Finite Group Databasebeta

← #27425349

Subgroups — #27425349

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810924C211yesmaximal, characteristic, complemented
2#20283025C211yesmaximal, characteristic, complemented
3C4×#13133191C211yesmaximal, characteristic
4#20276148C211yesmaximal, characteristic, complemented
5#20276138C211yesmaximal, characteristic, complemented
6#20275905C211yesmaximal, characteristic
7#20275898C211yesmaximal, characteristic
8#20275670C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275458C211yesmaximal, characteristic, complemented
11#20275453C211yesmaximal, characteristic, complemented
12#20275412C211yesmaximal, characteristic, complemented
13#20275412C211yesmaximal, characteristic, complemented
14#20274726C211yesmaximal, characteristic, complemented
15C2×C53×#432C211yesmaximal, characteristic, complemented
16C2×#13133209C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×C22⋊D53—21no—
20#16810924—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29#16810525C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810514—21no—
34#16810507—21no—
35#16810507—21no—
36#16810502—21no—
37#16810496—21no—
38#16810496—21no—
39#16810489C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×#11719921—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×C42×C53C2211yescharacteristic
54C53×#432—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×#97×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C4×D53—21no—
64#13133209—22no—
65#13133209—22no—
66#13133191—22no—
67#13133191—22no—
68C4×D212—22no—
69C4×D212—22no—
70C4×C22⋊D53—44no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×C4×Dic53—21no—
91#13133151—22no—
92#13133151—22no—
93#13133149—22no—
94#13133149—22no—
95#13133139—22no—
96#13133139—22no—
97#13133139—22no—
98#13133139—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53D411yescharacteristic
109C23×C4×C53D411yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C42×C53—22no—
116C2×C42×C53—22no—
117C2×C42×C53D411yescharacteristic
118C2×C42×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C4⋊C8×C53—22no—
122C4⋊C8×C53D414yescomplemented
123C22⋊C8×C53—22no—
124C22⋊C8×C53—22no—
125#97×C53—22no—
126#97×C53—22no—
127C24×C53C2×D411yescharacteristic
128C23×D53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—41no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C2×C22⋊D53—24no—
148C2×C22⋊D53—42no—
149C2×C22⋊D53—42no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212#4112yes—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
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×C53C2×D412yes—
177C2×C8×C53C2×D412yes—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53C2×D412yes—
183C42×C53#4112yes—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—42no—
187#11719921—24no—
188#11719921—42no—
189#11719921—42no—
190#11719921—42no—
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—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53#4112yes—
205#11719919—42no—
206#11719919—42no—
207#11719919—42no—
208#11719919—42no—
209#11719919—42no—
210C4×Dic53—42no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53AGL(1,ℤ/8)12yes—
218C23×C53C22×D411yescharacteristic
219C22×D53—22no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
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—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—44no—
240C2×C4×C53—44no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53#12112yes—
246C2×C4×C53#12112yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C22⋊D53—48no—
254C2×Dic53—22no—
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—41no—
275D212—22no—
276D212—22no—
277D212—41no—
278D212—41no—
279C4×D53—84no—
280C8×C53—22no—
281C8×C53—22no—
282C8×C53—41no—
283C8×C53—41no—
284#24706—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C2×#12111yescharacteristic
293C22×C53#52511yescharacteristic
294C22×C53#53611yescharacteristic
295C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C53#57511yescharacteristic
297C22×C53#57511yescharacteristic
298C22×C53#57511yescharacteristic
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53#54711yescharacteristic
312C4×C53#54811yescharacteristic
313C4×C53#55211yescharacteristic
314C4×C53#55611yescharacteristic
315Dic53—42no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322C2×#478—531no—
323C2×#432D5311yescharacteristic, complemented
324#2851—531no—
325C4×#121—531no—
326#2077—531no—
327#2075—531no—
328#2034—531no—
329#2033—531no—
330#1989—531no—
331#1949—531no—
332#1944—531no—
333#1943—531no—
334#1936—531no—
335#1936—531no—
336#1804—531no—
337C2×C53—42no—
338C2×C53#241511yescharacteristic
339C2×C53#305411yescharacteristic
340C2×C53#305511yescharacteristic
341C2×C53#311111yescharacteristic
342C2×C53#311411yescharacteristic
343C2×C53#323411yescharacteristic
344C2×C53#324011yescharacteristic
345D53—82no—
346C2×#125—531no—
347C2×#121—531no—
348C2×C4×D4—531no—
349C2×C4×D4—1061no—
350C22×C42C2×D5311yescharacteristic
351#478—1064no—
352#432—24no—
353C2×#107—531no—
354C2×#107—531no—
355C2×#106—531no—
356C2×#106—531no—
357C2×C4⋊C8C2×D5311yescharacteristic
358C2×C4⋊C8C2×D5311yescharacteristic
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×C22⋊C8C2×D5311yescharacteristic
364C2×C22⋊C8C2×D5311yescharacteristic
365C2×#97C2×D5311yescharacteristic
366C2×#97C2×D5311yescharacteristic
367#390—531no—
368#389—531no—
369#389—531no—
370#389—531no—
371#388—1061no—
372#387—1061no—
373#387—1061no—
374#386—1061no—
375#385—1061no—
376#385—1061no—
377#382—531no—
378#380—531no—
379#380—531no—
380#380—531no—
381C4×C4⋊C4—531no—
382C4×C22⋊C4—1061no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391C53#2470611yescharacteristic, complemented, Sylow
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—1061no—
395C23×C4D21211yescharacteristic
396C23×C4D21211yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402#125—1062no—
403#125—1062no—
404#121—1062no—
405#121—1062no—
406C4×D4—1062no—
407C4×D4—1062no—
408C4×D4—2124no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—22no—
429C2×C42—22no—
430C2×C42—1061no—
431C2×C42D21211yescharacteristic
432C2×C42D21211yescharacteristic
433C2×C42C22×D5311yescharacteristic
434C2×C42C22×D5311yescharacteristic
435#107—1062no—
436#107—1062no—
437#106—1062no—
438#106—1062no—
439C4⋊C8—22no—
440C4⋊C8C22⋊D5314yescomplemented
441#102—1062no—
442#102—1062no—
443#102—1062no—
444#102—1062no—
445C22⋊C8—22no—
446C22⋊C8—22no—
447#97—22no—
448#97—22no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458C24—1061no—
459C24C2×D21211yescharacteristic
460C2×D4—532no—
461C2×D4—1064no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—2122no—
468C2×D4—2122no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4C2×D21211yescharacteristic
492C22×C4C2×D21211yescharacteristic
493C22×C4C2×D21211yescharacteristic
494C22×C4C2×D21211yescharacteristic
495C22×C4C2×D21211yescharacteristic
496C22×C4C23×D5311yescharacteristic
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—21no—
506C2×C8—21no—
507C2×C8C2×C22⋊D5312yes—
508C2×C8C2×C22⋊D5312yes—
509C4⋊C4—532no—
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—1061no—
522C4⋊C4—1061no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C22⋊C4—1064no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—2122no—
540C42#1171994512yes—
541C42C2×C22⋊D5312yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—1062no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23#1313324512yes—
556C23C22×D21211yescharacteristic
557D4—1062no—
558D4—1062no—
559D4—2128no—
560D4—2121no—
561D4—2121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
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—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—1062no—
582C2×C4—4244no—
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—2121no—
603C2×C4#1313319112yes—
604C2×C4#1313319112yes—
605C2×C4C22×C22⋊D5311yescharacteristic
606C2×C4#41×D5311yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4C2×#1171994511yescharacteristic
609C2×C4#1313326411yescharacteristic
610C2×C4#1313326411yescharacteristic
611C8—22no—
612C8—22no—
613C8—41no—
614C8—41no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—4241no—
623C22—4241no—
624C22—4241no—
625C22—4241no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22C2×#1313319111yescharacteristic
630C22#1681113511yescharacteristic
631C22#1681119911yescharacteristic
632C22C2×#1313324511yescharacteristic
633C22#1681142311yescharacteristic
634C22#1681142311yescharacteristic
635C22#1681142311yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—4241no—
642C4—4241no—
643C4—4241no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—2122no—
648C4#1681127211yescharacteristic
649C4#1681128011yescharacteristic
650C4#1681131211yescharacteristic
651C4#1681134111yescharacteristic
652C2—42no—
653C2—4242no—
654C2#2027813811yescharacteristic
655C2#2028432411yescharacteristic
656C2#2028433411yescharacteristic
657C2#2028482611yescharacteristic
658C2#2028485511yescharacteristic
659C2#2028590711yescharacteristic
660C2#2028595711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#432C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#432D53
hypocenterC53#24706
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
1. C2×#16810924C2semidirect
2. #20283025C2semidirect
4. #20276148C2semidirect
5. #20276138C2semidirect
8. #20275670C2semidirect
9. #20275485C2semidirect
10. #20275458C2semidirect
11. #20275453C2semidirect
12. #20275412C2semidirect
13. #20275412C2semidirect
14. #20274726C2semidirect
15. C2×C53×#432C2semidirect
122. C4⋊C8×C53D4semidirect
323. C2×#432D53semidirect
391. C53#24706semidirect
440. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109242
#202830252
C4×#131331912
#202761482
#202761382
#202759052
#202758982
#202756702
#202754852
#202754582
#202754532
#202754122
#202747262
C2×C53×#4322
#2470653