Finite Group Databasebeta

← #27452211

Subgroups — #27452211

Order 13568.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279089C211yesmaximal, characteristic
2#20277718C211yesmaximal, characteristic
3#20276038C211yesmaximal, characteristic, complemented
4#20275919C211yesmaximal, characteristic
5#20275857C211yesmaximal, characteristic, complemented
6#20275725C211yesmaximal, characteristic
7#20275662C211yesmaximal, characteristic, complemented
8#20275391C211yesmaximal, characteristic, complemented
9#20275357C211yesmaximal, characteristic, complemented
10#20275308C211yesmaximal, characteristic, complemented
11#20275055C211yesmaximal, characteristic, complemented
12#20275020C211yesmaximal, characteristic
13#20274806C211yesmaximal, characteristic
14C4×#13133142C211yesmaximal, characteristic, complemented
15C53×#1923C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133196C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29#16810565—21no—
30#16810535—21no—
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810527—21no—
35#16810521C2211yescharacteristic
36#16810520—21no—
37#16810516—21no—
38#16810510—21no—
39#16810495—21no—
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C22⋊C4×Dic53—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C2×Dic212—22no—
64C2×Dic212—41no—
65C2×Dic212—41no—
66#13133217—22no—
67#13133217—41no—
68#13133196—22no—
69#13133196—22no—
70#13133196—22no—
71#13133196—41no—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—22no—
95#13133142—22no—
96#13133142—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—41no—
100#13133142—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C23×C4×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—41no—
121C22⋊C8×C53—41no—
122C22⋊C8×C53—41no—
123C4×C8×C53—41no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53C2×D411yescharacteristic
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×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—81no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×Dic106—81no—
164C2×Dic106#4112yes—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53Q1614yescomplemented
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C22⋊Dic53—24no—
184C22⋊Dic53—42no—
185C22⋊Dic53—81no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4⋊Dic53#4112yes—
198C4⋊Dic53#4112yes—
199C4⋊Dic53#4112yes—
200#11719919—24no—
201#11719919—42no—
202#11719919—42no—
203#11719919—81no—
204C4×Dic53—42no—
205C4×Dic53—81no—
206C4×Dic53—81no—
207Dic212—44no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53C2×Q1612yes—
215C23×C53C22×D411yescharacteristic
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—21no—
228C2×C4×C53—21no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C2×Q1612yes—
234C2×C4×C53C2×Q1612yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
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×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—84no—
266Dic106—22no—
267Dic106—22no—
268Dic106—41no—
269Dic106—41no—
270Dic106—41no—
271Dic106—41no—
272Dic106—41no—
273Dic106—41no—
274Dic106—84no—
275C8×C53—41no—
276C8×C53—41no—
277C8×C53—41no—
278C8×C53—41no—
279#26726—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53#52511yescharacteristic
288C22×C53#54611yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53C22×Q1611yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57711yescharacteristic
294C4×C53—22no—
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53#52911yescharacteristic
300C4×C53#53811yescharacteristic
301C4×C53#55511yescharacteristic
302C4×C53#55911yescharacteristic
303Dic53—42no—
304Dic53—42no—
305Dic53—42no—
306Dic53—42no—
307Dic53—81no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311#2492—531no—
312#2360—531no—
313#2056—531no—
314#2036—531no—
315#2024—531no—
316#2001—531no—
317#1987—531no—
318#1933—531no—
319#1928—531no—
320#1923D5311yescharacteristic
321#1921—531no—
322#1878—531no—
323#1872—531no—
324#1824—531no—
325C4×#103—531no—
326C2×C53—42no—
327C2×C53#245911yescharacteristic
328C2×C53#310411yescharacteristic
329C2×C53#312511yescharacteristic
330C2×C53#329911yescharacteristic
331C2×C53#329911yescharacteristic
332C2×C53#331111yescharacteristic
333C2×C53#332311yescharacteristic
334C22×Q16—531no—
335C2×C4⋊Q8—531no—
336C2×C22⋊Q8—531no—
337C2×C22⋊Q8—531no—
338C2×C4×Q8—531no—
339C22×C4⋊C4—531no—
340C2×#107—531no—
341C2×C4⋊C8C2×D5311yescharacteristic
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×C22⋊C8C2×D5311yescharacteristic
349C2×C22⋊C8C2×D5311yescharacteristic
350C2×C4×C8C2×D5311yescharacteristic
351#400—1061no—
352#393—1061no—
353#391—531no—
354#391—531no—
355#391—531no—
356#391—1061no—
357#389—531no—
358#389—1061no—
359#388—1061no—
360#387—1061no—
361#385—1061no—
362#380—531no—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—531no—
365C4×C22⋊C4—1061no—
366C4×C22⋊C4C2×D5311yescharacteristic
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#336C2×D5311yescharacteristic
376#336C2×D5311yescharacteristic
377C53#2672611yescharacteristic, complemented, Sylow
378C22×Q8—531no—
379C22×Q8—531no—
380C22×Q8—1061no—
381C23×C4—1061no—
382C23×C4C22×D5311yescharacteristic
383C2×Q16—1062no—
384C2×Q16—2121no—
385C2×Q16—2121no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390C4⋊Q8—1062no—
391C4⋊Q8—2121no—
392C22⋊Q8—1062no—
393C22⋊Q8—1062no—
394C22⋊Q8—1062no—
395C22⋊Q8—2121no—
396C4×Q8—1062no—
397C4×Q8—2121no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—2121no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4D21211yescharacteristic
416C2×C22⋊C4D21211yescharacteristic
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#107—1062no—
423C4⋊C8—41no—
424#103—1062no—
425#103—1062no—
426#103—1062no—
427#103—1062no—
428#103—2121no—
429#103—2121no—
430C22⋊C8—41no—
431C22⋊C8—41no—
432C4×C8—41no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95D21211yescharacteristic
444#95D21211yescharacteristic
445C24C2×D21211yescharacteristic
446C2×Q8—532no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—4241no—
459C2×Q8—2122no—
460C2×Q8—2122no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—4241no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4C2×D21211yescharacteristic
479C22×C4C2×D21211yescharacteristic
480C22×C4C2×D21211yescharacteristic
481C22×C4C2×D21211yescharacteristic
482C22×C4C2×D21211yescharacteristic
483C22×C4C23×D5311yescharacteristic
484Q16—2124no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C4⋊C4—532no—
498C4⋊C4—532no—
499C4⋊C4—532no—
500C4⋊C4—1064no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—4241no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C22⋊C4—41no—
516C22⋊C4—1064no—
517C22⋊C4—4241no—
518C22⋊C4—2122no—
519C22⋊C4Dic21214yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—4241no—
525C42—4241no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×Dic21212yes—
534C23C22×D21211yescharacteristic
535Q8—1062no—
536Q8—1062no—
537Q8—4244no—
538Q8—2121no—
539Q8—2121no—
540Q8—2121no—
541Q8—2121no—
542Q8—2121no—
543Q8—2121no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—4244no—
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×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
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×C4C2×Dic21212yes—
587C2×C4C2×Dic21212yes—
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4#41×D5311yescharacteristic
592C2×C4#1313326211yescharacteristic
593C2×C4#1313326211yescharacteristic
594C8—41no—
595C8—41no—
596C8—41no—
597C8—41no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1681113511yescharacteristic
606C22#1681126811yescharacteristic
607C22#1681126811yescharacteristic
608C22C22×Dic21211yescharacteristic
609C22#1681142311yescharacteristic
610C22#1681143511yescharacteristic
611C22#1681143511yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—4241no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—2122no—
622C4—2122no—
623C4—2122no—
624C4—2122no—
625C4#1681115611yescharacteristic
626C4#1681120511yescharacteristic
627C4#1681133511yescharacteristic
628C4#1681135711yescharacteristic
629C2—42no—
630C2#2027865011yescharacteristic
631C2#2028475511yescharacteristic
632C2#2028499411yescharacteristic
633C2#2028659211yescharacteristic
634C2#2028659211yescharacteristic
635C2#2028675311yescharacteristic
636C2#2028693311yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1923C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1923D53
hypocenterC53#26726
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #20276038C2semidirect
5. #20275857C2semidirect
7. #20275662C2semidirect
8. #20275391C2semidirect
9. #20275357C2semidirect
10. #20275308C2semidirect
11. #20275055C2semidirect
14. C4×#13133142C2semidirect
178. C22⋊C4×C53Q16semidirect
377. C53#26726semidirect
519. C22⋊C4Dic212semidirect

Maximal subgroups

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

subgroupindex
#202790892
#202777182
#202760382
#202759192
#202758572
#202757252
#202756622
#202753912
#202753572
#202753082
#202750552
#202750202
#202748062
C4×#131331422
C53×#19232
#2672653