Finite Group Databasebeta

← #27388296

Subgroups — #27388296

Order 13568.

898 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810708C211yesmaximal, characteristic
2#20277190C211yesmaximal, characteristic, complemented
3#20277173C211yesmaximal, characteristic
4#20275572C211yesmaximal, characteristic
5#20275566C211yesmaximal, characteristic
6#20275442C211yesmaximal, characteristic, complemented
7#20275440C211yesmaximal, characteristic, complemented
8#20275398C211yesmaximal, characteristic
9#20275396C211yesmaximal, characteristic
10#20275393C211yesmaximal, characteristic, complemented
11#20274603C211yesmaximal, characteristic, complemented
12#103×Dic53C211yesmaximal, characteristic, complemented
13#20274529C211yesmaximal, characteristic
14C4×C8×Dic53C211yesmaximal, characteristic, complemented
15C53×#1933C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133175—21no—
20#16810708C418yescomplemented
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C4×#88322C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810524C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810515—21no—
40#16810513—21no—
41#16810508—21no—
42#16810506—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45#16810474—21no—
46#16810470—21no—
47#16810468—21no—
48C42×Dic53C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C2×#11719924C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67Q8×Dic53—22no—
68Q8×Dic53C2×C414yes—
69#13133179—22no—
70#13133179C2×C414yes—
71#13133175—44no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133148—22no—
94#13133148—22no—
95#13133148C2×C414yes—
96#13133141—22no—
97#13133139—22no—
98#13133139C2×C414yes—
99#13133137—22no—
100C4×#88322—22no—
101C8×Dic53—22no—
102C8×Dic53—22no—
103C8×Dic53—22no—
104C8×Dic53C2×C414yes—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112C22×Q8×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×C42×C53C2311yescharacteristic
120#103×C53—22no—
121#103×C53C2×C414yes—
122C4×C8×C53—22no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C23×D53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212C22×C412yes—
153C2×C4×D53—24no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53C22×C412yes—
161C2×C8×C53#4112yes—
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—
169C42×C53—21no—
170C42×C53—21no—
171#11719924—28no—
172#11719921—24no—
173#11719921—42no—
174#11719921—42no—
175#11719921—42no—
176#11719921—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—24no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53C22×C412yes—
185#11719919—42no—
186#11719919—42no—
187C4×Dic53—22no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—24no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53SD1614yescomplemented
196C4×Dic53C22×C412yes—
197C4×Dic53#4112yes—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C22×C412yes—
201C2×#88322#4112yes—
202C2×#88322#4112yes—
203C2×#88322#4112yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
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×C53C2×SD1612yes—
233C2×C4×C53#13512yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—22no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53C4×D412yes—
256C2×Dic53C4×D412yes—
257C2×Dic53C2×SD1612yes—
258C2×Dic53C2×SD1612yes—
259C2×Dic53#13512yes—
260C2×Dic53#13512yes—
261D212—22no—
262D212—22no—
263D212—41no—
264D212—41no—
265C4×D53—48no—
266C8×C53—22no—
267C8×C53—22no—
268#88322—22no—
269#88322#11714yescomplemented
270#21274—531nomaximal, Sylow
271C22×C53C2×C4×D411yescharacteristic
272C22×C53#53211yescharacteristic
273C22×C53#53211yescharacteristic
274C22×C53C22×SD1611yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53C2×#13511yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53C2×#11711yescharacteristic
292C4×C53C4×#4111yescharacteristic
293C4×C53#52911yescharacteristic
294C4×C53#52911yescharacteristic
295Dic53—22no—
296Dic53—22no—
297Dic53—22no—
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301Dic53C4×SD1614yes—
302C2×C4×SD16—531no—
303#2295—531no—
304#2293—531no—
305#1968—531no—
306#1968—531no—
307#1941—531no—
308#1941—531no—
309#1934—531no—
310#1934—531no—
311#1933—531no—
312#1933D5311yescharacteristic, complemented
313C4×#106—531no—
314C4×#103—531no—
315C4×#102—531no—
316C42×C8—531no—
317C2×C53#229311yescharacteristic
318C2×C53C2×C4×SD1611yescharacteristic
319C2×C53C4×#13511yescharacteristic
320C2×C53#316711yescharacteristic
321C2×C53#316711yescharacteristic
322C2×C53#316811yescharacteristic
323C2×C53#316811yescharacteristic
324D53—82no—
325C22×SD16—531no—
326C2×C4×Q8—531no—
327C2×C4×D4—531no—
328C2×#117—1061no—
329C4×SD16—538no—
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×#103—531no—
334C2×#103C2×D5311yescharacteristic
335C2×#103C2×D5311yescharacteristic
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×C4×C8—531no—
340C2×C4×C8—531no—
341C2×C4×C8—531no—
342C2×C4×C8—531no—
343C2×C4×C8—531no—
344C2×C4×C8C2×D5311yescharacteristic
345#391—531no—
346#391C2×D5311yescharacteristic
347#390—531no—
348#390—531no—
349#389—531no—
350#389—531no—
351#389—531no—
352#389C2×D5311yescharacteristic
353#388—1061no—
354#388—1061no—
355#387—1061no—
356#387—1061no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—1061no—
360C4×C22⋊C4—1061no—
361#376—1061no—
362C43—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369C53#2127411yescharacteristic, complemented, Sylow
370C22×Q8C22×D5311yescharacteristic
371C22×D4—531no—
372C23×C4—1061no—
373C2×SD16—534no—
374C2×SD16—1062no—
375C2×SD16—1062no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C4×Q8—534no—
381C4×Q8—1062no—
382C4×D4—534no—
383C4×D4—1062no—
384#117—2124no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411#106—534no—
412#106—1062no—
413#106—1062no—
414#103—22no—
415#103—1062no—
416#103C4×D5314yes—
417#102—534no—
418#102—1062no—
419#102—1062no—
420C4×C8—22no—
421C4×C8—534no—
422C4×C8—1062no—
423C4×C8—1062no—
424C4×C8—1062no—
425C4×C8—1062no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—1061no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5312yes—
443C2×D4—532no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—1064no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466SD16—1068no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—532no—
472C2×C8—532no—
473C2×C8—532no—
474C2×C8—532no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8C2×C4×D5312yes—
478C2×C8#1171994712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—532no—
486C4⋊C4—1064no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4C2×C4×D5312yes—
496C22⋊C4—1064no—
497C22⋊C4—2122no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C22⋊C4—2122no—
501C42—21no—
502C42—21no—
503C42—534no—
504C42—532no—
505C42—532no—
506C42—1062no—
507C42—1062no—
508C42—1064no—
509C42—1062no—
510C42—1061no—
511C42—1061no—
512C42—2122no—
513C42—2122no—
514C23—1062no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2121no—
519C23—2121no—
520C23—2121no—
521C23C2×D4×D5311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—1062no—
527D4—1062no—
528D4—2121no—
529D4—2121no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1062no—
559C2×C4—2128no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4SD16×D5312yes—
569C2×C4#1313324012yes—
570C2×C4C22×C4×D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4#41×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994711yescharacteristic
576C8—22no—
577C8—22no—
578C8—534no—
579C8—1062no—
580C22—4241no—
581C22—4241no—
582C22—4241no—
583C22—4241no—
584C22—2122no—
585C22—2122no—
586C22—2122no—
587C22C4×D4×D5311yescharacteristic
588C22#1681116811yescharacteristic
589C22#1681117211yescharacteristic
590C22C2×SD16×D5311yescharacteristic
591C22#135×D5311yescharacteristic
592C22#135×D5311yescharacteristic
593C22C2×#1313324011yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—534no—
601C4—1062no—
602C4—1062no—
603C4—1062no—
604C4—4241no—
605C4—4241no—
606C4—2122no—
607C4#117×D5311yescharacteristic
608C4C4×#1171994711yescharacteristic
609C4#1681115011yescharacteristic
610C4#1681115611yescharacteristic
611C2—4242no—
612C2#2027717511yescharacteristic
613C2C4×SD16×D5311yescharacteristic
614C2C4×#1313324011yescharacteristic
615C2#2028526911yescharacteristic
616C2#2028527711yescharacteristic
617C2#2028528411yescharacteristic
618C2#2028528611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1933C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1933D53
hypocenterC53#21274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. #20277190C2semidirect
6. #20275442C2semidirect
7. #20275440C2semidirect
10. #20275393C2semidirect
11. #20274603C2semidirect
12. #103×Dic53C2semidirect
14. C4×C8×Dic53C2semidirect
15. C53×#1933C2semidirect
20. #16810708C4semidirect
195. C4×Dic53SD16semidirect
269. #88322#117semidirect
312. #1933D53semidirect
369. C53#21274semidirect

Maximal subgroups

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

subgroupindex
C2×#168107082
#202771902
#202771732
#202755722
#202755662
#202754422
#202754402
#202753982
#202753962
#202753932
#202746032
#103×Dic532
#202745292
C4×C8×Dic532
C53×#19332
#2127453