Finite Group Databasebeta

← #27394659

Subgroups — #27394659

Order 13568.

830 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
1C2×#16810677C211yesmaximal, characteristic
2#20277118C211yesmaximal, characteristic
3#20277117C211yesmaximal, characteristic
4#20275477C211yesmaximal, characteristic, complemented
5#20275475C211yesmaximal, characteristic, complemented
6#20275472C211yesmaximal, characteristic, complemented
7#20275470C211yesmaximal, characteristic, complemented
8#20274916C211yesmaximal, characteristic
9#20274914C211yesmaximal, characteristic
10#20274912C211yesmaximal, characteristic
11#20274882C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274847C211yesmaximal, characteristic, complemented
14#20274847C211yesmaximal, characteristic, complemented
15C53×#1854C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133173—21no—
18C22×C4×Dic53C2211yescharacteristic
19#16810677C418yescomplemented
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31#16810492C2211yescharacteristic
32#16810492C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810492C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810488C2211yescharacteristic
37#16810487—21no—
38#16810483C2211yescharacteristic
39#16810481—21no—
40#16810481—21no—
41#16810481—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719919—21no—
45C4×#11719919—21no—
46#16810474—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—41no—
65#13133176—41no—
66#13133173—44no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2×C414yes—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53Q814yes—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133151—41no—
91#13133150—41no—
92#13133150—41no—
93#13133149—41no—
94#13133148—41no—
95#13133148—41no—
96#13133146C2×C414yes—
97#13133146C2×C414yes—
98C22⋊#88322C2×C414yes—
99#13133127C2×C414yes—
100#13133127C2×C414yes—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111#117×C53—41no—
112#117×C53—41no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120#107×C53—41no—
121#106×C53—41no—
122C22⋊C8×C53C2×C414yes—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×Q812yes—
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—41no—
144C22×Dic53—41no—
145C22×Dic53C4⋊C414yes—
146C22×Dic53C4⋊C414yes—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C22×C412yes—
156C2×C8×C53C22×C412yes—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C42×C53—42no—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53C4⋊C414yes—
190C4×Dic53C4⋊C414yes—
191C4×Dic53C22×C412yes—
192C4×Dic53C22×C412yes—
193C4×Dic53C2×Q812yes—
194C4×Dic53C2×Q812yes—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C22×C412yes—
200C2×#88322C22×C412yes—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53C2×C4⋊C412yes—
208C23×C53C22×D411yescharacteristic
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
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×C53C2×C4⋊C412yes—
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×Q811yescharacteristic
230C2×C4×C53H(5,𝔽2)11yescharacteristic
231C2×C4×C53H(5,𝔽2)11yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—28no—
241C2×Dic53—44no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53C2×C4⋊C412yes—
255C2×Dic53C2×C4⋊C412yes—
256C2×Dic53C2×C4⋊C412yes—
257C2×Dic53C2×C4⋊C412yes—
258C2×Dic53AGL(1,ℤ/8)12yes—
259C2×Dic53#13712yes—
260C8×C53—22no—
261C8×C53—22no—
262#88322—22no—
263#88322—22no—
264#21988—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53C22×C4⋊C411yescharacteristic
273C22×C53#53511yescharacteristic
274C22×C53#53611yescharacteristic
275C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57511yescharacteristic
279C4×C53—22no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#52011yescharacteristic
285C4×C53#52011yescharacteristic
286C4×C53#54411yescharacteristic
287C4×C53#54411yescharacteristic
288Dic53—22no—
289Dic53—22no—
290Dic53—22no—
291Dic53—81no—
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295Dic53#42814yes—
296C2×#432—531no—
297#2285—531no—
298#2285—531no—
299#1948—531no—
300#1948—531no—
301#1947—531no—
302#1947—531no—
303#1854—531no—
304#1854—531no—
305#1854—531no—
306#1854D5311yescharacteristic
307#1847—531no—
308#1847—531no—
309#1836—531no—
310#1836—531no—
311C2×C53—42no—
312C2×C53#231711yescharacteristic
313C2×C53C2×#42811yescharacteristic
314C2×C53#291111yescharacteristic
315C2×C53#321411yescharacteristic
316C2×C53#321611yescharacteristic
317C2×C53#323511yescharacteristic
318C2×C53#323711yescharacteristic
319C2×#117—531no—
320C2×#117—1061no—
321C2×#117C2×D5311yescharacteristic
322C22×C42—531no—
323#432—538no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#107C2×D5311yescharacteristic
328C2×#106—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×#106C2×D5311yescharacteristic
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×C22⋊C8—531no—
335C2×C22⋊C8C2×D5311yescharacteristic
336C2×#97—531no—
337C2×#97—531no—
338#384—531no—
339#384—531no—
340#384—531no—
341#384—531no—
342#382—531no—
343#382—531no—
344#382—1061no—
345#380—531no—
346#380—1061no—
347#380—1061no—
348#380—1061no—
349#380C2×D5311yescharacteristic
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—1061no—
353C4×C4⋊C4—1061no—
354C4×C4⋊C4—1061no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340C2×D5311yescharacteristic
362#340C2×D5311yescharacteristic
363C53#2198811yescharacteristic, complemented, Sylow
364C23×C4—531no—
365C23×C4—531no—
366C23×C4C22×D5311yescharacteristic
367C22×C8—531no—
368C22×C8—531no—
369C22×C8C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371#117—41no—
372#117—41no—
373#117—2124no—
374#117—2121no—
375#117—2121no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C42—21no—
397C2×C42—534no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—534no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—1061no—
404C2×C42—1061no—
405C2×C42—1061no—
406#107—41no—
407#107—2121no—
408#107—2121no—
409#107—2121no—
410#106—41no—
411#106—2121no—
412#106—2121no—
413#106—2121no—
414C4⋊C8—534no—
415C4⋊C8—534no—
416C22⋊C8—534no—
417C22⋊C8C4×D5314yes—
418#97—534no—
419#97—534no—
420#95—21no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428C24D4×D5311yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—531no—
438C22×C4—534no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—534no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4C2×C4×D5312yes—
452C22×C4D4×D5311yescharacteristic
453C22×C4Q8×D5312yes—
454C22×C4C23×D5311yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—532no—
460C2×C8—532no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8C2×C4×D5312yes—
466C2×C8C2×C4×D5312yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—42no—
496C42—534no—
497C42—534no—
498C42—532no—
499C42—532no—
500C42—532no—
501C42—532no—
502C42—2122no—
503C42—2122no—
504C42—2122no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4⋊C4×D5312yes—
512C23C2×D4×D5311yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—1068no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—2124no—
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—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4C4⋊C4×D5312yes—
558C2×C4C22×C4×D5311yescharacteristic
559C2×C4C2×Q8×D5311yescharacteristic
560C2×C4#1313326211yescharacteristic
561C2×C4#1313326211yescharacteristic
562C2×C4#1313326411yescharacteristic
563C2×C4#1313326411yescharacteristic
564C8—22no—
565C8—22no—
566C8—1062no—
567C8—1062no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4⋊C4×D5311yescharacteristic
576C22#1681118811yescharacteristic
577C22#1681119511yescharacteristic
578C22AGL(1,ℤ/8)×D5311yescharacteristic
579C22#137×D5311yescharacteristic
580C22#1681142011yescharacteristic
581C22#1681142011yescharacteristic
582C4—22no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—534no—
588C4—1062no—
589C4—1062no—
590C4—1062no—
591C4—4241no—
592C4—4241no—
593C4—4241no—
594C4—4241no—
595C4#1681110211yescharacteristic
596C4#1681110211yescharacteristic
597C4#1681125211yescharacteristic
598C4#1681125211yescharacteristic
599C2—42no—
600C2#2027733411yescharacteristic
601C2#428×D5311yescharacteristic
602C2#2028330311yescharacteristic
603C2#2028570711yescharacteristic
604C2#2028572311yescharacteristic
605C2#2028591011yescharacteristic
606C2#2028592611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1854D53
hypocenterC53#21988
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #20275477C2semidirect
5. #20275475C2semidirect
6. #20275472C2semidirect
7. #20275470C2semidirect
11. #20274882C2semidirect
12. #20274880C2semidirect
13. #20274847C2semidirect
14. #20274847C2semidirect
19. #16810677C4semidirect
363. C53#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#168106772
#202771182
#202771172
#202754772
#202754752
#202754722
#202754702
#202749162
#202749142
#202749122
#202748822
#202748802
#202748472
C53×#18542
#2198853