Finite Group Databasebeta

← #27466292

Subgroups — #27466292

Order 13568.

872 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279186C211yesmaximal, characteristic
2#20278417C211yesmaximal, characteristic, complemented
3#20276274C211yesmaximal, characteristic, complemented
4#20276274C211yesmaximal, characteristic, complemented
5#20276120C211yesmaximal, characteristic, complemented
6#20275852C211yesmaximal, characteristic, complemented
7#20275676C211yesmaximal, characteristic, complemented
8#20275398C211yesmaximal, characteristic, complemented
9#20275398C211yesmaximal, characteristic, complemented
10#20275143C211yesmaximal, characteristic
11#20275030C211yesmaximal, characteristic
12#20274843C211yesmaximal, characteristic, complemented
13#20274802C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic, complemented
15C8×C22⋊C4×C53C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133201—21no—
18C2×#13133201—21no—
19C2×#13133194—21no—
20C2×#13133191C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27#16810560—21no—
28#16810560—21no—
29#16810555—21no—
30#16810544—21no—
31#16810525C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810498—21no—
36#16810493C2211yescharacteristic
37#16810493C2211yescharacteristic
38#16810493C2211yescharacteristic
39#16810483C2211yescharacteristic
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×C22⋊C8×C53C2211yescharacteristic
54C2×C4×C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C23×Dic53—21no—
63C23×Dic53—21no—
64#13133217—22no—
65#13133217—41no—
66#13133201—44no—
67#13133201—44no—
68#13133194—44no—
69#13133191—22no—
70#13133191—22no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—22no—
81C2×C4⋊Dic53—22no—
82C2×C4⋊Dic53—22no—
83C2×C4⋊Dic53—41no—
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×C22⋊Dic53—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×#11719921—21no—
99#13133139—22no—
100#13133139—22no—
101#13133139—41no—
102#13133139—41no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53—22no—
108C22×C8×C53—22no—
109C22×C8×C53—41no—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C22⋊C8×C53—22no—
119C4×C8×C53—22no—
120C4×C8×C53—41no—
121#95×C53D411yescharacteristic
122#95×C53D411yescharacteristic
123C24×C53C2×D411yescharacteristic
124C23×D53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—42no—
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—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×Dic106—42no—
160C2×Dic106—42no—
161C2×Dic106—81no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
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—44no—
174C2×C8×C53#4112yes—
175C22⋊C4×C53D814yes—
176C22⋊C4×C53SD1614yes—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C22⋊Dic53—42no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185#11719921—42no—
186#11719921—42no—
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—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—42no—
204C4⋊Dic53—42no—
205C4⋊Dic53—42no—
206C4⋊Dic53—42no—
207C4⋊Dic53—42no—
208C4⋊Dic53—81no—
209C4⋊Dic53#4112yes—
210#11719919—42no—
211#11719919—42no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53#12412yes—
217C23×C53C2×D812yes—
218C23×C53C2×SD1612yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C2×C4×C53—21no—
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—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53#12412yes—
242C2×C4×C53C2×D812yes—
243C2×C4×C53C2×D812yes—
244C2×C4×C53C2×SD1612yes—
245C2×C4×C53C2×SD1612yes—
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C22⋊D53—84no—
253C2×Dic53—22no—
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—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—84no—
275C2×Dic53—84no—
276D212—41no—
277D212—41no—
278D212—41no—
279D212—41no—
280Dic106—84no—
281C8×C53—22no—
282C8×C53—41no—
283C8×C53—41no—
284C8×C53—41no—
285#27695—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53#48614yes—
293C22×C53C2×#12411yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53C22×D811yescharacteristic
297C22×C53C22×SD1611yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57811yescharacteristic
300C2×D53—81no—
301C2×D53—81no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—22no—
311C4×C53—22no—
312C4×C53#52911yescharacteristic
313C4×C53#53911yescharacteristic
314C4×C53#55111yescharacteristic
315C4×C53#55211yescharacteristic
316Dic53—42no—
317Dic53—81no—
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323#2501—531no—
324#2440—531no—
325#2099—531no—
326#2099—531no—
327#2071—531no—
328#2023—531no—
329#1990—531no—
330#1934—531no—
331#1934—531no—
332#1893—531no—
333#1874—531no—
334#1834—531no—
335#1823—531no—
336C8×C22⋊C4D5311yescharacteristic, complemented
337C2×#340—531no—
338C2×C53—24no—
339C2×C53#248811yescharacteristic
340C2×C53C2×#48611yescharacteristic
341C2×C53#313711yescharacteristic
342C2×C53#329411yescharacteristic
343C2×C53#329811yescharacteristic
344C2×C53#333411yescharacteristic
345C2×C53#333511yescharacteristic
346D53—161no—
347C23×C8C2×D5311yescharacteristic
348C2×C4⋊Q8—531no—
349C2×#123—1061no—
350C2×#123—1061no—
351C2×C22⋊Q8—1061no—
352C2×#121—531no—
353C22×C4⋊C4—531no—
354C22×C4⋊C4—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×C4×C8C2×D5311yescharacteristic
361C2×C4×C8C2×D5311yescharacteristic
362#399—1061no—
363#399—1061no—
364#398—1061no—
365#395—1061no—
366#390—531no—
367#390—531no—
368#389—531no—
369#389—531no—
370#385—1061no—
371#384—531no—
372#384—531no—
373#384—531no—
374#380—531no—
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336C2×D5311yescharacteristic
389#336C2×D5311yescharacteristic
390C53#2769511yescharacteristic, complemented, Sylow
391C22×Q8—1061no—
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—1061no—
395C23×C4—1061no—
396C23×C4C22×D5311yescharacteristic
397C22×C8—22no—
398C22×C8—22no—
399C22×C8—41no—
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
404C4⋊Q8—1062no—
405C4⋊Q8—2121no—
406#123—2124no—
407#123—2124no—
408C22⋊Q8—2124no—
409#121—1062no—
410#121—1062no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4—1062no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—2121no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4D21211yescharacteristic
440C2×C22⋊C4D21211yescharacteristic
441C2×C42C22×D5311yescharacteristic
442C2×C42C22×D5311yescharacteristic
443#102—1062no—
444#102—1062no—
445#102—2121no—
446#102—2121no—
447C22⋊C8—22no—
448C4×C8—22no—
449C4×C8—41no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95D21211yescharacteristic
454#95D21211yescharacteristic
455C24—1061no—
456C24C2×D21211yescharacteristic
457C2×Q8—4241no—
458C2×Q8—2122no—
459C2×Q8—2122no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—2122no—
467C2×D4—2122no—
468C2×D4—2122no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—4241no—
481C22×C4—2122no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4C2×D21211yescharacteristic
487C22×C4C2×D21211yescharacteristic
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4#1171994512yes—
492C22×C4#1171994512yes—
493C22×C4C23×D5311yescharacteristic
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—44no—
506C2×C8#1171994712yes—
507C4⋊C4—532no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—4241no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C22⋊C4—2122no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C22⋊C4#1171991314yes—
539C22⋊C4D42414yes—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23#1313320512yes—
556C23C2×D42412yes—
557C23C2×#1171991312yes—
558C23C22×D21211yescharacteristic
559Q8—4244no—
560D4—4244no—
561D4—2121no—
562D4—2121no—
563D4—2121no—
564D4—2121no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—1062no—
580C2×C4—4244no—
581C2×C4—4244no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4#1313320512yes—
603C2×C4C2×D42412yes—
604C2×C4C2×D42412yes—
605C2×C4C2×#1171991312yes—
606C2×C4C2×#1171991312yes—
607C2×C4#41×D5311yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4C2×#1171994511yescharacteristic
610C2×C4C2×#1171994511yescharacteristic
611C2×C4#1313326211yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—22no—
614C8—41no—
615C8—41no—
616C8—41no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—4241no—
624C22—4241no—
625C22—4241no—
626C22—4241no—
627C22—4241no—
628C22—4241no—
629C22—4241no—
630C22#1681097014yes—
631C22C2×#1313320511yescharacteristic
632C22#1681126811yescharacteristic
633C22#1681127711yescharacteristic
634C22C22×D42411yescharacteristic
635C22C22×#1171991311yescharacteristic
636C22#1681143511yescharacteristic
637C22#1681144011yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—4241no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—4241no—
648C4—4241no—
649C4—2122no—
650C4#1681115311yescharacteristic
651C4#1681121511yescharacteristic
652C4#1681129911yescharacteristic
653C4#1681130611yescharacteristic
654C2—24no—
655C2—8481no—
656C2#2027903511yescharacteristic
657C2C2×#1681097011yescharacteristic
658C2#2028508211yescharacteristic
659C2#2028652911yescharacteristic
660C2#2028658111yescharacteristic
661C2#2028706311yescharacteristic
662C2#2028707311yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC8×C22⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC8×C22⋊C4D53
hypocenterC53#27695
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20278417C2semidirect
3. #20276274C2semidirect
4. #20276274C2semidirect
5. #20276120C2semidirect
6. #20275852C2semidirect
7. #20275676C2semidirect
8. #20275398C2semidirect
9. #20275398C2semidirect
12. #20274843C2semidirect
13. #20274802C2semidirect
14. C2×#16810386C2semidirect
15. C8×C22⋊C4×C53C2semidirect
336. C8×C22⋊C4D53semidirect
390. C53#27695semidirect

Maximal subgroups

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

subgroupindex
#202791862
#202784172
#202762742
#202761202
#202758522
#202756762
#202753982
#202751432
#202750302
#202748432
#202748022
C2×#168103862
C8×C22⋊C4×C532
#2769553