Finite Group Databasebeta

← #27389213

Subgroups — #27389213

Order 13568.

1518 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811168C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
4#20275661C212yesmaximal, complemented
5#20275640C212yesmaximal, complemented
6#20275033C211yesmaximal, characteristic, complemented
7#20275032C211yesmaximal, characteristic, complemented
8#20275030C211yesmaximal, characteristic, complemented
9#20274542C212yesmaximal, complemented
10#20274539C212yesmaximal, complemented
11C53×#1874C211yesmaximal, characteristic, complemented
12C22×#11719947—21no—
13#16811168—24no—
14#16811168C418yescomplemented
15C2×#13133216C2211yescharacteristic
16C2×#13133213C2211yescharacteristic
17C2×#13133203—21no—
18C2×#13133191C2211yescharacteristic
19C2×#13133189—21no—
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133179C2211yescharacteristic
22C2×#13133177—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×C4×Dic53—21no—
25C22×C4×Dic53C2211yescharacteristic
26#16810677—24no—
27C2×#13133146C2212yes—
28C2×#13133139C2212yes—
29C2×#13133138C2212yes—
30C2×#13133137C2212yes—
31C2×C22⋊#88322C2211yescharacteristic
32C2×#13133127C2212yes—
33C4⋊C4×Dic53—21no—
34C4⋊C4×Dic53—21no—
35C4⋊C4×Dic53C2212yes—
36C4⋊C4×Dic53C2212yes—
37C2×#13133124—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#121×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#102×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719947—28no—
47C2×#11719947—44no—
48C2×#11719947—44no—
49C22×Dic106—21no—
50C22×D212C2311yescharacteristic
51C22×C22⋊D53—21no—
52C22×C22⋊D53—21no—
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C23×Dic53—21no—
56C23×Dic53D411yescharacteristic
57C23×Dic53D411yescharacteristic
58C22×C4×D53—21no—
59C22×#88322C2312yes—
60#13133216—22no—
61#13133216C2×C414yescomplemented
62#13133213—22no—
63#13133213C2×C414yescomplemented
64#13133203—28no—
65#13133191—22no—
66#13133191C2×C414yes—
67#13133189—28no—
68D4×Dic53—22no—
69D4×Dic53C2×C414yescomplemented
70#13133179—22no—
71#13133179C2×C414yescomplemented
72#13133177—28no—
73C2×C4⋊Dic53—24no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—44no—
84C2×C4×Dic53C2×C414yes—
85C2×C4×Dic53D414yes—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133146—24no—
91#13133139—24no—
92#13133138—24no—
93#13133137—24no—
94C22⋊#88322—22no—
95#13133127—24no—
96#13133124—22no—
97#13133124—22no—
98C22×D4×C53—21no—
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102C22×C8×C53C2312yes—
103#121×C53—22no—
104#121×C53C2×C414yes—
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—21no—
115C2×D4×C53—24no—
116C2×D4×C53—21no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53C22×C412yes—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—24no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2411yescharacteristic
133C2×C22⋊D53—24no—
134C2×C22⋊D53—24no—
135C2×C22⋊D53—24no—
136C2×C22⋊D53—42no—
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—42no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—24no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53C22⋊C414yes—
155C22×Dic53C22⋊C414yes—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160#11719947—432no—
161C2×D212—22no—
162C2×D212—22no—
163C2×D212—21no—
164C2×D212C22×C412yes—
165C2×C4×D53—24no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×Dic106—24no—
169C2×Dic106—42no—
170C2×Dic106—42no—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53C22×C412yes—
178C4⋊C4×C53#4114yescomplemented
179C22⋊C4×C53—24no—
180C22⋊Dic53—24no—
181#11719921—24no—
182#11719921—24no—
183C4⋊Dic53—24no—
184C4⋊Dic53—21no—
185C4⋊Dic53—22no—
186C4⋊Dic53C22×C412yes—
187C4⋊Dic53#4114yescomplemented
188#11719919—24no—
189C4×Dic53—44no—
190C4×Dic53—44no—
191C4×Dic53C22⋊C414yes—
192C4×Dic53C22⋊C414yes—
193C4×Dic53C22×C412yes—
194C4×Dic53C22×C412yes—
195C4×Dic53C2×D412yes—
196C4×Dic53C2×D412yes—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
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×C53C2×C22⋊C412yes—
212C23×C53AGL(1,ℤ/8)12yes—
213C23×C53AGL(1,ℤ/8)12yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—22no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222D4×C53—24no—
223D4×C53—48no—
224D4×C53—42no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—48no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4112yes—
240C2×C4×C53C2×#4112yes—
241C22⋊D53—48no—
242C22⋊D53—48no—
243C22⋊D53—48no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—28no—
251C2×Dic53—28no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—48no—
255C2×Dic53—48no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53C2×C22⋊C412yes—
261C2×Dic53C2×C22⋊C412yes—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×C22⋊C412yes—
264C2×Dic53AGL(1,ℤ/8)12yes—
265C2×Dic53#13712yes—
266D212—24no—
267D212—42no—
268C4×D53—48no—
269Dic106—48no—
270C8×C53—42no—
271#88322—42no—
272#21395—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53#41814yes—
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53#52511yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C53C2×#13711yescharacteristic
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—81no—
298C2×D53—81no—
299C2×D53—81no—
300C2×D53—81no—
301C4×C53—22no—
302C4×C53—44no—
303C4×C53—42no—
304C4×C53C4×#4112yes—
305C4×C53#53212yes—
306Dic53—22no—
307Dic53—22no—
308Dic53—22no—
309Dic53—44no—
310Dic53—42no—
311Dic53#41714yes—
312C2×#532—531no—
313C2×#432—531no—
314C2×C4×C4⋊C4—531no—
315#1987—532no—
316#1982—532no—
317#1874—531no—
318#1874—531no—
319#1874—531no—
320#1874D5311yescharacteristic, complemented
321#1762—532no—
322#1762—532no—
323C2×C53—24no—
324C2×C53—82no—
325C2×C53#229711yescharacteristic
326C2×C53C2×#41811yescharacteristic
327C2×C53C2×#41711yescharacteristic
328C2×C53#310911yescharacteristic
329C2×C53#310911yescharacteristic
330C2×C53#311211yescharacteristic
331C2×C53#311211yescharacteristic
332D53—82no—
333C22×#41—1061no—
334#532—538no—
335#532—1064no—
336C2×C4⋊Q8—531no—
337C2×#127—531no—
338C2×#124—1061no—
339C2×#121—531no—
340C2×#121—1061no—
341C2×#121C2×D5311yescharacteristic
342C2×C4×D4—531no—
343C2×C4×D4—531no—
344C2×C4×D4—1061no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4C2×D5311yescharacteristic
347C22×C42—531no—
348C22×C42—1061no—
349#432—1064no—
350C2×C4⋊C8—532no—
351C2×#102—532no—
352C2×#102—532no—
353C2×#102—532no—
354C2×#102C2×D5312yes—
355C2×C22⋊C8—531no—
356C2×C22⋊C8C2×D5311yescharacteristic
357C2×#97—532no—
358C4×C4⋊C4—532no—
359C4×C4⋊C4—532no—
360C4×C4⋊C4—1061no—
361C4×C4⋊C4—1061no—
362C2×#95—1061no—
363#340—532no—
364#340—532no—
365#340—532no—
366#340C2×D5312yes—
367C53#2139511yescharacteristic, complemented, Sylow
368C2×#41—1068no—
369C2×#41—2124no—
370C2×#41—2124no—
371C22×Q8—1061no—
372C22×D4—21no—
373C22×D4—531no—
374C22×D4—1061no—
375C22×D4—1061no—
376C22×D4—1061no—
377C22×D4C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—1061no—
382C23×C4—1061no—
383C23×C4—1061no—
384C23×C4C22×D5311yescharacteristic
385C22×C8—532no—
386C22×C8C22×D5312yes—
387C4⋊Q8—534no—
388C4⋊Q8—1062no—
389#127—534no—
390#127—1062no—
391#124—1068no—
392#121—22no—
393#121—534no—
394#121—1068no—
395#121—1062no—
396#121C4×D5314yes—
397C4×D4—534no—
398C4×D4—534no—
399C4×D4—1068no—
400C4×D4—1062no—
401C4×D4—1062no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—532no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5312yes—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—534no—
416C2×C42—531no—
417C2×C42—531no—
418C2×C42—534no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1062no—
422C2×C42—1062no—
423C2×C42—2124no—
424C4⋊C8—1064no—
425#102—24no—
426#102—1064no—
427#102—1064no—
428#102—1064no—
429C22⋊C8—22no—
430C22⋊C8—1062no—
431#97—1064no—
432#95—1062no—
433#95—1062no—
434C24—21no—
435C24—1061no—
436C24D4×D5311yescharacteristic
437#41—21232no—
438C2×Q8—1064no—
439C2×Q8—2122no—
440C2×Q8—2122no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—532no—
448C2×D4—1064no—
449C2×D4—1064no—
450C2×D4—1064no—
451C2×D4—1061no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—2122no—
455C2×D4—2122no—
456C2×D4—2122no—
457C2×D4—2122no—
458C2×D4—2122no—
459C2×D4—2122no—
460C2×D4C2×C4×D5312yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—531no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—1064no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1064no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1064no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4C2×C4×D5312yes—
490C22×C4D4×D5311yescharacteristic
491C22×C4D4×D5312yes—
492C22×C4C23×D5311yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1062no—
497C2×C8—1062no—
498C2×C8—1062no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—532no—
503C4⋊C4—534no—
504C4⋊C4—1064no—
505C4⋊C4—1064no—
506C4⋊C4—1061no—
507C4⋊C4—1062no—
508C4⋊C4C2×C4×D5312yes—
509C4⋊C4#1171994714yescomplemented
510C22⋊C4—24no—
511C22⋊C4—1064no—
512C22⋊C4—1064no—
513C22⋊C4—1064no—
514C42—534no—
515C42—534no—
516C42—532no—
517C42—532no—
518C42—532no—
519C42—532no—
520C42—2124no—
521C42—2124no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—1062no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2121no—
539C23—2121no—
540C23C22⋊C4×D5312yes—
541C23#1313324112yes—
542C23#1313324412yes—
543C23C2×D4×D5311yescharacteristic
544Q8—2128no—
545D4—24no—
546D4—48no—
547D4—42no—
548D4—1064no—
549D4—2128no—
550D4—2128no—
551D4—2128no—
552D4—2122no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—532no—
565C2×C4—532no—
566C2×C4—532no—
567C2×C4—532no—
568C2×C4—532no—
569C2×C4—532no—
570C2×C4—1068no—
571C2×C4—1068no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1062no—
579C2×C4—1064no—
580C2×C4—2128no—
581C2×C4—2128no—
582C2×C4—2128no—
583C2×C4—2122no—
584C2×C4—2121no—
585C2×C4—2122no—
586C2×C4—2121no—
587C2×C4C22⋊C4×D5312yes—
588C2×C4C22×C4×D5311yescharacteristic
589C2×C4C2×D4×D5311yescharacteristic
590C2×C4#41×D5312yes—
591C2×C4C2×#1171994712yes—
592C8—42no—
593C8—2122no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—4241no—
608C22—4241no—
609C22—4241no—
610C22—4241no—
611C22—2122no—
612C22—2122no—
613C22—2122no—
614C22#1681062614yes—
615C22C2×C22⋊C4×D5311yescharacteristic
616C22#1681112911yescharacteristic
617C22#1681113111yescharacteristic
618C22AGL(1,ℤ/8)×D5311yescharacteristic
619C22C2×#1313324111yescharacteristic
620C22C2×#1313324411yescharacteristic
621C22#137×D5311yescharacteristic
622C4—22no—
623C4—42no—
624C4—44no—
625C4—534no—
626C4—1062no—
627C4—1062no—
628C4—1062no—
629C4—2122no—
630C4—2124no—
631C4C4×#1171994712yes—
632C4#1681116812yes—
633C2—24no—
634C2—82no—
635C2—4242no—
636C2#2027720011yescharacteristic
637C2C2×#1681062611yescharacteristic
638C2#417×D5311yescharacteristic
639C2#2028480111yescharacteristic
640C2#2028480311yescharacteristic
641C2#2028483211yescharacteristic
642C2#2028483411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#16811168C2
cosocleC22×C4×C53C24
hypercenter#1874D53
hypocenterC53#21395
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
2. C2×#16810677C2semidirect
3. C2×C4⋊C4×Dic53C2semidirect
4. #20275661C2semidirect
5. #20275640C2semidirect
6. #20275033C2semidirect
7. #20275032C2semidirect
8. #20275030C2semidirect
9. #20274542C2semidirect
10. #20274539C2semidirect
11. C53×#1874C2semidirect
14. #16811168C4semidirect
61. #13133216C2×C4semidirect
63. #13133213C2×C4semidirect
69. D4×Dic53C2×C4semidirect
71. #13133179C2×C4semidirect
178. C4⋊C4×C53#41semidirect
187. C4⋊Dic53#41semidirect
320. #1874D53semidirect
367. C53#21395semidirect
509. C4⋊C4#11719947semidirect

Maximal subgroups

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

subgroupindex
C2×#168111682
C2×#168106772
C2×C4⋊C4×Dic532
#202756612
#202756402
#202750332
#202750322
#202750302
#202745422
#202745392
C53×#18742
#2139553