Finite Group Databasebeta

← #27447204

Subgroups — #27447204

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810708C211yesmaximal, characteristic
2#20278583C211yesmaximal, characteristic
3Q8×#11719919C211yesmaximal, characteristic, complemented
4#20276240C211yesmaximal, characteristic
5#20276194C211yesmaximal, characteristic, complemented
6#20276102C211yesmaximal, characteristic
7#20276050C211yesmaximal, characteristic, complemented
8#20275446C211yesmaximal, characteristic, complemented
9#20275269C211yesmaximal, characteristic, complemented
10#20275070C211yesmaximal, characteristic
11#20275068C211yesmaximal, characteristic
12#20274973C211yesmaximal, characteristic
13#20274971C211yesmaximal, characteristic, complemented
14#20274967C211yesmaximal, characteristic, complemented
15C2×C53×#479C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20#16810708—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810558—21no—
30#16810545—21no—
31#16810525C2211yescharacteristic
32#16810524C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810521—21no—
35#16810521C2211yescharacteristic
36#16810519—21no—
37#16810519C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810494—21no—
40#16810492—21no—
41#16810492C2211yescharacteristic
42#16810480—21no—
43C4×#11719919—21no—
44C4×#11719919C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#125×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C53×#479—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C2×#11719924—22no—
64C2×#11719924—22no—
65C2×#11719924Q814yes—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68Q8×Dic53—22no—
69Q8×Dic53—22no—
70Q8×Dic53—22no—
71Q8×Dic53—41no—
72#13133179—22no—
73#13133179—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
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×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96#13133151—22no—
97#13133150—41no—
98#13133148—22no—
99#13133141—22no—
100#13133139—22no—
101#13133139—22no—
102#13133137—41no—
103C8×Dic53—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C22×Q8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113#125×C53—22no—
114#125×C53—41no—
115C4×Q8×C53—22no—
116C4×Q8×C53—41no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125#107×C53—22no—
126#106×C53—22no—
127C4⋊C8×C53—22no—
128#103×C53—22no—
129#103×C53D414yes—
130C23×D53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53C2×D412yes—
135C2×Q8×C53C2×Q812yes—
136C2×Q8×C53#4112yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212C2×Q812yes—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—81no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×D412yes—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53C2×D412yes—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—42no—
178#11719924—28no—
179#11719921—42no—
180#11719921—42no—
181#11719921—81no—
182#11719921—81no—
183#11719921—81no—
184C4⋊Dic53—24no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191C4⋊Dic53—81no—
192C4⋊Dic53#4112yes—
193#11719919—22no—
194#11719919—22no—
195#11719919—42no—
196#11719919—81no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—24no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322C2×Q812yes—
212C23×C53C22×D411yescharacteristic
213C22×D53—22no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220Q8×C53—22no—
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53C22⋊Q814yes—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×Q811yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53#13512yes—
268C2×Dic53#13712yes—
269D212—22no—
270D212—22no—
271D212—41no—
272D212—41no—
273C4×D53—84no—
274C8×C53—22no—
275C8×C53—41no—
276#88322—22no—
277#88322—41no—
278#26379—531nomaximal, Sylow
279C22×C53#52511yescharacteristic
280C22×C53D4211yescharacteristic
281C22×C53D4×Q811yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53C2×#13711yescharacteristic
284C22×C53#57611yescharacteristic
285C22×C53#57711yescharacteristic
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—81no—
290C2×D53—81no—
291C2×D53—81no—
292C2×D53—81no—
293C4×C53—24no—
294C4×C53—42no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53C2×C22⋊Q811yescharacteristic
300C4×C53#53911yescharacteristic
301C4×C53#54711yescharacteristic
302C4×C53#55411yescharacteristic
303Dic53—22no—
304Dic53—22no—
305Dic53—41no—
306Dic53—42no—
307Dic53—41no—
308Dic53—81no—
309Dic53—81no—
310C2×#479D5311yescharacteristic, complemented
311C2×C4×SD16—531no—
312#2454—531no—
313Q8×C4⋊C4—531no—
314#2093—531no—
315#2084—531no—
316#2067—531no—
317#2059—531no—
318#1942—531no—
319#1916—531no—
320#1880—531no—
321#1880—531no—
322#1864—531no—
323#1863—531no—
324#1863—531no—
325C2×C53#244911yescharacteristic
326C2×C53#310811yescharacteristic
327C2×C53#311911yescharacteristic
328C2×C53#326611yescharacteristic
329C2×C53#328711yescharacteristic
330C2×C53#338111yescharacteristic
331C2×C53#338511yescharacteristic
332D53—82no—
333C22×SD16—531no—
334C2×#125C2×D5311yescharacteristic
335C2×C4×Q8—531no—
336C2×C4×Q8—531no—
337C2×C4×Q8C2×D5311yescharacteristic
338C2×C4×D4—531no—
339#479—24no—
340C4×SD16—1064no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#107C2×D5311yescharacteristic
344C2×#106—531no—
345C2×#106C2×D5311yescharacteristic
346C2×C4⋊C8C2×D5311yescharacteristic
347C2×#103—531no—
348C2×#103C2×D5311yescharacteristic
349C2×#103C2×D5311yescharacteristic
350C2×#102—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×C4×C8—531no—
354#399—1061no—
355#396—1061no—
356#390—531no—
357#390—531no—
358#389—531no—
359#389—531no—
360#389—531no—
361#389—531no—
362#389—1061no—
363#389—1061no—
364#385—1061no—
365#384—531no—
366#384—1061no—
367#380—1061no—
368C4×C4⋊C4—531no—
369C4×C4⋊C4—1061no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—531no—
378C53#2637911yescharacteristic, complemented, Sylow
379C22×Q8C22×D5311yescharacteristic
380C22×D4—531no—
381C23×C4—1061no—
382C2×SD16—534no—
383C2×SD16—1062no—
384C2×SD16—1062no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389#125—22no—
390#125—41no—
391C4×Q8—22no—
392C4×Q8—41no—
393C4×Q8—1062no—
394C4×Q8—1062no—
395C4×Q8—1062no—
396C4×Q8—2121no—
397C4×D4—1062no—
398C4×D4—2121no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C42—21no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429#107—22no—
430#107—1062no—
431#107—2121no—
432#106—22no—
433#106—1062no—
434C4⋊C8—22no—
435#103—22no—
436#103—1062no—
437#103C22⋊D5314yes—
438#102—1062no—
439#102—1062no—
440#102—2121no—
441C4×C8—1062no—
442#95—531no—
443#95—531no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448C24—1061no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8Q8×D5312yes—
453C2×Q8#1171994912yes—
454C2×Q8C2×C22⋊D5312yes—
455C2×D4—532no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—4241no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—532no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8C2×C22⋊D5312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—532no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1064no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—4241no—
511C4⋊C4—4241no—
512C4⋊C4—4241no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4C2×C22⋊D5312yes—
516C22⋊C4—4241no—
517C22⋊C4—4241no—
518C22⋊C4—4241no—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—1064no—
525C42—1061no—
526C42—1061no—
527C42—1061no—
528C42—1061no—
529C42—2122no—
530C23—1062no—
531C23—2121no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23C2×D4×D5311yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8#1313319314yes—
542D4—1062no—
543D4—1062no—
544D4—2121no—
545D4—2121no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—532no—
565C2×C4—532no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1062no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—4244no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4C22×C22⋊D5311yescharacteristic
587C2×C4C2×Q8×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4C2×#1171994911yescharacteristic
590C2×C4#1313326211yescharacteristic
591C2×C4#1313326411yescharacteristic
592C8—22no—
593C8—41no—
594C8—1062no—
595C8—2121no—
596C22—4241no—
597C22—4241no—
598C22—4241no—
599C22—4241no—
600C22—2122no—
601C22—2122no—
602C22—2122no—
603C22#1681113611yescharacteristic
604C22D4×C22⋊D5311yescharacteristic
605C22#1681128611yescharacteristic
606C22#135×D5311yescharacteristic
607C22#137×D5311yescharacteristic
608C22#1681142511yescharacteristic
609C22#1681143211yescharacteristic
610C4—24no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—1062no—
617C4—1062no—
618C4—4241no—
619C4—4241no—
620C4—2121no—
621C4—2121no—
622C4—2122no—
623C4C2×#1313319311yescharacteristic
624C4#1681122211yescharacteristic
625C4#1681127211yescharacteristic
626C4#1681132711yescharacteristic
627C2—4242no—
628C2#2027852011yescharacteristic
629C2#2028480011yescharacteristic
630C2#2028492111yescharacteristic
631C2#2028621911yescharacteristic
632C2#2028646911yescharacteristic
633C2#2028759311yescharacteristic
634C2#2028762811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#479C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenterC2×#479D53
hypocenterC53#26379
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
3. Q8×#11719919C2semidirect
5. #20276194C2semidirect
7. #20276050C2semidirect
8. #20275446C2semidirect
9. #20275269C2semidirect
13. #20274971C2semidirect
14. #20274967C2semidirect
15. C2×C53×#479C2semidirect
310. C2×#479D53semidirect
378. C53#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#168107082
#202785832
Q8×#117199192
#202762402
#202761942
#202761022
#202760502
#202754462
#202752692
#202750702
#202750682
#202749732
#202749712
#202749672
C2×C53×#4792
#2637953