Finite Group Databasebeta

← #27314150

Subgroups — #27314150

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810516C212yesmaximal, complemented
2C2×#16810503C212yesmaximal
3#20272983C211yesmaximal, characteristic, complemented
4#20272972C212yesmaximal, complemented
5#20272960C212yesmaximal, complemented
6#20272947C211yesmaximal, characteristic, complemented
7#20272937C212yesmaximal, complemented
8#20272937C212yesmaximal, complemented
9C2×C53×#394C211yesmaximal, characteristic
10C24×Dic53C2212yes—
11C22×C4⋊Dic53C2212yes—
12C22×C22⋊Dic53C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C4×Dic53C2211yescharacteristic
16#16810516—28no—
17#16810503—28no—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C23×D4×C53C2211yescharacteristic
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C22⋊C4×C53C2212yes—
31C22×C22⋊C4×C53C2212yes—
32C53×#394C418yescomplemented
33C2×#95×C53C2211yescharacteristic
34C23×Dic53—24no—
35C23×Dic53—24no—
36C23×Dic53—24no—
37C23×Dic53—42no—
38C23×Dic53—42no—
39C23×Dic53—42no—
40C23×Dic53—42no—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—42no—
47C2×C4⋊Dic53—42no—
48C2×C22⋊Dic53—24no—
49C2×C22⋊Dic53—24no—
50C2×C22⋊Dic53—24no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—42no—
53C2×C22⋊Dic53—42no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53D418yes—
57C2×C4×Dic53—24no—
58C2×C4×Dic53—41no—
59#13133124—24no—
60#13133124—24no—
61#13133124—24no—
62#13133124—24no—
63#13133124—24no—
64#13133124—41no—
65#13133124—41no—
66#13133124—42no—
67#13133124—42no—
68#13133124—41no—
69C25×C53C2312yes—
70C22×D4×C53—24no—
71C22×D4×C53—24no—
72C22×D4×C53—41no—
73C22×D4×C53—41no—
74C22×D4×C53C2×C414yes—
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2312yes—
78C23×C4×C53C2311yescharacteristic
79C2×C4⋊C4×C53—41no—
80C2×C4⋊C4×C53—41no—
81C2×C4⋊C4×C53C2×C414yescomplemented
82C2×C22⋊C4×C53—24no—
83C2×C22⋊C4×C53—24no—
84C2×C22⋊C4×C53—42no—
85C2×C22⋊C4×C53—42no—
86C2×C22⋊C4×C53C2×C418yescomplemented
87C2×C22⋊C4×C53C2×C418yescomplemented
88#95×C53C2×C414yescomplemented
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53C22×C414yes—
102C24×C53C2×D414yes—
103C24×C53#4114yes—
104C24×C53C2411yescharacteristic
105C2×D4×C53—216no—
106C2×D4×C53—44no—
107C2×D4×C53—44no—
108C2×D4×C53—48no—
109C2×D4×C53—44no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53C22×C414yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—48no—
158C22×Dic53—48no—
159C22×Dic53—48no—
160C22×Dic53—48no—
161C22×Dic53—48no—
162C22×Dic53—48no—
163C22×Dic53—48no—
164C22×Dic53C2×D414yes—
165C22×Dic53C2×D414yes—
166C22×Dic53#4114yes—
167C4⋊C4×C53—44no—
168C22⋊C4×C53—216no—
169C22⋊C4×C53—48no—
170C22⋊Dic53—48no—
171C22⋊Dic53—48no—
172C22⋊Dic53—48no—
173C4⋊Dic53—48no—
174C4×Dic53—44no—
175C23×C53—24no—
176C23×C53—24no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53C4×D418yes—
210C23×C53C23×C411yescharacteristic
211C23×C53C22×D412yes—
212C23×C53C2×#4112yes—
213C23×C53C2×#4112yes—
214C23×C53H(5,𝔽2)12yes—
215C23×C53H(5,𝔽2)11yescharacteristic
216C23×C53H(5,𝔽2)11yescharacteristic
217C23×C53H(5,𝔽2)12yes—
218C23×C532−1+412yes—
219D4×C53—416no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—432no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271#14571—531nomaximal, Sylow
272C22×C53—28no—
273C22×C53—28no—
274C22×C53—28no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—44no—
281C22×C53—44no—
282C22×C53—44no—
283C22×C53—44no—
284C22×C53—44no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53C2×C4×D412yes—
288C22×C53C4×#4111yescharacteristic
289C22×C53#51811yescharacteristic
290C22×C53#51811yescharacteristic
291C22×C53#52012yes—
292C22×C53#53412yes—
293C22×C53#53712yes—
294C22×C53#53812yes—
295C22×C53#53812yes—
296C22×C53#53912yes—
297C22×C53#54012yes—
298C22×C53D4211yescharacteristic
299C22×C53#54612yes—
300C22×C53#54612yes—
301C22×C53#54712yes—
302C22×C53#55112yes—
303C22×C53#55111yescharacteristic
304C22×C53#55212yes—
305C22×C53H(3,𝔽4)11yescharacteristic
306C22×C53#56212yes—
307C22×C53#56311yescharacteristic
308C4×C53—44no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53—42no—
312Dic53—44no—
313Dic53—44no—
314Dic53—44no—
315Dic53—82no—
316C2×#394D5311yescharacteristic
317C2×#388—532no—
318C2×#386—532no—
319#1445—531no—
320#1442—532no—
321#1439—532no—
322#1436—531no—
323C22⋊#95—532no—
324C22⋊#95—532no—
325C2×C53—48no—
326C2×C53#232512yes—
327C2×C53#233311yescharacteristic
328C2×C53#235811yescharacteristic
329C2×C53#235912yes—
330C2×C53#237911yescharacteristic
331C2×C53#267112yes—
332C2×C53#268412yes—
333C2×C53#269512yes—
334C2×C53#274312yes—
335C23×D4C2×D5311yescharacteristic
336C24×C4—532no—
337C22×C4⋊C4—532no—
338C22×C4⋊C4C2×D5311yescharacteristic
339C22×C22⋊C4—532no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4C2×D5312yes—
343C22×C22⋊C4C2×D5312yes—
344C22×C42—531no—
345#394Dic5318yescomplemented
346#388—1068no—
347#386—1068no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—532no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—531no—
356C2×#95—532no—
357C2×#95—531no—
358C2×#95C2×D5311yescharacteristic
359C53#1457111yescharacteristic, complemented, Sylow
360C25C22×D5312yes—
361C22×D4—24no—
362C22×D4—24no—
363C22×D4—41no—
364C22×D4—41no—
365C22×D4C2×Dic5314yes—
366C23×C4—532no—
367C23×C4—532no—
368C23×C4—532no—
369C23×C4—532no—
370C23×C4—1064no—
371C23×C4—1064no—
372C23×C4—1064no—
373C23×C4—2122no—
374C23×C4—2122no—
375C23×C4—2122no—
376C23×C4—2122no—
377C23×C4C22×D5312yes—
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C2×C4⋊C4—41no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4—2122no—
386C2×C4⋊C4C2×Dic5314yescomplemented
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—538no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4—1064no—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—2122no—
397C2×C22⋊C4—2122no—
398C2×C22⋊C4—2122no—
399C2×C22⋊C4—2122no—
400C2×C22⋊C4C2×Dic5318yescomplemented
401C2×C22⋊C4C2×Dic5318yescomplemented
402C2×C42—1064no—
403C2×C42—2121no—
404#95—1064no—
405#95—1064no—
406#95—1064no—
407#95—1064no—
408#95—1064no—
409#95—2122no—
410#95—2121no—
411#95—2121no—
412#95—2121no—
413#95—2122no—
414#95C2×Dic5314yescomplemented
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24D4×D5314yes—
428C24#1171994714yes—
429C24C22×Dic5314yes—
430C24C23×D5311yescharacteristic
431C2×D4—216no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—48no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—534no—
455C22×C4—534no—
456C22×C4—534no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—2128no—
483C22×C4—2128no—
484C22×C4—2128no—
485C22×C4—2128no—
486C22×C4—2128no—
487C22×C4—2128no—
488C22×C4—2128no—
489C22×C4C22×Dic5314yes—
490C22×C4C22×Dic5312yes—
491C22×C4C22×Dic5312yes—
492C22×C4C22×Dic5312yes—
493C4⋊C4—44no—
494C4⋊C4—2128no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—2128no—
498C22⋊C4—2128no—
499C22⋊C4—2128no—
500C42—2124no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23D4×Dic5318yes—
536C23C23×Dic5311yescharacteristic
537C23C2×D4×D5312yes—
538C23#41×D5312yes—
539C23C2×#1171994712yes—
540C23#1313326211yescharacteristic
541C23#1313326211yescharacteristic
542C23#1313326212yes—
543C23#1313326312yes—
544C23#1313326412yes—
545D4—416no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—21232no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22C2×D4×Dic5312yes—
613C22#41×Dic5311yescharacteristic
614C22#1681109811yescharacteristic
615C22#1681109811yescharacteristic
616C22#1681110612yes—
617C22#1681118112yes—
618C22#1681120012yes—
619C22#1681120512yes—
620C22#1681121012yes—
621C22#1681121812yes—
622C22#1681122312yes—
623C22#1681125511yescharacteristic
624C22#1681126012yes—
625C22#1681126512yes—
626C22#1681127112yes—
627C22#1681129612yes—
628C22#1681130411yescharacteristic
629C22#1681131012yes—
630C22#1681136511yescharacteristic
631C22#1681136712yes—
632C22#1681137311yescharacteristic
633C4—44no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—4242no—
638C4—2124no—
639C4—2124no—
640C4—2124no—
641C2—48no—
642C2#2027742012yes—
643C2#2027748211yescharacteristic
644C2#2027769911yescharacteristic
645C2#2027771112yes—
646C2#2027788911yescharacteristic
647C2#2028094412yes—
648C2#2028109012yes—
649C2#2028124912yes—
650C2#2028185612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#394C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC2×#394D53
hypocenterC53#14571
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#16810516C2semidirect
3. #20272983C2semidirect
4. #20272972C2semidirect
5. #20272960C2semidirect
6. #20272947C2semidirect
7. #20272937C2semidirect
8. #20272937C2semidirect
32. C53×#394C4semidirect
81. C2×C4⋊C4×C53C2×C4semidirect
86. C2×C22⋊C4×C53C2×C4semidirect
87. C2×C22⋊C4×C53C2×C4semidirect
88. #95×C53C2×C4semidirect
345. #394Dic53semidirect
359. C53#14571semidirect
386. C2×C4⋊C4C2×Dic53semidirect
400. C2×C22⋊C4C2×Dic53semidirect
401. C2×C22⋊C4C2×Dic53semidirect
414. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105162
C2×#168105032
#202729832
#202729722
#202729602
#202729472
#202729372
C2×C53×#3942
#1457153