Finite Group Databasebeta

← #27428655

Subgroups — #27428655

Order 13568.

740 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279012C211yesmaximal, characteristic
2#20277664C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276188C211yesmaximal, characteristic, complemented
5#20276143C211yesmaximal, characteristic, complemented
6#20276138C211yesmaximal, characteristic, complemented
7#20275301C211yesmaximal, characteristic, complemented
8#20275287C211yesmaximal, characteristic, complemented
9#20274916C211yesmaximal, characteristic, complemented
10#20274916C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic, complemented
12#20274811C211yesmaximal, characteristic, complemented
13#20274744C211yesmaximal, characteristic, complemented
14#20274432C211yesmaximal, characteristic, complemented
15C53×#1869C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23#16810551—21no—
24#16810537—21no—
25#16810526—21no—
26#16810521C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810516—21no—
29#16810516—21no—
30#16810511—21no—
31#16810504—21no—
32#16810493C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38C4⋊C4×Dic53—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59#13133176—22no—
60#13133176—22no—
61#13133176—22no—
62#13133176—22no—
63#13133176—41no—
64#13133176—41no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89#13133151—22no—
90#13133151—41no—
91#13133151—41no—
92#13133149—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×D4×C53D411yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53—22no—
105C22×C8×C53—22no—
106C22×C8×C53—41no—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C4×D4×C53—22no—
112C4×D4×C53—22no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C22⋊C8×C53—22no—
118C22⋊C8×C53—22no—
119C22⋊C8×C53—41no—
120C24×C53C2×D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—41no—
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×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—44no—
162C2×C8×C53#4112yes—
163C4⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C42×C53—21no—
167C42×C53—21no—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C22⋊Dic53—81no—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—42no—
192C4⋊Dic53—81no—
193C4⋊Dic53#4112yes—
194#11719919—42no—
195#11719919—81no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53#12312yes—
214C23×C53C22×D411yescharacteristic
215D4×C53—44no—
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—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53#12312yes—
235C2×C4×C53#13512yes—
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—22no—
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×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—41no—
266C8×C53—22no—
267C8×C53—41no—
268C8×C53—41no—
269C8×C53—41no—
270#24948—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53C2×#12311yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53#54711yescharacteristic
288C22×C53C2×#13511yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57611yescharacteristic
291C22×C53#57811yescharacteristic
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53#52911yescharacteristic
297C4×C53#53911yescharacteristic
298C4×C53#55111yescharacteristic
299C4×C53#55211yescharacteristic
300Dic53—42no—
301Dic53—81no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308#2485—531no—
309#2354—531no—
310#2100—531no—
311#2083—531no—
312#2076—531no—
313#2075—531no—
314#1920—531no—
315#1918—531no—
316#1869D5311yescharacteristic
317#1854—531no—
318#1854—531no—
319#1825—531no—
320#1825—531no—
321#1808—531no—
322#1735—531no—
323C2×C53—42no—
324C2×C53—81no—
325C2×C53#241911yescharacteristic
326C2×C53#309111yescharacteristic
327C2×C53#309811yescharacteristic
328C2×C53#330211yescharacteristic
329C2×C53#330911yescharacteristic
330C2×C53#334111yescharacteristic
331C2×C53#334911yescharacteristic
332C23×C8C2×D5311yescharacteristic
333C2×C4×D4C2×D5311yescharacteristic
334C2×#117—531no—
335C2×#117—531no—
336C2×#117—531no—
337C2×#107—531no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#106—531no—
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×C22⋊C8C2×D5311yescharacteristic
343C2×C22⋊C8C2×D5311yescharacteristic
344#397—1061no—
345#394—1061no—
346#390—1061no—
347#389—531no—
348#389—531no—
349#388—1061no—
350#388—1061no—
351#388—1061no—
352#387—1061no—
353#384—531no—
354#380—531no—
355#380—531no—
356#380—531no—
357#380—531no—
358#380—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#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#336C2×D5311yescharacteristic
374#336C2×D5311yescharacteristic
375C53#2494811yescharacteristic, complemented, Sylow
376C22×D4D21211yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C23×C4C22×D5311yescharacteristic
379C22×C8—22no—
380C22×C8—22no—
381C22×C8—41no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386C4×D4—22no—
387C4×D4—22no—
388#117—1062no—
389#117—1062no—
390#117—1062no—
391#117—1062no—
392#117—2121no—
393#117—2121no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4D21211yescharacteristic
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
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×C42—1061no—
421C2×C42C22×D5311yescharacteristic
422#107—1062no—
423#107—2121no—
424#107—2121no—
425#106—1062no—
426C22⋊C8—22no—
427C22⋊C8—22no—
428C22⋊C8—41no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24C2×D21211yescharacteristic
438C24C2×D21211yescharacteristic
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—41no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4C2×D21211yescharacteristic
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4#1171994712yes—
465C22×C4#1171994712yes—
466C22×C4C23×D5311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—44no—
479C2×C8#1171994712yes—
480C4⋊C4—22no—
481C4⋊C4—532no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—4241no—
504C22⋊C4—4241no—
505C22⋊C4—4241no—
506C22⋊C4—4241no—
507C22⋊C4—2122no—
508C22⋊C4—2122no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—21no—
512C42—21no—
513C42—2122no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1313320112yes—
531C23C22×D21211yescharacteristic
532D4—44no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—1062no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
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×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4#1313320112yes—
576C2×C4#1313323912yes—
577C2×C4#41×D5311yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4#1313326211yescharacteristic
582C2×C4#1313326411yescharacteristic
583C8—22no—
584C8—41no—
585C8—41no—
586C8—41no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22C2×#1313320111yescharacteristic
602C22#1681126811yescharacteristic
603C22#1681127711yescharacteristic
604C22C2×#1313323911yescharacteristic
605C22#1681142311yescharacteristic
606C22#1681142811yescharacteristic
607C22#1681144011yescharacteristic
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—4241no—
613C4—4241no—
614C4—4241no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2122no—
620C4#1681115411yescharacteristic
621C4#1681121811yescharacteristic
622C4#1681130411yescharacteristic
623C4#1681131011yescharacteristic
624C2—42no—
625C2—81no—
626C2#2027819011yescharacteristic
627C2#2028463011yescharacteristic
628C2#2028469511yescharacteristic
629C2#2028663311yescharacteristic
630C2#2028672311yescharacteristic
631C2#2028713811yescharacteristic
632C2#2028724711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1869C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1869D53
hypocenterC53#24948
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276188C2semidirect
5. #20276143C2semidirect
6. #20276138C2semidirect
7. #20275301C2semidirect
8. #20275287C2semidirect
9. #20274916C2semidirect
10. #20274916C2semidirect
11. #20274811C2semidirect
12. #20274811C2semidirect
13. #20274744C2semidirect
14. #20274432C2semidirect
375. C53#24948semidirect

Maximal subgroups

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

subgroupindex
#202790122
#202776642
#202762792
#202761882
#202761432
#202761382
#202753012
#202752872
#202749162
#202748112
#202747442
#202744322
C53×#18692
#2494853