Finite Group Databasebeta

← #27468151

Subgroups — #27468151

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279736C211yesmaximal, characteristic, complemented
2#20276277C211yesmaximal, characteristic, complemented
3#20276272C211yesmaximal, characteristic, complemented
4#20275861C211yesmaximal, characteristic
5#20275851C211yesmaximal, characteristic
6#20275689C211yesmaximal, characteristic, complemented
7#20275683C211yesmaximal, characteristic, complemented
8#20275675C211yesmaximal, characteristic, complemented
9#20275664C211yesmaximal, characteristic, complemented
10#20275415C211yesmaximal, characteristic
11#20275401C211yesmaximal, characteristic
12#20274725C211yesmaximal, characteristic
13#20274590C211yesmaximal, characteristic, complemented
14#20274540C211yesmaximal, characteristic
15C53×#2542C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133138C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133128C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25C2×C4×#88322C2211yescharacteristic
26#16810556—21no—
27#16810543—21no—
28#16810527—21no—
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810520—21no—
32#16810489C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×C22⋊Q8×C53—21no—
50C2×C22⋊Q8×C53—21no—
51C2×C4×D4×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#399—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#384C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×Dic106—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133217—22no—
65#13133217—41no—
66#13133209—22no—
67#13133209—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C4×Dic53—21no—
78C4⋊#88322—22no—
79#13133138—22no—
80#13133138—41no—
81#13133138—41no—
82#13133138Q814yescomplemented
83#13133128—22no—
84#13133128—41no—
85C4×#88322—41no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53—21no—
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94C4⋊Q8×C53—22no—
95C4⋊Q8×C53—22no—
96C22⋊Q8×C53—44no—
97C22⋊Q8×C53—44no—
98C4×D4×C53—22no—
99C4×D4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53—21no—
121C2×Q8×C53—42no—
122C2×Q8×C53—42no—
123C2×Q8×C53—42no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53C2×Q812yes—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C2×Dic106—42no—
148C2×Dic106—42no—
149C2×Dic106—81no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53#4112yes—
173C42×C53#4112yes—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—42no—
181C4⋊Dic53C2×Q812yes—
182C4⋊Dic53#4112yes—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186#11719919—81no—
187C4×Dic53—81no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×Q812yes—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208Q8×C53—84no—
209D4×C53—22no—
210D4×C53—22no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
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×C53#12412yes—
238C2×C4×C53#12412yes—
239C2×C4×C53AGL(1,ℤ/8)12yes—
240C2×C4×C53AGL(1,ℤ/8)12yes—
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259Dic106—84no—
260#88322—22no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#27830—531nomaximal, Sylow
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53C2×#12411yescharacteristic
273C22×C53#54611yescharacteristic
274C22×C53D4×Q811yescharacteristic
275C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C53#57711yescharacteristic
278C22×C53#57811yescharacteristic
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#53311yescharacteristic
289C4×C53#54211yescharacteristic
290C4×C53#55411yescharacteristic
291C4×C53#55611yescharacteristic
292Dic53—42no—
293Dic53—42no—
294Dic53—81no—
295Dic53—81no—
296#2550—531no—
297#2542D5311yescharacteristic
298#2100—531no—
299#2099—531no—
300#2025—531no—
301#2023—531no—
302#1993—531no—
303#1992—531no—
304#1990—531no—
305#1987—531no—
306#1936—531no—
307#1934—531no—
308#1804—531no—
309#1771—531no—
310#1762—531no—
311C2×C53—82no—
312C2×C53#249211yescharacteristic
313C2×C53#313711yescharacteristic
314C2×C53#313811yescharacteristic
315C2×C53#330611yescharacteristic
316C2×C53#330811yescharacteristic
317C2×C53#338411yescharacteristic
318C2×C53#338611yescharacteristic
319C2×C4⋊Q8—531no—
320C2×C4⋊Q8C2×D5311yescharacteristic
321C2×#125—531no—
322C2×C22⋊Q8—21no—
323C2×C22⋊Q8—21no—
324C2×C4×D4C2×D5311yescharacteristic
325C2×C4⋊C8—531no—
326C2×#102—531no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#102—531no—
330C2×#97—531no—
331C2×#97—531no—
332C2×C4×C8—531no—
333#399—21no—
334#399—21no—
335#398—1061no—
336#395—1061no—
337#391—1061no—
338#390C2×D5311yescharacteristic
339#390C2×D5311yescharacteristic
340#389—531no—
341#389—531no—
342#389—1061no—
343#384C2×D5311yescharacteristic
344#384C2×D5311yescharacteristic
345#382—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4C2×D5311yescharacteristic
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#336—531no—
361#336—531no—
362C53#2783011yescharacteristic, complemented, Sylow
363C22×Q8—21no—
364C22×Q8—1061no—
365C22×D4C22×D5311yescharacteristic
366C23×C4—21no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371C4⋊Q8—22no—
372C4⋊Q8—22no—
373C4⋊Q8—1062no—
374C4⋊Q8—2121no—
375#125—1062no—
376#125—2121no—
377C22⋊Q8—44no—
378C22⋊Q8—44no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22⋊D5311yescharacteristic
396C2×C4⋊C4C22⋊D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42—1061no—
406C2×C42C22×D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409C4⋊C8—1062no—
410#102—534no—
411#102—1062no—
412#102—2121no—
413#102—2121no—
414#97—1062no—
415#97—2121no—
416C4×C8—2121no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95C22⋊D5311yescharacteristic
423#95C22⋊D5311yescharacteristic
424C24—21no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—4241no—
429C2×Q8—2122no—
430C2×Q8—2122no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D5312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—532no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—532no—
480C4⋊C4—532no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—4241no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—4241no—
502C42#1171994712yes—
503C42#1171994712yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C22×C22⋊D5311yescharacteristic
512Q8—84no—
513Q8—4244no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4#1313320412yes—
555C2×C4#1313320412yes—
556C2×C4#1313324312yes—
557C2×C4#1313324312yes—
558C2×C4C2×Q8×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4C2×#1171994711yescharacteristic
561C2×C4C2×#1171994711yescharacteristic
562C2×C4#1313326311yescharacteristic
563C2×C4#1313326511yescharacteristic
564C8—1062no—
565C8—2121no—
566C8—2121no—
567C8—2121no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C2×#1313320411yescharacteristic
576C22#1681126111yescharacteristic
577C22Q8×C22⋊D5311yescharacteristic
578C22C2×#1313324311yescharacteristic
579C22C2×#1313324311yescharacteristic
580C22#1681143111yescharacteristic
581C22#1681143811yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—4241no—
592C4—4241no—
593C4—2122no—
594C4—2122no—
595C4#1681117511yescharacteristic
596C4#1681124611yescharacteristic
597C4#1681133111yescharacteristic
598C4#1681134411yescharacteristic
599C2—82no—
600C2#2027909111yescharacteristic
601C2#2028508011yescharacteristic
602C2#2028508911yescharacteristic
603C2#2028667011yescharacteristic
604C2#2028670011yescharacteristic
605C2#2028761011yescharacteristic
606C2#2028763611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2542C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2542D53
hypocenterC53#27830
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. #20279736C2semidirect
2. #20276277C2semidirect
3. #20276272C2semidirect
6. #20275689C2semidirect
7. #20275683C2semidirect
8. #20275675C2semidirect
9. #20275664C2semidirect
13. #20274590C2semidirect
82. #13133138Q8semidirect
362. C53#27830semidirect

Maximal subgroups

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

subgroupindex
#202797362
#202762772
#202762722
#202758612
#202758512
#202756892
#202756832
#202756752
#202756642
#202754152
#202754012
#202747252
#202745902
#202745402
C53×#25422
#2783053