×C53#309311yescharacteristic
343C2×C53#309411yescharacteristic
344C2×C53#311111yescharacteristic
345C2×C53#311511yescharacteristic
346D53—82no—
347C2×#121—531no—
348C2×C4×D4—531no—
349C2×C4×D4—1061no—
350C2×#117—531no—
351C2×#117C2×D5311yescharacteristic
352C22×C42—531no—
353#482—538no—
354C2×#107—531no—
355C2×#107C2×D5311yescharacteristic
356C2×#106—531no—
357C2×#106C2×D5311yescharacteristic
358C2×#102—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C22⋊C8—531no—
363C2×C22⋊C8C2×D5311yescharacteristic
364C2×#97—531no—
365C2×#97—531no—
366#390—531no—
367#389—531no—
368#388—1061no—
369#386—1061no—
370#380C2×D5311yescharacteristic
371C4×C4⋊C4—531no—
372C4×C4⋊C4—531no—
373C4×C4⋊C4—531no—
374C4×C4⋊C4—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—1061no—
377C4×C22⋊C4—1061no—
378C4×C22⋊C4—1061no—
379#376—1061no—
380C43—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340C2×D5311yescharacteristic
388#340C2×D5311yescharacteristic
389#336—531no—
390#336—531no—
391C53#2158211yescharacteristic, complemented, Sylow
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—1061no—
397C23×C4C22×D5311yescharacteristic
398C22×C8—531no—
399C22×C8—531no—
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402#121—534no—
403#121—1062no—
404C4×D4—1062no—
405C4×D4—1062no—
406C4×D4—2124no—
407#117—22no—
408#117—22no—
409#117—534no—
410#117—1062no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C42—21no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—1062no—
432C2×C42—1062no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42—1061no—
439#107—22no—
440#107—534no—
441#106—22no—
442#106—534no—
443#102—534no—
444#102—534no—
445#102—1062no—
446#102—1062no—
447C22⋊C8—1062no—
448C22⋊C8C4×D5314yescomplemented
449#97—1062no—
450#97—1062no—
451#95—21no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458C24—1061no—
459C24D4×D5311yescharacteristic
460C2×D4—532no—
461C2×D4—1064no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—2122no—
468C2×D4—2122no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4—2121no—
491C22×C4—2121no—
492C22×C4—2121no—
493C22×C4C2×C4×D5312yes—
494C22×C4D4×D5311yescharacteristic
495C22×C4#1171994912yes—
496C22×C4C23×D5311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8C2×C4×D5312yes—
508C2×C8C2×C4×D5312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—532no—
516C4⋊C4—532no—
517C4⋊C4—532no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—2122no—
522C4⋊C4#1171994712yes—
523C4⋊C4#1171994712yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—1064no—
527C22⋊C4—1064no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C42—24no—
532C42—532no—
533C42—532no—
534C42—1062no—
535C42—1062no—
536C42—1064no—
537C42—1061no—
538C42—1061no—
539C42—2122no—
540C42—2122no—
541C42—2122no—
542C42—2122no—
543C42—2122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—1062no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23#1313317912yes—
558C23C2×D4×D5311yescharacteristic
559D4—1062no—
560D4—1062no—
561D4—2128no—
562D4—2121no—
563D4—2121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—4244no—
598C2×C4—2124no—
599C2×C4—2124no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4#1313317912yes—
607C2×C4C22×C4×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4#41×D5311yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4C2×#1171994711yescharacteristic
612C2×C4C2×#1171994911yescharacteristic
613C8—22no—
614C8—22no—
615C8—2121no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—4241no—
625C22—4241no—
626C22—4241no—
627C22—4241no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22C2×#1313317911yescharacteristic
632C22#123×D5311yescharacteristic
633C22#1681112911yescharacteristic
634C22AGL(1,ℤ/8)×D5311yescharacteristic
635C22#1681142011yescharacteristic
636C22#1681142011yescharacteristic
637C22#1681142211yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—1062no—
644C4—1062no—
645C4—1062no—
646C4—2122no—
647C4—2122no—
648C4—2122no—
649C4—2121no—
650C4C4×#1171994711yescharacteristic
651C4C4×#1171994711yescharacteristic
652C4#1681115911yescharacteristic
653C4#1681115911yescharacteristic
654C2—42no—
655C2—4242no—
656C2#2027723411yescharacteristic
657C2#2028351511yescharacteristic
658C2#2028352211yescharacteristic
659C2#2028464711yescharacteristic
660C2#2028465411yescharacteristic
661C2#2028482511yescharacteristic
662C2#2028485911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1854D53
hypocenterC53#21582
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #117×Dic53C2semidirect
4. #20275341C2semidirect
5. #20275334C2semidirect
6. #20275292C2semidirect
7. #20275285C2semidirect
9. #20274807C2semidirect
10. #20274615C2semidirect
11. #20274614C2semidirect
12. #20274539C2semidirect
13. #20274539C2semidirect
14. #20274479C2semidirect
15. C53×#1854C2semidirect
21. #16810948C4semidirect
73. #13133176C2×C4semidirect
95. #13133151C2×C4semidirect
96. #13133149C2×C4semidirect
97. #13133139C2×C4semidirect
98. #13133139C2×C4semidirect
124. C22⋊C8×C53C2×C4semidirect
331. #1854D53semidirect
391. C53#21582semidirect
448. C22⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109482
#202772212
#117×Dic532
#202753412
#202753342
#202752922
#202752852
#202750372
#202748072
#202746152
#202746142
#202745392
#202744792
C53×#18542
#2158253