Finite Group Databasebeta

← #27440652

Subgroups — #27440652

Order 13568.

1442 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810982C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3#20279888C211yesmaximal, characteristic, complemented
4#20275915C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275648C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274655C212yesmaximal, complemented
9#20274646C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#546C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic, complemented
13#16810982—24no—
14#16810903—24no—
15C2×#13133150C2212yescomplemented
16C2×C4⋊#88322C2211yescharacteristic, complemented
17C2×#13133141C2212yescomplemented
18C2×#13133141C2212yescomplemented
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2212yescomplemented
25C2×#13133128C2211yescharacteristic, complemented
26#16810535—21no—
27#16810535—21no—
28#16810526—21no—
29#16810488—21no—
30#16810483C2212yes—
31#16810483C2212yescomplemented
32C4×C4⋊Dic53C2211yescharacteristic, complemented
33#16810385C2212yescomplemented
34C22×#41×C53C2211yescharacteristic
35C23×D4×C53C2211yescharacteristic
36C53×#546—24no—
37C53×#546—24no—
38C2×#124×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53—21no—
42C2×#121×C53C2211yescharacteristic
43C2×#120×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C22×C22⋊C4×C53—21no—
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C4×Dic53—21no—
58#13133150—24no—
59C4⋊#88322—22no—
60#13133141—24no—
61#13133141—42no—
62#13133138—24no—
63#13133138—24no—
64#13133138—24no—
65#13133138—42no—
66C22⋊#88322—42no—
67C22⋊#88322—42no—
68#13133128—22no—
69#13133124—21no—
70#13133124—22no—
71#13133124—22no—
72C25×C53—21no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—24no—
76C2×#41×C53—42no—
77C22×Q8×C53C2311yescharacteristic
78C22×D4×C53—21no—
79C22×D4×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53—22no—
82C22×D4×C53—24no—
83C22×D4×C53—21no—
84C22×D4×C53—42no—
85C22×D4×C53D411yescharacteristic
86C22×D4×C53C2311yescharacteristic
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92#124×C53—22no—
93#124×C53D414yes—
94#123×C53—28no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97#121×C53—28no—
98#121×C53—22no—
99#121×C53—22no—
100#120×C53—28no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53D414yes—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—28no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53—24no—
119C24×C53—21no—
120C24×C53—44no—
121C24×C53—42no—
122C24×C53—42no—
123C24×C53—82no—
124C24×C53C2×D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126#41×C53—48no—
127#41×C53—44no—
128#41×C53—44no—
129C2×Q8×C53—21no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53C2×D412yes—
133C2×D4×C53—24no—
134C2×D4×C53—24no—
135C2×D4×C53—22no—
136C2×D4×C53—24no—
137C2×D4×C53—22no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—41no—
144C2×D4×C53—44no—
145C2×D4×C53—48no—
146C2×D4×C53—44no—
147C2×D4×C53—44no—
148C2×D4×C53—44no—
149C2×D4×C53—82no—
150C2×D4×C53C2×D412yes—
151C2×D4×C53C2×D412yes—
152C2×D4×C53#4114yes—
153C22×C4×C53—24no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—22no—
161C22×C4×C53—22no—
162C22×C4×C53—22no—
163C22×C4×C53—22no—
164C22×C4×C53—24no—
165C22×C4×C53—44no—
166C22×C4×C53—82no—
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2×D411yescharacteristic
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2411yescharacteristic
172C22×Dic53—22no—
173C22×Dic53—22no—
174C2×Dic106—42no—
175C2×Dic106—82no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53C2×D412yes—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—22no—
187C42×C53C2×D412yes—
188C42×C53C2×D412yes—
189C22⋊Dic53—82no—
190C22⋊Dic53—82no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—42no—
196C4×Dic53—42no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C23×C53—22no—
204C23×C53—21no—
205C23×C53—22no—
206C23×C53—24no—
207C23×C53—21no—
208C23×C53—22no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—22no—
215C23×C53—48no—
216C23×C53—48no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—42no—
220C23×C53—42no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—42no—
224C23×C53—88no—
225C23×C53—84no—
226C23×C53—88no—
227C23×C53#13712yes—
228C23×C53C22×D411yescharacteristic
229Q8×C53—24no—
230Q8×C53—41no—
231Q8×C53—41no—
232D4×C53—22no—
233D4×C53—24no—
234D4×C53—24no—
235D4×C53—44no—
236D4×C53—48no—
237D4×C53—48no—
238D4×C53—41no—
239D4×C53—41no—
240D4×C53—84no—
241D4×C53#12414yes—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—21no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—48no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—84no—
261C2×C4×C53#12012yes—
262C2×C4×C53#12012yes—
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2×#4112yes—
266C2×C4×C53H(5,𝔽2)12yes—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273Dic106—84no—
274#88322—42no—
275#88322—42no—
276#25897—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—22no—
279C22×C53—24no—
280C22×C53—41no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—41no—
284C22×C53—44no—
285C22×C53—42no—
286C22×C53—44no—
287C22×C53—41no—
288C22×C53—42no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—41no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—816no—
296C22×C53—82no—
297C22×C53—82no—
298C22×C53C2×#12011yescharacteristic
299C22×C53#54612yes—
300C22×C53C2×#13711yescharacteristic
301C22×C53#57511yescharacteristic
302C22×C53#57612yes—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—42no—
308C4×C53—42no—
309C4×C53C2×#12411yescharacteristic
310C4×C53#53511yescharacteristic
311C4×C53#54612yes—
312Dic53—82no—
313Dic53—82no—
314C2×#546D5311yescharacteristic
315C2×#489—531no—
316C2×#474—531no—
317#2563—531no—
318#2036—532no—
319#2033—532no—
320#1983—531no—
321#1955—531no—
322#1788—532no—
323#1786—532no—
324#1762—531no—
325C2×C53—42no—
326C2×C53—44no—
327C2×C53—81no—
328C2×C53—82no—
329C2×C53#243511yescharacteristic
330C2×C53#300711yescharacteristic
331C2×C53#300811yescharacteristic
332C2×C53#331412yes—
333C2×C53#331912yes—
334C22×#41C2×D5311yescharacteristic
335C23×D4C2×D5311yescharacteristic
336#546—24no—
337#546—24no—
338C2×C4⋊Q8—531no—
339C2×#124C2×D5311yescharacteristic
340C2×#123—21no—
341C2×C22⋊Q8C2×D5311yescharacteristic
342C2×#121—21no—
343C2×#121C2×D5311yescharacteristic
344C2×#120—21no—
345C2×C4×D4C2×D5311yescharacteristic
346C2×C4×D4C2×D5311yescharacteristic
347C22×C22⋊C4—21no—
348#489—1064no—
349#474—1064no—
350C2×#107—532no—
351C2×C4⋊C8—531no—
352C2×#103—532no—
353C2×#103—532no—
354C2×#102—532no—
355C2×#102—532no—
356C2×#102—532no—
357C2×#102—532no—
358C2×C22⋊C8—532no—
359C2×C22⋊C8—532no—
360C2×#97—531no—
361#393—1061no—
362#393—1061no—
363#390—1061no—
364#382—1061no—
365#380—532no—
366#380—532no—
367C4×C4⋊C4—531no—
368#340—532no—
369C53#2589711yescharacteristic, complemented, Sylow
370C25—21no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—24no—
374C2×#41—42no—
375C22×Q8—1061no—
376C22×Q8C22×D5311yescharacteristic
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—21no—
383C22×D4—42no—
384C22×D4C22⋊D5311yescharacteristic
385C22×D4C22×D5311yescharacteristic
386C22×D4C22×D5311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C22×C8—532no—
392C22×C8—532no—
393C4⋊Q8—1062no—
394C4⋊Q8—1062no—
395#124—22no—
396#124C22⋊D5314yes—
397#123—28no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400#121—28no—
401#121—22no—
402#121—22no—
403#120—28no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4C22⋊D5314yes—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—532no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4C22⋊D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—28no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4C22⋊D5311yescharacteristic
423C2×C22⋊C4C22⋊D5311yescharacteristic
424C2×C42—1061no—
425C2×C42C22×D5311yescharacteristic
426#107—1064no—
427C4⋊C8—1062no—
428#103—1064no—
429#103—2122no—
430#102—1064no—
431#102—1064no—
432#102—1064no—
433#102—2122no—
434C22⋊C8—2122no—
435C22⋊C8—2122no—
436#97—1062no—
437#95—1061no—
438#95—1062no—
439#95—1062no—
440C24—21no—
441C24—24no—
442C24—21no—
443C24—44no—
444C24—42no—
445C24—42no—
446C24—82no—
447C24C2×C22⋊D5311yescharacteristic
448C24C2×C22⋊D5311yescharacteristic
449#41—48no—
450#41—44no—
451#41—44no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—4242no—
456C2×Q8—2122no—
457C2×Q8C2×C22⋊D5312yes—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—41no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—82no—
475C2×D4#1171994714yes—
476C2×D4C2×C22⋊D5312yes—
477C2×D4C2×C22⋊D5312yes—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499C2×C8—1062no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—2122no—
513C4⋊C4C2×C22⋊D5312yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—4242no—
522C22⋊C4—4242no—
523C42—2122no—
524C42C2×C22⋊D5312yes—
525C42C2×C22⋊D5312yes—
526C23—22no—
527C23—24no—
528C23—21no—
529C23—22no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—88no—
548C23—84no—
549C23—88no—
550C23#1313324712yes—
551C23C22×C22⋊D5311yescharacteristic
552Q8—24no—
553Q8—41no—
554Q8—41no—
555Q8—4244no—
556D4—24no—
557D4—24no—
558D4—22no—
559D4—44no—
560D4—48no—
561D4—48no—
562D4—41no—
563D4—41no—
564D4—84no—
565D4#1313320414yes—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4#1313318612yes—
592C2×C4#1313318612yes—
593C2×C4C22×C22⋊D5311yescharacteristic
594C2×C4C22×C22⋊D5311yescharacteristic
595C2×C4C2×#1171994712yes—
596C2×C4#1313326212yes—
597C8—2122no—
598C8—2122no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—44no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—42no—
617C22—816no—
618C22—82no—
619C22—82no—
620C22C2×#1313318611yescharacteristic
621C22#1681126712yes—
622C22C2×#1313324711yescharacteristic
623C22#1681142111yescharacteristic
624C22#1681142612yes—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—4242no—
632C4—4242no—
633C4C2×#1313320411yescharacteristic
634C4#1681119011yescharacteristic
635C4#1681126712yes—
636C2—42no—
637C2—44no—
638C2—81no—
639C2—82no—
640C2#2027836211yescharacteristic
641C2#2028397511yescharacteristic
642C2#2028398311yescharacteristic
643C2#2028679612yes—
644C2#2028687112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#546C2
socleC23×C53C22×D4
hypercenterC2×#546D53
hypocenterC53#25897
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#16810982C2semidirect
2. C2×#16810903C2semidirect
3. #20279888C2semidirect
4. #20275915C2semidirect
5. #20275894C2semidirect
6. #20275648C2semidirect
7. #20275514C2semidirect
8. #20274655C2semidirect
9. #20274646C2semidirect
10. #20274540C2semidirect
12. C2×#13133217C22semidirect
15. C2×#13133150C22semidirect
16. C2×C4⋊#88322C22semidirect
17. C2×#13133141C22semidirect
18. C2×#13133141C22semidirect
24. C2×C22⋊#88322C22semidirect
25. C2×#13133128C22semidirect
31. #16810483C22semidirect
32. C4×C4⋊Dic53C22semidirect
33. #16810385C22semidirect
369. C53#25897semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
C2×#168109032
#202798882
#202759152
#202758942
#202756482
#202755142
#202746552
#202746462
#202745402
C2×C53×#5462
#2589753