Finite Group Databasebeta

← #27389666

Subgroups — #27389666

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×#13133179C211yesmaximal, characteristic, complemented
2C2×#16810756C211yesmaximal, characteristic
3C2×#16810703C212yesmaximal, complemented
4#121×Dic53C211yesmaximal, characteristic, complemented
5#20275060C212yesmaximal, complemented
6#20274963C212yesmaximal, complemented
7#20274903C211yesmaximal, characteristic, complemented
8#20274647C211yesmaximal, characteristic, complemented
9#20274529C212yesmaximal, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1874C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C23×C4×D53—21no—
14C22×#11719922C2212yes—
15C2×D4×Dic53—21no—
16C2×D4×Dic53C2211yescharacteristic, complemented
17C2×#13133179—28no—
18C2×#13133179—21no—
19C2×#13133179—21no—
20C2×#13133179C2212yes—
21C2×#13133179C2212yes—
22C22×#11719921—21no—
23C22×C4×Dic53C2211yescharacteristic
24#16810756C418yescomplemented
25#16810703—28no—
26C2×#13133150C2212yescomplemented
27C2×#13133139C2212yes—
28C2×#13133139C2212yes—
29C2×#13133137C2212yes—
30C2×C22⋊#88322C2211yescharacteristic
31C2×C8×Dic53C2212yescomplemented
32#16810526C2211yescharacteristic, complemented
33#16810522C2211yescharacteristic, complemented
34#16810516—21no—
35#16810503—21no—
36#16810483C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic, complemented
38C22⋊C4×Dic53—21no—
39#16810385C2212yescomplemented
40#16810377C2212yescomplemented
41C2×#121×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#102×C53C2212yescomplemented
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yescomplemented
46C24×D53—21no—
47C22×D212—21no—
48C22×D212—22no—
49C22×D212—24no—
50C22×D212—24no—
51C22×D212C2×C414yescomplemented
52C22×D212C2312yes—
53C23×Dic53—21no—
54C23×Dic53D411yescharacteristic
55C23×Dic53D411yescharacteristic
56C22×C4×D53—28no—
57C22×C4×D53—22no—
58C22×C4×D53—22no—
59C22×C4×D53—44no—
60C22×C4×D53—44no—
61C2×#11719922—24no—
62C2×#11719922—24no—
63C2×#11719922C2×C418yescomplemented
64C22×#88322C2312yes—
65D4×Dic53—24no—
66D4×Dic53—44no—
67#13133179—28no—
68#13133179—28no—
69#13133179—24no—
70#13133179—24no—
71#13133179—24no—
72#13133179D418yescomplemented
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×#11719921—28no—
81C2×#11719921—22no—
82C2×#11719921—22no—
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#13133139—24no—
92#13133137C2×C418yescomplemented
93C22⋊#88322C2×C414yescomplemented
94C8×Dic53—24no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C22×D4×C53—21no—
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102C22×C8×C53C2312yes—
103#121×C53—22no—
104#121×C53C2×C414yescomplemented
105C2×C4⋊C4×C53—24no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2312yes—
109C2×C22⋊C4×C53—21no—
110#102×C53—24no—
111C22⋊C8×C53—22no—
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C23×D53—24no—
115C23×D53—24no—
116C23×D53—24no—
117C23×D53—22no—
118C23×D53—42no—
119C23×D53—42no—
120C23×D53—42no—
121C23×D53—42no—
122C2×D4×C53—24no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—21no—
126C2×D4×C53—42no—
127C2×D4×C53—42no—
128C2×D4×C53C22×C412yescomplemented
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—41no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53D814yescomplemented
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×D212—24no—
156C2×D212—24no—
157C2×D212—216no—
158C2×D212—24no—
159C2×D212—22no—
160C2×D212—22no—
161C2×D212—22no—
162C2×D212—216no—
163C2×D212C22×C414yes—
164C2×D212C2×D414yes—
165C2×D212#4114yes—
166C2×C4×D53—28no—
167C2×C4×D53—28no—
168C2×C4×D53—44no—
169C2×C4×D53—432no—
170C2×C4×D53—44no—
171C2×C4×D53—44no—
172C2×C4×D53—44no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53C22×C412yescomplemented
180C4⋊C4×C53C2×D414yes—
181C22⋊C4×C53—24no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186#11719922—216no—
187#11719921—28no—
188#11719921—28no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53—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—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53C4×D412yes—
212C23×C53C2×D812yes—
213C23×C53#13512yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—216no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—48no—
220C22×D53—48no—
221C22×D53—48no—
222C22×D53—48no—
223C22×D53—48no—
224C22×D53—42no—
225C22×D53—42no—
226C22×D53—42no—
227C22×D53—42no—
228C22×D53—48no—
229D4×C53—24no—
230D4×C53—48no—
231D4×C53—42no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—48no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53C4×D412yes—
244C2×C4×C53C23×C411yescharacteristic
245C2×C4×C53C22×D412yes—
246C2×C4×C53C2×#4112yes—
247C2×C4×C53C2×#4111yescharacteristic
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—28no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—44no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—84no—
264C2×Dic53#12012yes—
265C2×Dic53#12012yes—
266C2×Dic53C2×D812yes—
267C2×Dic53C2×D812yes—
268C2×Dic53AGL(1,ℤ/8)12yes—
269D212—216no—
270D212—24no—
271D212—24no—
272D212—24no—
273D212C4×D418yescomplemented
274C4×D53—416no—
275C4×D53—416no—
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—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53C4×D814yescomplemented
293C22×C53C2×C4×D411yescharacteristic
294C22×C53C2×#12011yescharacteristic
295C22×C53#52511yescharacteristic
296C22×C53C22×D811yescharacteristic
297C22×C53C2×#13511yescharacteristic
298C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C53#57511yescharacteristic
300C2×D53—28no—
301C2×D53—432no—
302C2×D53—44no—
303C2×D53—44no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—44no—
307C2×D53—44no—
308C4×C53—22no—
309C4×C53—42no—
310C4×C53—44no—
311C4×C53C2×C4×D412yes—
312C4×C53#52512yes—
313Dic53—22no—
314Dic53—22no—
315Dic53—41no—
316Dic53—81no—
317Dic53—81no—
318Dic53#44714yescomplemented
319C22×C4×D4—531no—
320C2×#447—531no—
321C2×C4×D8—532no—
322C4×#121—531no—
323#1879—532no—
324#1874D5311yescharacteristic, complemented
325#1862—532no—
326#1852—531no—
327#1786—531no—
328C4×#102—532no—
329C4×C22⋊C8—531no—
330C2×C53—24no—
331C2×C53—82no—
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—48no—
340C23×D4—531no—
341C24×C4—1061no—
342C22×D8—532no—
343C2×#121C2×D5311yescharacteristic
344C2×C4×D4—531no—
345C2×C4×D4—532no—
346C2×C4×D4—532no—
347C2×C4×D4—1061no—
348C2×C4×D4—1061no—
349C2×C4×D4—1068no—
350C2×C4×D4—1061no—
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—1061no—
353C22×C42—531no—
354#447—538no—
355C4×D8—1068no—
356C2×#107—532no—
357C2×#102—532no—
358C2×#102—532no—
359C2×#102—532no—
360C2×#102C2×D5312yescomplemented
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#380—531no—
369C4×C4⋊C4—531no—
370C4×C22⋊C4—1061no—
371#340—532no—
372#340C2×D5312yescomplemented
373#336—532no—
374C53C4×#44711yescharacteristic, complemented, Sylow
375C25—1061no—
376C22×D4—21no—
377C22×D4—534no—
378C22×D4—532no—
379C22×D4—1064no—
380C22×D4—1064no—
381C22×D4—1061no—
382C22×D4—1062no—
383C22×D4C22×D5311yescharacteristic
384C23×C4—21no—
385C23×C4—531no—
386C23×C4—531no—
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—22no—
400#121C4×D5314yescomplemented
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—21no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5312yes—
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—538no—
433#102—1064no—
434#102—1064no—
435C22⋊C8—22no—
436C22⋊C8—534no—
437C4×C8—1064no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441C24—21no—
442C24—1064no—
443C24—1064no—
444C24—1064no—
445C24—1062no—
446C24—2122no—
447C24—2122no—
448C24—2122no—
449C24—2122no—
450C24D4×D5311yescharacteristic
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—534no—
458C2×D4—534no—
459C2×D4—534no—
460C2×D4—10616no—
461C2×D4—1064no—
462C2×D4—1064no—
463C2×D4—10616no—
464C2×D4—1064no—
465C2×D4—1062no—
466C2×D4—1062no—
467C2×D4—1062no—
468C2×D4C2×C4×D5312yescomplemented
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—531no—
477C22×C4—534no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—1068no—
482C22×C4—1068no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—2124no—
490C22×C4—2124no—
491C22×C4—2124no—
492C22×C4—21232no—
493C22×C4—2124no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2121no—
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—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—2122no—
515C4⋊C4C2×C4×D5312yescomplemented
516C4⋊C4D4×D5314yes—
517C22⋊C4—24no—
518C22⋊C4—1068no—
519C22⋊C4—1068no—
520C22⋊C4—2122no—
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—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—10616no—
541C23—1064no—
542C23—1064no—
543C23—1064no—
544C23—2128no—
545C23—2128no—
546C23—2128no—
547C23—2128no—
548C23—2128no—
549C23—2122no—
550C23—2122no—
551C23—2122no—
552C23—2122no—
553C23—2128no—
554C23#1313317712yes—
555C23D8×D5312yes—
556C23#1313323812yes—
557C23C2×D4×D5311yescharacteristic
558D4—24no—
559D4—48no—
560D4—42no—
561D4—538no—
562D4—10616no—
563D4—1064no—
564D4—1064no—
565D4—1064no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—532no—
582C2×C4—1068no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—4244no—
591C2×C4—21216no—
592C2×C4—21216no—
593C2×C4—2124no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4#1313317712yes—
601C2×C4C22×C4×D5311yescharacteristic
602C2×C4C2×D4×D5312yes—
603C2×C4#41×D5312yes—
604C2×C4C2×#1171994711yescharacteristic
605C8—42no—
606C8—1064no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—1068no—
621C22—21232no—
622C22—2124no—
623C22—2124no—
624C22—2124no—
625C22—2124no—
626C22—2124no—
627C22—2124no—
628C22#1681070314yescomplemented
629C22C2×#1313317711yescharacteristic
630C22#120×D5311yescharacteristic
631C22#1681113111yescharacteristic
632C22C2×D8×D5311yescharacteristic
633C22C2×#1313323811yescharacteristic
634C22AGL(1,ℤ/8)×D5311yescharacteristic
635C22#1681142211yescharacteristic
636C4—22no—
637C4—42no—
638C4—44no—
639C4—534no—
640C4—1062no—
641C4—1062no—
642C4—4241no—
643C4—4241no—
644C4—2121no—
645C4C4×D4×D5312yes—
646C4#1681113312yes—
647C2—24no—
648C2—82no—
649C2—2128no—
650C2#2027722011yescharacteristic
651C2C2×#1681070311yescharacteristic
652C2#2028352811yescharacteristic
653C2#447×D5311yescharacteristic
654C2#2028393211yescharacteristic
655C2#2028471211yescharacteristic
656C2#2028475911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#16810756C2
cosocleC22×C4×C53C24
hypercenter#1874D53
hypocenterC53C4×#447
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#13133179C2semidirect
3. C2×#16810703C2semidirect
4. #121×Dic53C2semidirect
5. #20275060C2semidirect
6. #20274963C2semidirect
7. #20274903C2semidirect
8. #20274647C2semidirect
9. #20274529C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1874C2semidirect
16. C2×D4×Dic53C22semidirect
24. #16810756C4semidirect
26. C2×#13133150C22semidirect
31. C2×C8×Dic53C22semidirect
32. #16810526C22semidirect
33. #16810522C22semidirect
37. C4⋊C4×Dic53C22semidirect
39. #16810385C22semidirect
40. #16810377C22semidirect
43. C2×#102×C53C22semidirect
45. C53×#340C22semidirect
51. C22×D212C2×C4semidirect
63. C2×#11719922C2×C4semidirect
72. #13133179D4semidirect
92. #13133137C2×C4semidirect
93. C22⋊#88322C2×C4semidirect
104. #121×C53C2×C4semidirect
128. C2×D4×C53C22×C4semidirect
150. C22×Dic53D8semidirect
179. C4⋊C4×C53C22×C4semidirect
199. C2×#88322C22×C4semidirect
273. D212C4×D4semidirect
292. C22×C53C4×D8semidirect
318. Dic53#447semidirect
324. #1874D53semidirect
360. C2×#102C2×D53semidirect
372. #340C2×D53semidirect
374. C53C4×#447semidirect
400. #121C4×D53semidirect
468. C2×D4C2×C4×D53semidirect
515. C4⋊C4C2×C4×D53semidirect
628. C22#16810703semidirect

Maximal subgroups

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

subgroupindex
C22×#131331792
C2×#168107562
C2×#168107032
#121×Dic532
#202750602
#202749632
#202749032
#202746472
#202745292
C22⋊C8×Dic532
C53×#18742
C4×#44753