Finite Group Databasebeta

← #27447932

Subgroups — #27447932

Order 13568.

840 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
1#20278411C211yesmaximal, characteristic, complemented
2#20277757C211yesmaximal, characteristic
3#20276276C211yesmaximal, characteristic, complemented
4#20276215C211yesmaximal, characteristic, complemented
5#20276087C211yesmaximal, characteristic, complemented
6#20275849C211yesmaximal, characteristic, complemented
7#20275661C211yesmaximal, characteristic, complemented
8#20275306C211yesmaximal, characteristic, complemented
9#20275154C211yesmaximal, characteristic
10#20274964C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic, complemented
12#20274897C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic, complemented
14#20274434C211yesmaximal, characteristic, complemented
15C53×#1874C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196—21no—
19C2×D4×Dic53C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27#16810556—21no—
28#16810543—21no—
29#16810526—21no—
30#16810526C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810516—21no—
35#16810503—21no—
36#16810496—21no—
37#16810492C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C2×#11719917—22no—
62C2×#11719917—41no—
63C2×#11719917—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—41no—
68#13133196—44no—
69D4×Dic53—22no—
70D4×Dic53—41no—
71#13133176—22no—
72#13133176—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53D411yescharacteristic
89C2×C22⋊Dic53D411yescharacteristic
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133151—22no—
95#13133149—41no—
96#13133146—41no—
97#13133146—41no—
98#13133138—41no—
99#13133138—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C22×D4×C53—21no—
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113#121×C53—22no—
114#121×C53Q814yescomplemented
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—22no—
117C2×C4⋊C4×C53—41no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122#102×C53—22no—
123#102×C53—41no—
124C22⋊C8×C53—22no—
125C24×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×D4×C53—24no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—81no—
134C2×D4×C53—81no—
135C2×D4×C53C2×Q812yes—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—81no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×Q812yes—
148C22×C4×C53#4112yes—
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—81no—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×Dic106—81no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53C2×Q812yes—
176C4⋊C4×C53#4112yes—
177C22⋊C4×C53—24no—
178C22⋊Dic53—22no—
179C22⋊Dic53—22no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—81no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—42no—
204C4×Dic53—81no—
205#11719917—44no—
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×C53—22no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53C22⋊Q812yes—
226C23×C53C22×D411yescharacteristic
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—41no—
230D4×C53—41no—
231D4×C53—84no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—84no—
248C2×C4×C53C22⋊Q812yes—
249C2×C4×C53C22×Q811yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—22no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—84no—
278C2×Dic53AGL(1,ℤ/8)12yes—
279C2×Dic53#13712yes—
280Dic106—84no—
281C8×C53—41no—
282C8×C53—41no—
283#88322—41no—
284#88322—41no—
285#26427—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53C2×C22⋊Q811yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53#54611yescharacteristic
303C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C53C2×#13711yescharacteristic
305C22×C53#57611yescharacteristic
306C22×C53#57711yescharacteristic
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—81no—
311C4×C53#52911yescharacteristic
312C4×C53#53811yescharacteristic
313C4×C53#55411yescharacteristic
314C4×C53#55411yescharacteristic
315Dic53—22no—
316Dic53—22no—
317Dic53—41no—
318Dic53—42no—
319Dic53—41no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323#2440—531no—
324#2363—531no—
325#2099—531no—
326#2088—531no—
327#2065—531no—
328#2023—531no—
329#1987—531no—
330#1921—531no—
331#1895—531no—
332#1874D5311yescharacteristic
333#1862—531no—
334#1854—531no—
335#1851—531no—
336#1802—531no—
337#1736—531no—
338C2×C53—42no—
339C2×C53—82no—
340C2×C53#245011yescharacteristic
341C2×C53#307911yescharacteristic
342C2×C53#308111yescharacteristic
343C2×C53#330611yescharacteristic
344C2×C53#332011yescharacteristic
345C2×C53#332511yescharacteristic
346C2×C53#333311yescharacteristic
347C22×M16—531no—
348C2×C4⋊Q8—531no—
349C2×C22⋊Q8—1061no—
350C2×#121C2×D5311yescharacteristic
351C2×C4×D4—531no—
352C2×#117—531no—
353C22×C4⋊C4C2×D5311yescharacteristic
354C2×#107—531no—
355C2×#106—531no—
356C2×C4⋊C8—531no—
357C2×C4⋊C8—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×#102C2×D5311yescharacteristic
361C2×#102C2×D5311yescharacteristic
362C2×C22⋊C8C2×D5311yescharacteristic
363#398—1061no—
364#395—1061no—
365#390—531no—
366#390—1061no—
367#389—531no—
368#389—531no—
369#389—531no—
370#388—1061no—
371#386—1061no—
372#385—1061no—
373#384—531no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—531no—
377C4×C22⋊C4—1061no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340C2×D5311yescharacteristic
387#340C2×D5311yescharacteristic
388#336—531no—
389#336—531no—
390C53#2642711yescharacteristic, complemented, Sylow
391C22×Q8—1061no—
392C22×D4—21no—
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4—1061no—
396C23×C4C22×D5311yescharacteristic
397C2×M16—1062no—
398C2×M16—2121no—
399C2×M16—2121no—
400C22×C8—531no—
401C22×C8—531no—
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
404C4⋊Q8—1062no—
405C4⋊Q8—2121no—
406C22⋊Q8—2124no—
407#121—22no—
408#121Dic10614yescomplemented
409C4×D4—1062no—
410C4×D4—2121no—
411#117—1062no—
412#117—2121no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—41no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—531no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—1061no—
440C2×C42—1061no—
441#107—1062no—
442#106—2121no—
443C4⋊C8—2121no—
444C4⋊C8—2121no—
445#102—22no—
446#102—41no—
447#102—2121no—
448#102—2121no—
449C22⋊C8—22no—
450#95—531no—
451#95—531no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457C24—21no—
458C24D4×D5311yescharacteristic
459C2×Q8—4241no—
460C2×Q8—2122no—
461C2×Q8—2122no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—81no—
469C2×D4—81no—
470C2×D4C2×Dic10612yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—81no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—4241no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4C2×Dic10612yes—
493C22×C4D4×D5311yescharacteristic
494C22×C4#1171994712yes—
495C22×C4C23×D5311yescharacteristic
496M16—2124no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—532no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—4241no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4C2×Dic10612yes—
529C4⋊C4#1171994712yes—
530C22⋊C4—24no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—4241no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C42—1061no—
539C42—1061no—
540C42—1061no—
541C42—1061no—
542C42—4241no—
543C42—2122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23#1313319412yes—
558C23C2×D4×D5311yescharacteristic
559Q8—4244no—
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1062no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—4244no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4#1313319412yes—
607C2×C4C22×Dic10611yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4#1313326211yescharacteristic
612C2×C4#1313326311yescharacteristic
613C8—41no—
614C8—41no—
615C8—2121no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#1313319411yescharacteristic
632C22#1681126011yescharacteristic
633C22#1681126511yescharacteristic
634C22AGL(1,ℤ/8)×D5311yescharacteristic
635C22#137×D5311yescharacteristic
636C22#1681142411yescharacteristic
637C22#1681142911yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—1062no—
643C4—1062no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—2121no—
648C4—2122no—
649C4—2121no—
650C4#1681115411yescharacteristic
651C4#1681121011yescharacteristic
652C4#1681133011yescharacteristic
653C4#1681133211yescharacteristic
654C2—42no—
655C2—82no—
656C2#2027854311yescharacteristic
657C2#2028452611yescharacteristic
658C2#2028454211yescharacteristic
659C2#2028667911yescharacteristic
660C2#2028687911yescharacteristic
661C2#2028695011yescharacteristic
662C2#2028705111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1874D53
hypocenterC53#26427
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. #20278411C2semidirect
3. #20276276C2semidirect
4. #20276215C2semidirect
5. #20276087C2semidirect
6. #20275849C2semidirect
7. #20275661C2semidirect
8. #20275306C2semidirect
10. #20274964C2semidirect
11. #20274916C2semidirect
12. #20274897C2semidirect
13. #20274713C2semidirect
14. #20274434C2semidirect
114. #121×C53Q8semidirect
390. C53#26427semidirect
408. #121Dic106semidirect

Maximal subgroups

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

subgroupindex
#202784112
#202777572
#202762762
#202762152
#202760872
#202758492
#202756612
#202753062
#202751542
#202749642
#202749162
#202748972
#202747132
#202744342
C53×#18742
#2642753