Finite Group Databasebeta

← #27389656

Subgroups — #27389656

Order 13568.

1964 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic53C211yesmaximal, characteristic, complemented
2C2×#16810757C211yesmaximal, characteristic
3C2×#16810703C212yesmaximal, complemented
4#20277221C211yesmaximal, characteristic, complemented
5#20275061C212yesmaximal, complemented
6#20275030C211yesmaximal, characteristic, complemented
7#20274960C212yesmaximal, complemented
8#20274902C211yesmaximal, characteristic, complemented
9#102×Dic53C212yesmaximal, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1786C211yesmaximal, characteristic, complemented
12C24×Dic53—21no—
13C22×#11719922C2212yes—
14C2×#13133191C2211yescharacteristic
15C2×D4×Dic53—28no—
16C2×D4×Dic53—21no—
17C2×D4×Dic53—21no—
18C2×D4×Dic53C2212yes—
19C2×D4×Dic53C2212yes—
20C2×#13133179C2211yescharacteristic, complemented
21C2×#13133177—21no—
22C22×C4⋊Dic53C2211yescharacteristic
23C22×C22⋊Dic53—21no—
24C22×C4×Dic53C2211yescharacteristic
25#16810757C418yescomplemented
26#16810703—28no—
27C2×#13133150C2212yescomplemented
28C2×#13133139C2212yescomplemented
29C2×#13133138C2212yes—
30C2×C22⋊#88322C2211yescharacteristic
31C2×C8×Dic53C2212yescomplemented
32#16810524C2211yescharacteristic, complemented
33#16810519C2211yescharacteristic, complemented
34#16810512—21no—
35#16810500—21no—
36C4⋊C4×Dic53C2211yescharacteristic, complemented
37#16810470—21no—
38#16810386C2212yescomplemented
39#16810385C2212yescomplemented
40#16810377C2212yescomplemented
41C23×D4×C53C2211yescharacteristic
42C2×#102×C53C2212yes—
43C2×#102×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#380C2211yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C23×Dic53—28no—
49C23×Dic53—21no—
50C23×Dic53—22no—
51C23×Dic53—22no—
52C23×Dic53—44no—
53C23×Dic53—44no—
54C23×Dic53D411yescharacteristic
55C23×Dic53D411yescharacteristic
56C22×C4×D53—21no—
57C2×#11719922—24no—
58C2×#11719922—24no—
59C2×#11719922C2×C418yescomplemented
60C22×#88322C2312yes—
61#13133191—22no—
62#13133191C2×C414yescomplemented
63D4×Dic53—28no—
64D4×Dic53—28no—
65D4×Dic53—24no—
66D4×Dic53—24no—
67D4×Dic53—24no—
68D4×Dic53D418yescomplemented
69#13133179—24no—
70#13133177—44no—
71C2×C4⋊Dic53—24no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2312yes—
75C2×#11719919—21no—
76C2×C22⋊Dic53—28no—
77C2×C22⋊Dic53—22no—
78C2×C22⋊Dic53—22no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—24no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133150—24no—
90#13133139—24no—
91#13133138C2×C418yescomplemented
92C22⋊#88322C2×C414yescomplemented
93C8×Dic53—24no—
94#13133124—21no—
95#13133124—21no—
96C25×C53—21no—
97C22×D4×C53—24no—
98C22×D4×C53—24no—
99C22×D4×C53—21no—
100C22×D4×C53—22no—
101C22×D4×C53C2×C414yescomplemented
102C22×D4×C53C2312yes—
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53C2312yes—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108#102×C53—24no—
109#102×C53—24no—
110C22⋊C8×C53—22no—
111#95×C53—21no—
112C24×C53—24no—
113C24×C53—24no—
114C24×C53—24no—
115C24×C53—22no—
116C24×C53—42no—
117C24×C53—42no—
118C24×C53—42no—
119C24×C53—42no—
120C24×C53C2×D411yescharacteristic
121C23×D53—21no—
122C2×D4×C53—24no—
123C2×D4×C53—216no—
124C2×D4×C53—24no—
125C2×D4×C53—216no—
126C2×D4×C53—24no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53C22×C414yes—
131C2×D4×C53C2×D414yes—
132C2×D4×C53#4114yes—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4112yes—
140C22×C4×C53C2411yescharacteristic
141C2×C22⋊D53—24no—
142C2×C22⋊D53—42no—
143C2×C22⋊D53—42no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—24no—
147C22×Dic53—28no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—28no—
153C22×Dic53—41no—
154C22×Dic53—44no—
155C22×Dic53—432no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—44no—
159C22×Dic53—44no—
160C22×Dic53—44no—
161C22×Dic53D814yescomplemented
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C2×D212—22no—
167C2×D212—22no—
168C2×D212—21no—
169C2×D212C22×C412yescomplemented
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C22⋊C4×C53—42no—
179C22⋊Dic53—28no—
180C22⋊Dic53—28no—
181#11719922—216no—
182#11719921—24no—
183#11719921—42no—
184#11719921—42no—
185#11719921—42no—
186C4⋊Dic53—24no—
187C4⋊Dic53—22no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C412yescomplemented
190C4⋊Dic53C2×D414yes—
191#11719919—42no—
192C4×Dic53—22no—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—42no—
196C4×Dic53C2×D414yes—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322C22×C414yescomplemented
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—216no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—48no—
209C23×C53—48no—
210C23×C53—48no—
211C23×C53—48no—
212C23×C53—48no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—48no—
218C23×C53C4×D412yes—
219C23×C53C2×D812yes—
220C23×C53#13512yes—
221C23×C53C22×D411yescharacteristic
222C22×D53—22no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229D4×C53—216no—
230D4×C53—24no—
231D4×C53—24no—
232D4×C53—24no—
233D4×C53C4×D418yescomplemented
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
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×C53C4×D412yes—
245C2×C4×C53C23×C411yescharacteristic
246C2×C4×C53C22×D412yes—
247C2×C4×C53C2×#4112yes—
248C2×C4×C53C2×#4111yescharacteristic
249C22⋊D53—48no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—28no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—44no—
261C2×Dic53—416no—
262C2×Dic53—48no—
263C2×Dic53—42no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53—416no—
268C2×Dic53#12012yes—
269C2×Dic53#12012yes—
270C2×Dic53C2×D812yes—
271C2×Dic53C2×D812yes—
272C2×Dic53AGL(1,ℤ/8)12yes—
273D212—24no—
274D212—42no—
275C4×D53—84no—
276C8×C53—42no—
277#88322—24no—
278C4×#447—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—28no—
286C22×C53—432no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—44no—
290C22×C53—44no—
291C22×C53—44no—
292C22×C53—44no—
293C22×C53C4×D814yescomplemented
294C22×C53C2×C4×D411yescharacteristic
295C22×C53C2×#12011yescharacteristic
296C22×C53#52511yescharacteristic
297C22×C53C22×D811yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53#57511yescharacteristic
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C2×D53—81no—
308C4×C53—22no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53C2×C4×D412yes—
312C4×C53#52512yes—
313Dic53—22no—
314Dic53—22no—
315Dic53—42no—
316Dic53—44no—
317Dic53—41no—
318Dic53#44714yescomplemented
319C22×C4×D4—531no—
320C2×#447—531no—
321C2×C4×D8—532no—
322C4×#121—531no—
323#1879—532no—
324#1874—531no—
325#1862—532no—
326#1852—531no—
327#1786D5311yescharacteristic, complemented
328C4×#102—532no—
329C4×C22⋊C8—531no—
330C2×C53—24no—
331C2×C53—48no—
332C2×C53C4×#12011yescharacteristic
333C2×C53C2×C4×D811yescharacteristic
334C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
335C2×C53C2×#44711yescharacteristic
336C2×C53#300211yescharacteristic
337C2×C53#310011yescharacteristic
338C2×C53#310511yescharacteristic
339D53—82no—
340C23×D4C2×D5311yescharacteristic
341C24×C4—1061no—
342C22×D8—532no—
343C2×#121—531no—
344C2×C4×D4—532no—
345C2×C4×D4—532no—
346C2×C4×D4—531no—
347C2×C4×D4—1061no—
348C2×C4×D4—1061no—
349C2×C4×D4—1061no—
350C2×C4×D4—1068no—
351C22×C4⋊C4—531no—
352C22×C22⋊C4—1061no—
353C22×C42—531no—
354#447—538no—
355C4×D8—1068no—
356C2×#107—532no—
357C2×#102—532no—
358C2×#102—532no—
359C2×#102C2×D5312yes—
360C2×#102C2×D5312yes—
361C2×C22⋊C8—531no—
362C2×C22⋊C8C2×D5311yescharacteristic
363C2×C4×C8—532no—
364#390—531no—
365#389—531no—
366#388—1061no—
367#386—1061no—
368#380C2×D5311yescharacteristic
369C4×C4⋊C4—531no—
370C4×C22⋊C4—1061no—
371#340—532no—
372#340—532no—
373#336—532no—
374C53C4×#44711yescharacteristic, complemented, Sylow
375C25—21no—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—21no—
379C22×D4—22no—
380C22×D4—531no—
381C22×D4—1061no—
382C22×D4C4×D5314yescomplemented
383C22×D4C22×D5312yes—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—1061no—
387C23×C4—1062no—
388C23×C4—1062no—
389C23×C4—1068no—
390C23×C4—1061no—
391C23×C4—2124no—
392C23×C4—2124no—
393C23×C4C22×D5311yescharacteristic
394C2×D8—538no—
395C2×D8—1064no—
396C2×D8—1064no—
397C22×C8—532no—
398C22×C8C22×D5312yes—
399#121—534no—
400#121—1062no—
401C4×D4—538no—
402C4×D4—1064no—
403C4×D4—1064no—
404C4×D4—1064no—
405C4×D4—1068no—
406C4×D4—1068no—
407C4×D4—1064no—
408C4×D4—2124no—
409C2×C4⋊C4—532no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1068no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1064no—
429C2×C42—1061no—
430#107—1064no—
431#102—24no—
432#102—24no—
433#102—538no—
434#102—1064no—
435C22⋊C8—22no—
436C22⋊C8—534no—
437C4×C8—1064no—
438#95—21no—
439#95—1061no—
440#95—1061no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—22no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—1061no—
450C24D4×D5311yescharacteristic
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—216no—
454C2×D4—216no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—532no—
460C2×D4—1064no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1061no—
464C2×D4—2122no—
465C2×D4—2122no—
466C2×D4C2×C4×D5314yes—
467C2×D4D4×D5314yes—
468C2×D4#1171994714yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—531no—
474C22×C4—534no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1068no—
482C22×C4—1064no—
483C22×C4—1068no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1062no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4—2124no—
491C22×C4—2124no—
492C22×C4—2124no—
493C22×C4—2121no—
494C22×C4—21232no—
495C22×C4—2124no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4C2×C4×D5312yes—
499C22×C4D4×D5311yescharacteristic
500C22×C4#1171994712yes—
501C22×C4C23×D5311yescharacteristic
502D8—10616no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—534no—
507C2×C8—1062no—
508C2×C8—1062no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—532no—
512C4⋊C4—534no—
513C4⋊C4—1064no—
514C4⋊C4—1062no—
515C4⋊C4—1061no—
516C4⋊C4—2122no—
517C22⋊C4—42no—
518C22⋊C4—1068no—
519C22⋊C4—1068no—
520C22⋊C4—1064no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C42—534no—
525C42—1062no—
526C42—1062no—
527C42—1062no—
528C42—2122no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—48no—
547C23—1062no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23#1313317712yes—
555C23D8×D5312yes—
556C23#1313323812yes—
557C23C2×D4×D5311yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—1064no—
563D4—2128no—
564D4—2122no—
565D4#1313317718yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—1068no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1062no—
587C2×C4—1061no—
588C2×C4—1064no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—4244no—
592C2×C4—21216no—
593C2×C4—2124no—
594C2×C4—21216no—
595C2×C4—2128no—
596C2×C4—2121no—
597C2×C4—2122no—
598C2×C4—2121no—
599C2×C4—2122no—
600C2×C4#1313317712yes—
601C2×C4C22×C4×D5311yescharacteristic
602C2×C4C2×D4×D5312yes—
603C2×C4C2×#1171994712yes—
604C2×C4C2×#1171994711yescharacteristic
605C8—42no—
606C8—1064no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—4241no—
622C22—4241no—
623C22—4241no—
624C22—4241no—
625C22—2122no—
626C22—2122no—
627C22—2122no—
628C22#1681070314yescomplemented
629C22C2×#1313317711yescharacteristic
630C22#120×D5311yescharacteristic
631C22#1681112911yescharacteristic
632C22C2×D8×D5311yescharacteristic
633C22C2×#1313323811yescharacteristic
634C22AGL(1,ℤ/8)×D5311yescharacteristic
635C22#1681141911yescharacteristic
636C4—22no—
637C4—81no—
638C4—81no—
639C4—534no—
640C4—1062no—
641C4—1062no—
642C4—2121no—
643C4—2122no—
644C4—2124no—
645C4C2×#1313317712yes—
646C4#1681112912yes—
647C2—24no—
648C2—48no—
649C2—4242no—
650C2#2027721411yescharacteristic
651C2C2×#1681070311yescharacteristic
652C2#2028351911yescharacteristic
653C2#447×D5311yescharacteristic
654C2#2028393411yescharacteristic
655C2#2028471011yescharacteristic
656C2#2028476311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1786C2
socleC23×C53C22×D4
prime radicalC2×#16810757C2
cosocleC22×C4×C53C24
hypercenter#1786D53
hypocenterC53C4×#447
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic53C2semidirect
3. C2×#16810703C2semidirect
4. #20277221C2semidirect
5. #20275061C2semidirect
6. #20275030C2semidirect
7. #20274960C2semidirect
8. #20274902C2semidirect
9. #102×Dic53C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1786C2semidirect
20. C2×#13133179C22semidirect
25. #16810757C4semidirect
27. C2×#13133150C22semidirect
28. C2×#13133139C22semidirect
31. C2×C8×Dic53C22semidirect
32. #16810524C22semidirect
33. #16810519C22semidirect
36. C4⋊C4×Dic53C22semidirect
38. #16810386C22semidirect
39. #16810385C22semidirect
40. #16810377C22semidirect
59. C2×#11719922C2×C4semidirect
62. #13133191C2×C4semidirect
68. D4×Dic53D4semidirect
91. #13133138C2×C4semidirect
92. C22⋊#88322C2×C4semidirect
101. C22×D4×C53C2×C4semidirect
161. C22×Dic53D8semidirect
169. C2×D212C22×C4semidirect
189. C4⋊Dic53C22×C4semidirect
199. C2×#88322C22×C4semidirect
233. D4×C53C4×D4semidirect
293. C22×C53C4×D8semidirect
318. Dic53#447semidirect
327. #1786D53semidirect
374. C53C4×#447semidirect
382. C22×D4C4×D53semidirect
565. D4#13133177semidirect
628. C22#16810703semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic532
C2×#168107572
C2×#168107032
#202772212
#202750612
#202750302
#202749602
#202749022
#102×Dic532
C22⋊C8×Dic532
C53×#17862
C4×#44753