Finite Group Databasebeta

← #27403535

Subgroups — #27403535

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810720C211yesmaximal, characteristic
2#20277663C211yesmaximal, characteristic
3#20277634C211yesmaximal, characteristic, complemented
4#20275400C211yesmaximal, characteristic
5#20275304C211yesmaximal, characteristic, complemented
6#20275289C211yesmaximal, characteristic, complemented
7#20275073C211yesmaximal, characteristic
8#20275067C211yesmaximal, characteristic
9#20274847C211yesmaximal, characteristic, complemented
10#20274723C211yesmaximal, characteristic
11#20274719C211yesmaximal, characteristic, complemented
12#20274591C211yesmaximal, characteristic, complemented
13#20274546C211yesmaximal, characteristic, complemented
14#20274534C211yesmaximal, characteristic, complemented
15C53×#1936C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133182C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133173—21no—
22#16810720C418yescomplemented
23C2×#13133150C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810520—21no—
34#16810518—21no—
35#16810517—21no—
36#16810511—21no—
37#16810492C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810487—21no—
40#16810481—21no—
41C4⋊C4×Dic53—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C4×Dic106—22no—
68C4×Dic106—41no—
69#13133182—22no—
70#13133182—22no—
71#13133182—41no—
72#13133182C2×C414yes—
73D4×Dic53—22no—
74D4×Dic53C2×C414yes—
75#13133173—44no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919D411yescharacteristic
86C2×#11719919D411yescharacteristic
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133150C2×C414yes—
98#13133146—22no—
99#13133142—22no—
100#13133142C2×C414yes—
101#13133140—22no—
102#13133138—41no—
103#13133127C2×C414yes—
104C8×Dic53—41no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×D4×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#102×C53—41no—
123#102×C53C2×C414yes—
124#97×C53—41no—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—24no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106C22×C412yes—
154C2×Dic106#4112yes—
155C2×Dic106#4112yes—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C412yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53C22×C412yes—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—24no—
174C22⋊Dic53—42no—
175C22⋊Dic53—81no—
176#11719923—28no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53C22×C412yes—
184#11719919—24no—
185#11719919—22no—
186#11719919—41no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—81no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—24no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53C22×C412yes—
199C4×Dic53#4112yes—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322C22×C412yes—
206C23×C53—22no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53C22×D411yescharacteristic
214D4×C53—22no—
215D4×C53—22no—
216D4×C53—41no—
217D4×C53—41no—
218C2×C4×C53—22no—
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—41no—
226C2×C4×C53—41no—
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×C53C23×C411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—48no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53C4×D412yes—
262C2×Dic53C4×D412yes—
263C2×Dic53AGL(1,ℤ/8)12yes—
264Dic106—22no—
265Dic106—22no—
266Dic106—22no—
267Dic106#11714yescomplemented
268C8×C53—22no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272#22781—531nomaximal, Sylow
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53C2×C4×D411yescharacteristic
281C22×C53#54611yescharacteristic
282C22×C53#54811yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57711yescharacteristic
286C22×C53#57811yescharacteristic
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53C2×#11711yescharacteristic
293C4×C53#52011yescharacteristic
294C4×C53#55211yescharacteristic
295C4×C53#55311yescharacteristic
296Dic53—24no—
297Dic53—22no—
298Dic53—22no—
299Dic53—22no—
300Dic53—42no—
301Dic53—41no—
302Dic53—81no—
303Dic53—81no—
304C2×#440—531no—
305#2354—531no—
306#2351—531no—
307#1936D5311yescharacteristic
308#1934—531no—
309#1921—531no—
310#1918—531no—
311#1880—531no—
312#1880—531no—
313#1836—531no—
314#1804—531no—
315#1803—531no—
316#1771—531no—
317#1763—531no—
318C4×#103—531no—
319C2×C53—82no—
320C2×C53#235211yescharacteristic
321C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
322C2×C53#294711yescharacteristic
323C2×C53#330811yescharacteristic
324C2×C53#332111yescharacteristic
325C2×C53#336811yescharacteristic
326C2×C53#337711yescharacteristic
327C22×SD16—531no—
328C2×C4×Q8—531no—
329C2×C4×Q8—531no—
330C2×C4×Q8—531no—
331C2×C4×D4—531no—
332C2×#117—1061no—
333#440—538no—
334C2×#107—531no—
335C2×C4⋊C8—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#102—531no—
340C2×#102C2×D5311yescharacteristic
341C2×#102C2×D5311yescharacteristic
342C2×#97—531no—
343C2×#97C2×D5311yescharacteristic
344C2×C4×C8—531no—
345#390—531no—
346#390C2×D5311yescharacteristic
347#389—531no—
348#389—1061no—
349#389—1061no—
350#389C2×D5311yescharacteristic
351#388—1061no—
352#388—1061no—
353#384—531no—
354#382—531no—
355#382—1061no—
356#380—1061no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—1061no—
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#2278111yescharacteristic, complemented, Sylow
372C22×Q8—531no—
373C22×D4C22×D5311yescharacteristic
374C23×C4—1061no—
375C2×SD16—534no—
376C2×SD16—1062no—
377C2×SD16—1062no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C4×Q8—534no—
383C4×Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—1062no—
386C4×Q8—2121no—
387C4×Q8—2121no—
388C4×D4—534no—
389C4×D4—1062no—
390#117—2124no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42C22×D5311yescharacteristic
419#107—534no—
420C4⋊C8—1062no—
421#103—534no—
422#103—1062no—
423#103—1062no—
424#102—41no—
425#102—2121no—
426#102C4×D5314yes—
427#97—41no—
428#97—534no—
429C4×C8—2121no—
430#95—21no—
431#95—531no—
432#95—531no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24—21no—
439C2×Q8—532no—
440C2×Q8—532no—
441C2×Q8—532no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D5312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—1064no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4C23×D5311yescharacteristic
468SD16—1068no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—532no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8C2×C4×D5312yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—532no—
488C4⋊C4—1062no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—2121no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4C2×C4×D5312yes—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—1064no—
505C22⋊C4—4241no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—532no—
510C42—532no—
511C42—1064no—
512C42—1061no—
513C42—1061no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C42—2122no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D5311yescharacteristic
526Q8—534no—
527Q8—1062no—
528Q8—1062no—
529Q8—1062no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
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—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—2128no—
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×C4C22×C4×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#1313326211yescharacteristic
579C2×C4#1313326411yescharacteristic
580C8—22no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C4×D4×D5311yescharacteristic
592C22#1681126211yescharacteristic
593C22#1681128111yescharacteristic
594C22AGL(1,ℤ/8)×D5311yescharacteristic
595C22#1681141911yescharacteristic
596C22#1681143311yescharacteristic
597C22#1681143911yescharacteristic
598C4—42no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—1062no—
604C4—1064no—
605C4—1062no—
606C4—1062no—
607C4—4241no—
608C4—4241no—
609C4—2122no—
610C4—2121no—
611C4#117×D5311yescharacteristic
612C4#1681110211yescharacteristic
613C4#1681130911yescharacteristic
614C4#1681131811yescharacteristic
615C2—82no—
616C2#2027764311yescharacteristic
617C2#2028352611yescharacteristic
618C2#2028355511yescharacteristic
619C2#2028670111yescharacteristic
620C2#2028689611yescharacteristic
621C2#2028744111yescharacteristic
622C2#2028755611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1936C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1936D53
hypocenterC53#22781
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
3. #20277634C2semidirect
5. #20275304C2semidirect
6. #20275289C2semidirect
9. #20274847C2semidirect
11. #20274719C2semidirect
12. #20274591C2semidirect
13. #20274546C2semidirect
14. #20274534C2semidirect
22. #16810720C4semidirect
267. Dic106#117semidirect
371. C53#22781semidirect

Maximal subgroups

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

subgroupindex
C2×#168107202
#202776632
#202776342
#202754002
#202753042
#202752892
#202750732
#202750672
#202748472
#202747232
#202747192
#202745912
#202745462
#202745342
C53×#19362
#2278153