Finite Group Databasebeta

← #27402509

Subgroups — #27402509

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719924C211yesmaximal, characteristic
2C2×#16810708C211yesmaximal, characteristic
3C2×#16810708C211yesmaximal, characteristic
4#20277633C211yesmaximal, characteristic, complemented
5#20277624C211yesmaximal, characteristic
6#20275442C211yesmaximal, characteristic, complemented
7#20275440C211yesmaximal, characteristic, complemented
8#20274971C211yesmaximal, characteristic, complemented
9#20274965C211yesmaximal, characteristic
10#20274959C211yesmaximal, characteristic
11#20274878C211yesmaximal, characteristic, complemented
12#20274728C211yesmaximal, characteristic, complemented
13#20274724C211yesmaximal, characteristic
14#20274576C211yesmaximal, characteristic, complemented
15C53×#1863C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×#13133179C2211yescharacteristic
22C4×#11719924—24no—
23#16810708—24no—
24#16810708C418yescomplemented
25C2×#13133151C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810521—21no—
39#16810521C2211yescharacteristic
40#16810519—21no—
41#16810519C2211yescharacteristic
42#16810497—21no—
43#16810494—21no—
44#16810488—21no—
45#16810488C2211yescharacteristic
46#16810488C2211yescharacteristic
47#16810480—21no—
48C4×#11719919—21no—
49C4×#11719919C2211yescharacteristic
50#95×D53—21no—
51#16810386C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810377C2211yescharacteristic
55C2×C4×Q8×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#336C2211yescharacteristic
62C22×D212C2311yescharacteristic
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×C4×D53—21no—
66C2×#11719924C2×C414yes—
67C2×#11719924D414yes—
68C2×#11719924Q814yes—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71Q8×Dic53—22no—
72Q8×Dic53—22no—
73Q8×Dic53—41no—
74Q8×Dic53C2×C414yescomplemented
75C4×D212—22no—
76C4×D212—41no—
77#13133179—22no—
78#13133179—22no—
79#13133179—41no—
80#13133179C2×C414yes—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919D411yescharacteristic
90C2×#11719919D411yescharacteristic
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53C2311yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98#13133151—41no—
99#13133148—22no—
100#13133148—22no—
101#13133148C2×C414yescomplemented
102#13133146—22no—
103#13133141—22no—
104#13133139—22no—
105#13133139C2×C414yes—
106#13133137—22no—
107C4×#88322—22no—
108C8×Dic53—22no—
109C8×Dic53C2×C414yescomplemented
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124D411yescharacteristic
118#13133124D411yescharacteristic
119C22×Q8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C4×Q8×C53—22no—
123C4×Q8×C53—41no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C42×C53—21no—
130C2×C42×C53C2311yescharacteristic
131#107×C53—41no—
132#103×C53—22no—
133#103×C53C2×C414yescomplemented
134#95×C53—21no—
135C23×D53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53C22×C412yes—
140C2×Q8×C53C2×D412yes—
141C2×Q8×C53C2×Q812yes—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212C22×C412yes—
160C2×D212C2×D412yes—
161C2×D212C2×Q812yes—
162C2×C4×D53—24no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—81no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C2×C4×D53—81no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53C22×C412yes—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—42no—
183C4⋊C4×C53—81no—
184C4⋊C4×C53C22×C412yes—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—42no—
188#11719924C4⋊C4116yescomplemented
189#11719921—24no—
190#11719921—42no—
191#11719921—42no—
192C4⋊Dic53—21no—
193C4⋊Dic53—24no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53—81no—
200C4⋊Dic53C22×C412yes—
201#11719919—22no—
202#11719919—22no—
203#11719919—42no—
204#11719919—42no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—24no—
208C4×Dic53—42no—
209C4×Dic53C22×C412yes—
210C4×Dic53#4112yes—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C22×C412yes—
215C2×#88322C2×D412yes—
216C2×#88322C2×Q812yes—
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225Q8×C53—22no—
226Q8×C53—22no—
227Q8×C53—22no—
228Q8×C53C2×C4⋊C414yes—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53#13512yes—
247C2×C4×C53C23×C411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×Q811yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—22no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
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×Dic53C4×D412yes—
272C2×Dic53C4×D412yes—
273C2×Dic53#13512yes—
274D212—22no—
275D212—22no—
276D212—22no—
277D212C2×C4⋊C414yes—
278C4×D53—84no—
279C4×D53—84no—
280C4×D53—84no—
281C8×C53—22no—
282C8×C53—41no—
283#88322—41no—
284#88322C2×C4⋊C414yes—
285#22683—531nomaximal, Sylow
286C22×C53C2×C4×D411yescharacteristic
287C22×C53D4211yescharacteristic
288C22×C53D4×Q811yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53#57711yescharacteristic
292C22×C53#57711yescharacteristic
293C2×D53—24no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C4×C53—24no—
301C4×C53—22no—
302C4×C53—42no—
303C4×C53—41no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53C22×C4⋊C411yescharacteristic
307C4×C53#52011yescharacteristic
308C4×C53#54711yescharacteristic
309C4×C53#55411yescharacteristic
310Dic53—22no—
311Dic53—22no—
312Dic53—22no—
313Dic53—42no—
314Dic53—41no—
315Dic53—81no—
316Dic53—81no—
317C2×C4×SD16—531no—
318C2×C4×SD16—531no—
319C2×C4×SD16—531no—
320#2351—531no—
321#2349—531no—
322#1941—531no—
323#1941—531no—
324#1863—531no—
325#1863D5311yescharacteristic, complemented
326#1862—531no—
327#1862—531no—
328#1846—531no—
329#1805—531no—
330#1804—531no—
331C8×C4⋊C4—531no—
332C2×C53D4×C4⋊C411yescharacteristic
333C2×C53C4×#13511yescharacteristic
334C2×C53#294911yescharacteristic
335C2×C53#328211yescharacteristic
336C2×C53#328211yescharacteristic
337C2×C53#338111yescharacteristic
338C2×C53#338111yescharacteristic
339D53—44no—
340C22×SD16—531no—
341C2×C4×Q8—531no—
342C2×C4×Q8—531no—
343C2×C4×Q8C2×D5311yescharacteristic
344C2×C4×D4—531no—
345C2×C4×D4—531no—
346C2×C4×D4—531no—
347C4×SD16—538no—
348C4×SD16—1064no—
349C4×SD16—1064no—
350C2×#107—531no—
351C2×#107C2×D5311yescharacteristic
352C2×#106—531no—
353C2×#106—531no—
354C2×#106—531no—
355C2×C4⋊C8—531no—
356C2×#103—531no—
357C2×#103C2×D5311yescharacteristic
358C2×#103C2×D5311yescharacteristic
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C4×C8—531no—
363C2×C4×C8—531no—
364C2×C4×C8—531no—
365#389—531no—
366#389—531no—
367#389—531no—
368#389—1061no—
369#389—1061no—
370#389C2×D5311yescharacteristic
371#385—1061no—
372#385—1061no—
373#382—531no—
374#382—531no—
375#382—1061no—
376#380—1061no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4—1061no—
379C2×#95—1061no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340C2×D5311yescharacteristic
384#336—531no—
385#336C2×D5311yescharacteristic
386C53#2268311yescharacteristic, complemented, Sylow
387C22×Q8C22×D5311yescharacteristic
388C22×D4—531no—
389C23×C4—1061no—
390C23×C4—1061no—
391C23×C4—1061no—
392C2×SD16—534no—
393C2×SD16—534no—
394C2×SD16—534no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—534no—
402C4×Q8—1062no—
403C4×Q8—1062no—
404C4×Q8—2121no—
405C4×D4—534no—
406C4×D4—1062no—
407C4×D4—1062no—
408C4×D4—1062no—
409C4×D4—2121no—
410C4×D4—2121no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C42—21no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42C22×D5311yescharacteristic
435#107—41no—
436#107—2121no—
437#106—534no—
438#106—1062no—
439#106—1062no—
440C4⋊C8—1062no—
441#103—22no—
442#103—1062no—
443#103C4×D5314yescomplemented
444#102—534no—
445#102—1062no—
446#102—1062no—
447C4×C8—534no—
448C4×C8—1062no—
449C4×C8—1062no—
450#95—21no—
451#95—531no—
452#95—531no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460C24—1061no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C4×D5312yes—
465C2×Q8D4×D5312yes—
466C2×Q8Q8×D5312yes—
467C2×D4—532no—
468C2×D4—532no—
469C2×D4—532no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—1064no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—4241no—
486C22×C4—4241no—
487C22×C4—4241no—
488C22×C4—4241no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4D4×D5311yescharacteristic
494C22×C4D4×D5311yescharacteristic
495C22×C4C23×D5311yescharacteristic
496SD16—5316no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—532no—
503C2×C8—532no—
504C2×C8—532no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8C2×C4×D5312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—532no—
517C4⋊C4—1062no—
518C4⋊C4—1064no—
519C4⋊C4—1062no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—4241no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4C2×C4×D5312yes—
530C22⋊C4—1064no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—532no—
537C42—532no—
538C42—1064no—
539C42—1061no—
540C42—1061no—
541C42—2122no—
542C23—1062no—
543C23—1062no—
544C23—1062no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2121no—
549C23C2×D4×D5311yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C4⋊C4×D5314yes—
554D4—534no—
555D4—1062no—
556D4—1062no—
557D4—1062no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—532no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1062no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—4244no—
589C2×C4—4244no—
590C2×C4—4244no—
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#1313324012yes—
600C2×C4C22×C4×D5311yescharacteristic
601C2×C4C2×D4×D5311yescharacteristic
602C2×C4C2×Q8×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C2×C4#1313326211yescharacteristic
605C2×C4#1313326411yescharacteristic
606C8—22no—
607C8—41no—
608C8—534no—
609C8—2121no—
610C22—1064no—
611C22—2122no—
612C22—2122no—
613C22—2122no—
614C22—2122no—
615C22—2122no—
616C22—2122no—
617C22C4×D4×D5311yescharacteristic
618C22#1681125511yescharacteristic
619C22#1681128611yescharacteristic
620C22#135×D5311yescharacteristic
621C22C2×#1313324011yescharacteristic
622C22#1681143211yescharacteristic
623C22#1681143311yescharacteristic
624C4—24no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—1062no—
631C4—1062no—
632C4—1062no—
633C4—4241no—
634C4—4241no—
635C4—2122no—
636C4—2121no—
637C4C2×C4⋊C4×D5311yescharacteristic
638C4#1681110211yescharacteristic
639C4#1681127111yescharacteristic
640C4#1681132711yescharacteristic
641C2—2124no—
642C2#2027761311yescharacteristic
643C2C4×#1313324011yescharacteristic
644C2#2028357111yescharacteristic
645C2#2028641511yescharacteristic
646C2#2028642511yescharacteristic
647C2#2028758911yescharacteristic
648C2#2028759311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1863C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1863D53
hypocenterC53#22683
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
4. #20277633C2semidirect
6. #20275442C2semidirect
7. #20275440C2semidirect
8. #20274971C2semidirect
11. #20274878C2semidirect
12. #20274728C2semidirect
14. #20274576C2semidirect
15. C53×#1863C2semidirect
24. #16810708C4semidirect
74. Q8×Dic53C2×C4semidirect
101. #13133148C2×C4semidirect
109. C8×Dic53C2×C4semidirect
133. #103×C53C2×C4semidirect
188. #11719924C4⋊C4semidirect
325. #1863D53semidirect
386. C53#22683semidirect
443. #103C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199242
C2×#168107082
#202776332
#202776242
#202754422
#202754402
#202749712
#202749652
#202749592
#202748782
#202747282
#202747242
#202745762
C53×#18632
#2268353