Finite Group Databasebeta

← #27425423

Subgroups — #27425423

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810904C211yesmaximal, characteristic, complemented
2#20280536C211yesmaximal, characteristic, complemented
3#20277408C211yesmaximal, characteristic
4#20276143C211yesmaximal, characteristic, complemented
5#20276138C211yesmaximal, characteristic, complemented
6#20275905C211yesmaximal, characteristic
7#20275898C211yesmaximal, characteristic
8#20275662C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275458C211yesmaximal, characteristic, complemented
11#20275448C211yesmaximal, characteristic, complemented
12#20275412C211yesmaximal, characteristic, complemented
13#20275398C211yesmaximal, characteristic, complemented
14#20274726C211yesmaximal, characteristic, complemented
15C2×C53×#433C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19#16810904—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28#16810562—21no—
29#16810559—21no—
30#16810543—21no—
31#16810535—21no—
32#16810525C2211yescharacteristic
33#16810523—21no—
34#16810521C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810513—21no—
38#16810499—21no—
39#16810494—21no—
40#16810489C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C53×#433—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×D212C2311yescharacteristic
63C22×C22⋊D53—21no—
64C22×C4×D53—21no—
65#13133217—22no—
66#13133217—22no—
67#13133191—22no—
68#13133191—22no—
69C4×D212—22no—
70C4×D212—22no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—21no—
90#13133151—22no—
91#13133151—22no—
92#13133151—22no—
93#13133149—22no—
94#13133139—22no—
95#13133139—22no—
96#13133139—22no—
97#13133139—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#117×C53—22no—
113#117×C53—22no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—22no—
121C4⋊C8×C53D414yescomplemented
122C22⋊C8×C53—22no—
123C22⋊C8×C53—22no—
124#97×C53—22no—
125C4×C8×C53—22no—
126C24×C53C2×D411yescharacteristic
127C23×D53—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×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—24no—
141C2×C22⋊D53—81no—
142C2×C22⋊D53—81no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212#4112yes—
156C2×C4×D53—42no—
157C2×C4×D53—81no—
158C2×C4×D53—81no—
159C2×Dic106—42no—
160C2×Dic106—81no—
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×C53C2×D412yes—
173C2×C8×C53C2×D412yes—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53C2×D412yes—
181C42×C53#4112yes—
182C22⋊Dic53—81no—
183C22⋊Dic53—81no—
184C22⋊Dic53—81no—
185#11719921—24no—
186#11719921—42no—
187#11719921—81no—
188#11719921—81no—
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—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53#4112yes—
204#11719919—81no—
205#11719919—81no—
206#11719919—81no—
207C4×Dic53—81no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53C22×D411yescharacteristic
216C22×D53—22no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
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—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—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53#12112yes—
241C2×C4×C53#12112yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C22⋊D53—84no—
249C2×Dic53—22no—
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—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270D212—22no—
271D212—22no—
272D212—41no—
273D212—41no—
274C4×D53—84no—
275Dic106—84no—
276C8×C53—22no—
277C8×C53—22no—
278C8×C53—41no—
279C8×C53—41no—
280#24712—531nomaximal, Sylow
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×#12111yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53#53611yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57611yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57811yescharacteristic
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—81no—
299C2×D53—81no—
300C2×D53—81no—
301C2×D53—81no—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—41no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53#54711yescharacteristic
308C4×C53#54811yescharacteristic
309C4×C53#55211yescharacteristic
310C4×C53#55611yescharacteristic
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318C2×#474—531no—
319C2×#433D5311yescharacteristic, complemented
320#2628—531no—
321#2324—531no—
322#2076—531no—
323#2075—531no—
324#2034—531no—
325#2033—531no—
326#1987—531no—
327#1949—531no—
328#1944—531no—
329#1942—531no—
330#1936—531no—
331#1934—531no—
332#1804—531no—
333C2×C53—81no—
334C2×C53#241511yescharacteristic
335C2×C53#305811yescharacteristic
336C2×C53#306311yescharacteristic
337C2×C53#311811yescharacteristic
338C2×C53#312411yescharacteristic
339C2×C53#324111yescharacteristic
340C2×C53#324311yescharacteristic
341D53—82no—
342C2×C4⋊Q8—531no—
343C2×#121—531no—
344C2×C4×D4—531no—
345C2×#117C2×D5311yescharacteristic
346#474—1064no—
347#433—24no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#107—531no—
351C2×#106—531no—
352C2×C4⋊C8C2×D5311yescharacteristic
353C2×C4⋊C8C2×D5311yescharacteristic
354C2×#102—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×C22⋊C8C2×D5311yescharacteristic
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×#97C2×D5311yescharacteristic
361C2×C4×C8C2×D5311yescharacteristic
362#400—1061no—
363#399—1061no—
364#395—1061no—
365#393—1061no—
366#390—531no—
367#390—1061no—
368#389—531no—
369#389—531no—
370#389—531no—
371#388—1061no—
372#386—1061no—
373#385—1061no—
374#382—531no—
375#380—531no—
376#380—531no—
377#380—531no—
378C4×C4⋊C4—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—
387C53#2471211yescharacteristic, complemented, Sylow
388C22×Q8—1061no—
389C22×D4—531no—
390C22×D4—1061no—
391C23×C4—1061no—
392C23×C4C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C22×C8C22×D5311yescharacteristic
396C22×C8C22×D5311yescharacteristic
397C4⋊Q8—1062no—
398C4⋊Q8—1062no—
399#121—1062no—
400#121—1062no—
401C4×D4—1062no—
402C4×D4—1062no—
403#117—22no—
404#117—22no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4D21211yescharacteristic
417C2×C4⋊C4D21211yescharacteristic
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4D21211yescharacteristic
426C2×C22⋊C4D21211yescharacteristic
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#107—1062no—
432#107—1062no—
433#106—1062no—
434C4⋊C8—22no—
435C4⋊C8C22⋊D5314yescomplemented
436#102—1062no—
437#102—1062no—
438#102—1062no—
439#102—1062no—
440C22⋊C8—22no—
441C22⋊C8—22no—
442#97—22no—
443C4×C8—22no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453C24—1061no—
454C24C2×D21211yescharacteristic
455C2×Q8—4241no—
456C2×Q8—4241no—
457C2×Q8—2122no—
458C2×D4—532no—
459C2×D4—1064no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—4241no—
466C2×D4—4241no—
467C22×C4—21no—
468C22×C4—21no—
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—4241no—
482C22×C4—2122no—
483C22×C4C2×D21211yescharacteristic
484C22×C4C2×D21211yescharacteristic
485C22×C4C2×D21211yescharacteristic
486C22×C4C2×D21211yescharacteristic
487C22×C4C2×D21211yescharacteristic
488C22×C4C23×D5311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8C2×C22⋊D5312yes—
500C2×C8C2×C22⋊D5312yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—532no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
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—4241no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—2122no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—1064no—
524C22⋊C4—4241no—
525C22⋊C4—4241no—
526C22⋊C4—4241no—
527C22⋊C4—4241no—
528C22⋊C4—4241no—
529C22⋊C4—2122no—
530C42—21no—
531C42—21no—
532C42—4241no—
533C42#1171994512yes—
534C42C2×C22⋊D5312yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—1062no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2121no—
549C23C22×D21211yescharacteristic
550Q8—4244no—
551D4—1062no—
552D4—1062no—
553D4—4244no—
554D4—2121no—
555D4—2121no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1062no—
574C2×C4—4244no—
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×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#1313319112yes—
596C2×C4#1313319112yes—
597C2×C4C22×C22⋊D5311yescharacteristic
598C2×C4#41×D5311yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C2×C4C2×#1171994511yescharacteristic
601C2×C4#1313326411yescharacteristic
602C2×C4#1313326411yescharacteristic
603C8—22no—
604C8—22no—
605C8—41no—
606C8—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4241no—
615C22—4241no—
616C22—4241no—
617C22—4241no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22C2×#1313319111yescharacteristic
622C22#1681113511yescharacteristic
623C22#1681119911yescharacteristic
624C22#1681142811yescharacteristic
625C22#1681142811yescharacteristic
626C22#1681143511yescharacteristic
627C22#1681144011yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—2122no—
640C4#1681127211yescharacteristic
641C4#1681128011yescharacteristic
642C4#1681131211yescharacteristic
643C4#1681134111yescharacteristic
644C2—81no—
645C2—4242no—
646C2#2027813811yescharacteristic
647C2#2028436311yescharacteristic
648C2#2028440611yescharacteristic
649C2#2028490111yescharacteristic
650C2#2028497911yescharacteristic
651C2#2028597011yescharacteristic
652C2#2028600011yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#433C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#433D53
hypocenterC53#24712
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#16810904C2semidirect
2. #20280536C2semidirect
4. #20276143C2semidirect
5. #20276138C2semidirect
8. #20275662C2semidirect
9. #20275485C2semidirect
10. #20275458C2semidirect
11. #20275448C2semidirect
12. #20275412C2semidirect
13. #20275398C2semidirect
14. #20274726C2semidirect
15. C2×C53×#433C2semidirect
121. C4⋊C8×C53D4semidirect
319. C2×#433D53semidirect
387. C53#24712semidirect
435. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
#202805362
#202774082
#202761432
#202761382
#202759052
#202758982
#202756622
#202754852
#202754582
#202754482
#202754122
#202753982
#202747262
C2×C53×#4332
#2471253