Finite Group Databasebeta

← #27391697

Subgroups — #27391697

Order 13568.

948 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810926C211yesmaximal, characteristic
2#20277274C211yesmaximal, characteristic, complemented
3C4×#13133179C211yesmaximal, characteristic
4#20275669C211yesmaximal, characteristic, complemented
5#20275486C211yesmaximal, characteristic, complemented
6#20275293C211yesmaximal, characteristic, complemented
7#20274975C211yesmaximal, characteristic
8#20274965C211yesmaximal, characteristic
9#20274724C211yesmaximal, characteristic
10#107×Dic53C211yesmaximal, characteristic, complemented
11#20274603C211yesmaximal, characteristic, complemented
12#20274563C211yesmaximal, characteristic, complemented
13#20274529C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1921C211yesmaximal, characteristic, complemented
16C2×#13133208C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×C42×D53—21no—
21#16810926C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810508—21no—
35#16810504—21no—
36#16810488C2211yescharacteristic
37C4⋊C4×Dic53—21no—
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42#16810474—21no—
43C4×#11719921—21no—
44#16810470—21no—
45#16810470—21no—
46C42×Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208C2×C414yescomplemented
68C4×D212—22no—
69C4×D212—22no—
70#13133179—22no—
71#13133179—22no—
72#13133179C2×C414yes—
73#13133179Q814yes—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C42×D53—44no—
94#13133151—22no—
95#13133148—22no—
96#13133148C2×C414yescomplemented
97#13133146—22no—
98#13133146C2×C414yescomplemented
99#13133139—22no—
100#13133139C2×C414yes—
101#13133137—22no—
102#13133137C2×C414yes—
103C8×Dic53—22no—
104C8×Dic53—22no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#107×C53C2×C414yescomplemented
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—24no—
147C2×C4×D53—24no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53C22×C412yes—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53C22×C412yes—
167C4⋊C4×C53C22×C412yes—
168C4⋊C4×C53C2×Q812yes—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—42no—
172#11719921—24no—
173#11719921—24no—
174#11719921—24no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—24no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C22×C412yes—
181C4⋊Dic53#4112yes—
182#11719919—24no—
183#11719919—42no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—41no—
188C4×Dic53—42no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53C22×C412yes—
193C4×Dic53C2×Q812yes—
194C4×Dic53#4112yes—
195C4×Dic53#4112yes—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C22×C412yes—
202C23×C53C22×D411yescharacteristic
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—41no—
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#13512yes—
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×Q811yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53C4×D412yes—
250C2×Dic53C4×D412yes—
251C2×Dic53C22⋊Q812yes—
252C2×Dic53C22⋊Q812yes—
253C2×Dic53#13512yes—
254C2×Dic53AGL(1,ℤ/8)12yes—
255D212—22no—
256D212—22no—
257D212—22no—
258D212C4×Q814yes—
259C4×D53—48no—
260C4×D53—48no—
261C4×D53—48no—
262C8×C53—22no—
263C8×C53—41no—
264#88322—22no—
265#88322—41no—
266C4×#478—531nomaximal, Sylow
267C22×C53C2×C4×D411yescharacteristic
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53#52511yescharacteristic
270C22×C53C2×#13511yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53#57511yescharacteristic
274C2×D53—24no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C4×C53—22no—
282C4×C53—42no—
283C4×C53—41no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—81no—
287C4×C53C2×C4×Q811yescharacteristic
288C4×C53C4×#4111yescharacteristic
289C4×C53#52911yescharacteristic
290C4×C53#53311yescharacteristic
291Dic53—22no—
292Dic53—22no—
293Dic53—22no—
294Dic53—42no—
295Dic53—42no—
296Dic53—81no—
297Dic53#47814yescomplemented
298C2×#478—531no—
299C4×#125—531no—
300C42×D4—531no—
301#1989—531no—
302#1949—531no—
303#1921D5311yescharacteristic, complemented
304#1919—531no—
305#1864—531no—
306#1862—531no—
307#1804—531no—
308C4×#107—531no—
309C4×#106—531no—
310C4×C4⋊C8—531no—
311C4×#102—531no—
312C4×#102—531no—
313C2×C53C4×C22⋊Q811yescharacteristic
314C2×C53C4×#13511yescharacteristic
315C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C53C2×#47811yescharacteristic
317C2×C53#307611yescharacteristic
318C2×C53#310211yescharacteristic
319C2×C53#310511yescharacteristic
320D53—44no—
321C2×#125—531no—
322C2×C4×D4—531no—
323C2×C4×D4—531no—
324C2×C4×D4—531no—
325C22×C42—1061no—
326#478—538no—
327C2×#107—531no—
328C2×#107C2×D5311yescharacteristic
329C2×#106—531no—
330C2×#106—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×#102—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×C4×C8—531no—
338C2×C4×C8—531no—
339#389—531no—
340#389C2×D5311yescharacteristic
341#387—1061no—
342#387—1061no—
343#382—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348C4×C4⋊C4—1061no—
349C4×C4⋊C4—1061no—
350C4×C4⋊C4C2×D5311yescharacteristic
351C4×C22⋊C4—1061no—
352C4×C22⋊C4—1061no—
353C4×C22⋊C4—1061no—
354C43—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340C2×D5311yescharacteristic
361#340C2×D5311yescharacteristic
362#340C2×D5311yescharacteristic
363#336—531no—
364#336C2×D5311yescharacteristic
365C53C4×#47811yescharacteristic, complemented, Sylow
366C22×D4—531no—
367C23×C4—1061no—
368C23×C4—1061no—
369C23×C4—1061no—
370C22×C8—531no—
371C22×C8—531no—
372C22×C8C22×D5311yescharacteristic
373C22×C8C22×D5311yescharacteristic
374#125—534no—
375#125—1062no—
376C4×D4—534no—
377C4×D4—534no—
378C4×D4—1062no—
379C4×D4—1062no—
380C4×D4—1062no—
381C4×D4—1062no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C42—21no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—1061no—
404C2×C42—1061no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42—2124no—
408C2×C42C22×D5311yescharacteristic
409#107—1062no—
410#107C4×D5314yescomplemented
411#106—534no—
412#106—1062no—
413C4⋊C8—534no—
414C4⋊C8—1062no—
415#102—534no—
416#102—534no—
417#102—1062no—
418#102—1062no—
419C4×C8—1062no—
420C4×C8—1062no—
421#95—21no—
422#95—21no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430C24—1061no—
431C2×D4—532no—
432C2×D4—532no—
433C2×D4—532no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—1064no—
446C22×C4—1064no—
447C22×C4—1064no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—532no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8C2×C4×D5312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—532no—
479C4⋊C4—532no—
480C4⋊C4—1064no—
481C4⋊C4—1064no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—2122no—
487C4⋊C4C2×C4×D5312yes—
488C4⋊C4C2×C4×D5312yes—
489C4⋊C4Q8×D5312yes—
490C22⋊C4—1064no—
491C22⋊C4—1064no—
492C22⋊C4—1064no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—532no—
497C42—532no—
498C42—532no—
499C42—532no—
500C42—1062no—
501C42—1062no—
502C42—1062no—
503C42—2121no—
504C42—2122no—
505C42—2122no—
506C42—2122no—
507C42—2122no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23C2×D4×D5311yescharacteristic
516D4—534no—
517D4—1062no—
518D4—1062no—
519D4—1062no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—532no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—2128no—
552C2×C4—2128no—
553C2×C4—2128no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4#1313324012yes—
562C2×C4C22×C4×D5311yescharacteristic
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4#41×D5311yescharacteristic
568C8—22no—
569C8—41no—
570C8—1062no—
571C8—2121no—
572C22—1064no—
573C22—2122no—
574C22—2122no—
575C22—2122no—
576C22—2122no—
577C22—2122no—
578C22—2122no—
579C22C4×D4×D5311yescharacteristic
580C22C22⋊Q8×D5311yescharacteristic
581C22#1681113111yescharacteristic
582C22#135×D5311yescharacteristic
583C22C2×#1313324011yescharacteristic
584C22AGL(1,ℤ/8)×D5311yescharacteristic
585C22#1681141911yescharacteristic
586C4—22no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—534no—
593C4—1062no—
594C4—1062no—
595C4—1062no—
596C4—4241no—
597C4—2122no—
598C4—2122no—
599C4C4×Q8×D5311yescharacteristic
600C4#1681109111yescharacteristic
601C4#1681115611yescharacteristic
602C4#1681117911yescharacteristic
603C2—2124no—
604C2#2027724911yescharacteristic
605C2C4×#1313324011yescharacteristic
606C2#2028352611yescharacteristic
607C2#478×D5311yescharacteristic
608C2#2028450611yescharacteristic
609C2#2028472811yescharacteristic
610C2#2028476511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1921C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1921D53
hypocenterC53C4×#478
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
2. #20277274C2semidirect
4. #20275669C2semidirect
5. #20275486C2semidirect
6. #20275293C2semidirect
10. #107×Dic53C2semidirect
11. #20274603C2semidirect
12. #20274563C2semidirect
15. C53×#1921C2semidirect
21. #16810926C4semidirect
67. #13133208C2×C4semidirect
96. #13133148C2×C4semidirect
98. #13133146C2×C4semidirect
121. #107×C53C2×C4semidirect
297. Dic53#478semidirect
303. #1921D53semidirect
365. C53C4×#478semidirect
410. #107C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109262
#202772742
C4×#131331792
#202756692
#202754862
#202752932
#202749752
#202749652
#202747242
#107×Dic532
#202746032
#202745632
#202745292
C53×#19212
C4×#47853