Finite Group Databasebeta

← #27447992

Subgroups — #27447992

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278411C211yesmaximal, characteristic
2#20277746C211yesmaximal, characteristic
3#20276276C211yesmaximal, characteristic, complemented
4#20276205C211yesmaximal, characteristic, complemented
5#20276108C211yesmaximal, characteristic, complemented
6#20275854C211yesmaximal, characteristic, complemented
7#20275661C211yesmaximal, characteristic, complemented
8#20275299C211yesmaximal, characteristic, complemented
9#20275149C211yesmaximal, characteristic
10#20274985C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274897C211yesmaximal, characteristic
13#20274718C211yesmaximal, characteristic, complemented
14#20274434C211yesmaximal, characteristic
15C53×#1872C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196—21no—
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27#16810556—21no—
28#16810543—21no—
29#16810530C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810516—21no—
34#16810516—21no—
35#16810509—21no—
36#16810496—21no—
37#16810492C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C2×#11719917—22no—
61C2×#11719917—41no—
62C2×#11719917—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216—41no—
67#13133196—44no—
68Q8×Dic53—22no—
69Q8×Dic53—41no—
70#13133176—22no—
71#13133176—41no—
72C2×C4⋊Dic53—21no—
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⋊Dic53D411yescharacteristic
86C2×C22⋊Dic53D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133151—41no—
93#13133149—22no—
94#13133146—41no—
95#13133146—41no—
96#13133141—41no—
97#13133141—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C22×Q8×C53C2311yescharacteristic
108C23×C4×C53—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22⋊Q8×C53—22no—
113C22⋊Q8×C53Q814yescomplemented
114C2×C4⋊C4×C53—21no—
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#103×C53—22no—
123#103×C53—41no—
124C22⋊C8×C53—22no—
125C24×C53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53—81no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×Q812yes—
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×Dic106—42no—
156C2×Dic106—42no—
157C2×Dic106—81no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53C2×Q812yes—
171C4⋊C4×C53#4112yes—
172C22⋊C4×C53—24no—
173C22⋊Dic53—22no—
174C22⋊Dic53—22no—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53#4112yes—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199#11719917—44no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53C22⋊Q812yes—
213C23×C53C22×D411yescharacteristic
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—41no—
217Q8×C53—41no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—41no—
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—41no—
236C2×C4×C53—84no—
237C2×C4×C53C22⋊Q812yes—
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—22no—
250C2×Dic53—21no—
251C2×Dic53—21no—
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×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53AGL(1,ℤ/8)12yes—
267C2×Dic53#13712yes—
268Dic106—84no—
269C8×C53—41no—
270C8×C53—41no—
271#88322—41no—
272#88322—41no—
273#26431—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C2×C22⋊Q811yescharacteristic
282C22×C53#54611yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C53C2×#13711yescharacteristic
286C22×C53#57711yescharacteristic
287C22×C53#57811yescharacteristic
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53#52911yescharacteristic
294C4×C53#53811yescharacteristic
295C4×C53#55411yescharacteristic
296C4×C53#55411yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—41no—
300Dic53—42no—
301Dic53—41no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305#2440—531no—
306#2362—531no—
307#2099—531no—
308#2086—531no—
309#2068—531no—
310#2024—531no—
311#1987—531no—
312#1920—531no—
313#1894—531no—
314#1872D5311yescharacteristic
315#1865—531no—
316#1854—531no—
317#1851—531no—
318#1803—531no—
319#1736—531no—
320C2×C53—42no—
321C2×C53#245011yescharacteristic
322C2×C53#307911yescharacteristic
323C2×C53#308211yescharacteristic
324C2×C53#330811yescharacteristic
325C2×C53#331211yescharacteristic
326C2×C53#332411yescharacteristic
327C2×C53#333111yescharacteristic
328C22×M16—531no—
329C2×C4⋊Q8—531no—
330C2×C22⋊Q8—1061no—
331C2×C22⋊Q8C2×D5311yescharacteristic
332C2×C4×Q8—531no—
333C2×#117—531no—
334C22×C4⋊C4C2×D5311yescharacteristic
335C2×#107—531no—
336C2×#106—531no—
337C2×C4⋊C8—531no—
338C2×C4⋊C8—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103C2×D5311yescharacteristic
342C2×#103C2×D5311yescharacteristic
343C2×C22⋊C8C2×D5311yescharacteristic
344#398—1061no—
345#395—1061no—
346#391—531no—
347#389—531no—
348#389—531no—
349#389—531no—
350#388—1061no—
351#388—1061no—
352#387—1061no—
353#385—1061no—
354#384—531no—
355#380—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C22⋊C4—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371C53#2643111yescharacteristic, complemented, Sylow
372C22×Q8—1061no—
373C22×Q8C22×D5311yescharacteristic
374C23×C4—21no—
375C23×C4C22×D5311yescharacteristic
376C2×M16—1062no—
377C2×M16—2121no—
378C2×M16—2121no—
379C22×C8—531no—
380C22×C8—531no—
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C4⋊Q8—1062no—
384C4⋊Q8—2121no—
385C22⋊Q8—22no—
386C22⋊Q8—2124no—
387C22⋊Q8Dic10614yescomplemented
388C4×Q8—1062no—
389C4×Q8—2121no—
390#117—1062no—
391#117—2121no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—41no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—531no—
416C2×C42—531no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420#107—2121no—
421#106—1062no—
422C4⋊C8—2121no—
423C4⋊C8—2121no—
424#103—22no—
425#103—41no—
426#103—2121no—
427#103—2121no—
428C22⋊C8—22no—
429#95—531no—
430#95—531no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24D4×D5311yescharacteristic
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—4241no—
445C2×Q8—2122no—
446C2×Q8—2122no—
447C2×Q8C2×Dic10612yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4C2×Dic10612yes—
468C22×C4D4×D5311yescharacteristic
469C22×C4#1171994712yes—
470C22×C4C23×D5311yescharacteristic
471M16—2124no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—532no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—4241no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4C2×Dic10612yes—
506C4⋊C4#1171994712yes—
507C22⋊C4—24no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C42—1061no—
513C42—1061no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C42—2122no—
518C42—2122no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#1313319412yes—
526C23C2×D4×D5311yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531Q8—4244no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1062no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4#1313319412yes—
576C2×C4C22×Dic10611yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4#1313326211yescharacteristic
581C2×C4#1313326311yescharacteristic
582C8—41no—
583C8—41no—
584C8—2121no—
585C8—2121no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1313319411yescharacteristic
594C22#1681126011yescharacteristic
595C22#1681126511yescharacteristic
596C22AGL(1,ℤ/8)×D5311yescharacteristic
597C22#137×D5311yescharacteristic
598C22#1681142911yescharacteristic
599C22#1681143611yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—1062no—
606C4—1062no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—2121no—
611C4—2122no—
612C4—2121no—
613C4#1681115411yescharacteristic
614C4#1681121011yescharacteristic
615C4#1681133011yescharacteristic
616C4#1681133211yescharacteristic
617C2—42no—
618C2#2027854311yescharacteristic
619C2#2028452611yescharacteristic
620C2#2028455311yescharacteristic
621C2#2028670911yescharacteristic
622C2#2028675911yescharacteristic
623C2#2028693911yescharacteristic
624C2#2028702911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1872D53
hypocenterC53#26431
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. #20276276C2semidirect
4. #20276205C2semidirect
5. #20276108C2semidirect
6. #20275854C2semidirect
7. #20275661C2semidirect
8. #20275299C2semidirect
10. #20274985C2semidirect
13. #20274718C2semidirect
113. C22⋊Q8×C53Q8semidirect
371. C53#26431semidirect
387. C22⋊Q8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202784112
#202777462
#202762762
#202762052
#202761082
#202758542
#202756612
#202752992
#202751492
#202749852
#202749162
#202748972
#202747182
#202744342
C53×#18722
#2643153