Finite Group Databasebeta

← #27391294

Subgroups — #27391294

Order 13568.

1048 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810943C211yesmaximal, characteristic
2#20277223C211yesmaximal, characteristic
3C4×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#20275336C211yesmaximal, characteristic, complemented
5#20275335C211yesmaximal, characteristic, complemented
6#20275297C211yesmaximal, characteristic, complemented
7#20275031C211yesmaximal, characteristic
8#20274904C211yesmaximal, characteristic, complemented
9#20274804C211yesmaximal, characteristic, complemented
10#20274615C211yesmaximal, characteristic, complemented
11#20274611C211yesmaximal, characteristic, complemented
12#20274539C211yesmaximal, characteristic, complemented
13#20274539C211yesmaximal, characteristic, complemented
14#20274476C211yesmaximal, characteristic, complemented
15C53×#1920C211yesmaximal, characteristic, complemented
16C2×#13133192C2211yescharacteristic
17C2×#13133179—21no—
18C2×#13133179C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C42×D53—21no—
22C2×C42×D53C2211yescharacteristic
23#16810943C418yescomplemented
24C2×#13133151C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133129C2211yescharacteristic
32C2×#13133129C2211yescharacteristic
33C2×#13133127C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35#16810525C2211yescharacteristic
36#16810513—21no—
37#16810501—21no—
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719919—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44#16810474—21no—
45#16810470—21no—
46#95×D53—21no—
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×D212—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×C4×D53D411yescharacteristic
66C22×C4×D53D411yescharacteristic
67C22×C4×D53C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133192—22no—
71#13133192C2×C414yes—
72#13133179—22no—
73#13133179—22no—
74#13133179—44no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53—22no—
82C4⋊C4×D53—22no—
83C4⋊C4×D53C2×C414yescomplemented
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C42×D53—22no—
97C42×D53—22no—
98C42×D53—44no—
99#13133151—22no—
100#13133150—22no—
101#13133148C2×C414yescomplemented
102#13133139—22no—
103#13133139C2×C414yescomplemented
104#13133137—22no—
105#13133137C2×C414yescomplemented
106#13133129—22no—
107#13133129C2×C414yescomplemented
108#13133127—22no—
109#13133127—22no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53C2311yescharacteristic
124#106×C53C2×C414yescomplemented
125#95×C53—21no—
126#95×C53—21no—
127C23×D53—21no—
128C23×D53C2×D411yescharacteristic
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—24no—
147C2×D212—21no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212C22×C412yes—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—22no—
158C2×C4×D53—24no—
159C2×C4×D53—24no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—41no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170C2×C4×D53C22×C412yes—
171C2×C4×D53#4112yes—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53C22×C412yes—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—42no—
184C4⋊C4×C53C22×C412yes—
185C4⋊C4×C53C22×C412yes—
186C4⋊C4×C53#4112yes—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—42no—
190#11719921—24no—
191#11719921—42no—
192#11719921—42no—
193#11719921—42no—
194#11719921—42no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53C22×C412yes—
200C4⋊Dic53#4112yes—
201#11719919—24no—
202#11719919—24no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53—42no—
209C4×Dic53#4112yes—
210C4×Dic53#4112yes—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322C22×C412yes—
217C23×C53C22×D411yescharacteristic
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—22no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53C4×D412yes—
230C22×D53AGL(1,ℤ/8)12yes—
231C22×D53#13712yes—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53AGL(1,ℤ/8)12yes—
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53C4×D412yes—
272C2×Dic53#12312yes—
273C2×Dic53#12312yes—
274D212—22no—
275D212—22no—
276D212—48no—
277D212—41no—
278D212—41no—
279C4×D53—28no—
280C4×D53—22no—
281C4×D53—22no—
282C4×D53—22no—
283C4×D53—22no—
284C4×D53—44no—
285C4×D53—48no—
286C4×D53—48no—
287C4×D53—84no—
288C8×C53—22no—
289C8×C53—41no—
290#88322—22no—
291#88322—41no—
292#21580—531nomaximal, Sylow
293C22×C53C2×C4×D411yescharacteristic
294C22×C53C2×#12311yescharacteristic
295C22×C53#52511yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C53C2×#13711yescharacteristic
299C22×C53#57511yescharacteristic
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53—22no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—81no—
310C2×D53—81no—
311C2×D53—81no—
312C2×D53—81no—
313C2×D53#44014yes—
314C4×C53—22no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53—81no—
320C4×C53C4×#4111yescharacteristic
321C4×C53C4×#4111yescharacteristic
322C4×C53#52911yescharacteristic
323C4×C53#52911yescharacteristic
324Dic53—22no—
325Dic53—22no—
326Dic53—22no—
327Dic53—42no—
328Dic53—42no—
329Dic53—81no—
330C2×#481—531no—
331C4×#121—531no—
332C2×C4×C4⋊C4—531no—
333#1925—531no—
334#1925—531no—
335#1920—531no—
336#1920D5311yescharacteristic, complemented
337#1874—531no—
338#1852—531no—
339#1824—531no—
340#1776—531no—
341#1776—531no—
342#1762—531no—
343#1762—531no—
344#1749—531no—
345C2×C53C4×#12311yescharacteristic
346C2×C53C2×#44011yescharacteristic
347C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
348C2×C53#309211yescharacteristic
349C2×C53#309411yescharacteristic
350C2×C53#310911yescharacteristic
351C2×C53#311711yescharacteristic
352D53—24no—
353D53—82no—
354C2×#121—531no—
355C2×C4×D4—531no—
356C2×C4×D4—1061no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359C22×C42—531no—
360C22×C42—1061no—
361#481—538no—
362C2×#107—531no—
363C2×#107—531no—
364C2×#106—531no—
365C2×#106C2×D5311yescharacteristic
366C2×#102—531no—
367C2×#102—531no—
368C2×#102—531no—
369C2×#102—531no—
370C2×C22⋊C8—531no—
371C2×C22⋊C8—531no—
372C2×#97—531no—
373C2×#97—531no—
374#390—531no—
375#389C2×D5311yescharacteristic
376#388—1061no—
377#386—1061no—
378#380—531no—
379C4×C4⋊C4—531no—
380C4×C4⋊C4—531no—
381C4×C4⋊C4—531no—
382C4×C4⋊C4—531no—
383C4×C4⋊C4—1061no—
384C4×C4⋊C4—1061no—
385C4×C4⋊C4C2×D5311yescharacteristic
386C4×C22⋊C4—1061no—
387C2×#95—1061no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340C2×D5311yescharacteristic
394#340C2×D5311yescharacteristic
395#340C2×D5311yescharacteristic
396#336—531no—
397#336C2×D5311yescharacteristic
398C53#2158011yescharacteristic, complemented, Sylow
399C22×D4—531no—
400C22×D4—1061no—
401C23×C4—531no—
402C23×C4—531no—
403C23×C4—531no—
404C23×C4—1061no—
405C23×C4—1061no—
406C23×C4—1061no—
407C22×C8—531no—
408C22×C8—531no—
409C22×C8C22×D5311yescharacteristic
410C22×C8C22×D5311yescharacteristic
411#121—534no—
412#121—1062no—
413C4×D4—1062no—
414C4×D4—1062no—
415C4×D4—2124no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—534no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1062no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C42—21no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—1062no—
440C2×C42—1062no—
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—2124no—
446C2×C42C22×D5311yescharacteristic
447#107—1062no—
448#107—1062no—
449#106—534no—
450#106C4×D5314yescomplemented
451#102—534no—
452#102—534no—
453#102—1062no—
454#102—1062no—
455C22⋊C8—534no—
456C22⋊C8—1062no—
457#97—1062no—
458#97—1062no—
459#95—21no—
460#95—21no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466C24—531no—
467C24—1061no—
468C2×D4—532no—
469C2×D4—1064no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4—2122no—
476C2×D4—2122no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1064no—
490C22×C4—1062no—
491C22×C4—1064no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—2121no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4—2122no—
508C22×C4D4×D5311yescharacteristic
509C22×C4D4×D5311yescharacteristic
510C22×C4C23×D5311yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—532no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8C2×C4×D5312yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—532no—
530C4⋊C4—532no—
531C4⋊C4—1064no—
532C4⋊C4—1064no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4C2×C4×D5312yes—
538C4⋊C4C2×C4×D5312yes—
539C4⋊C4#1171994912yes—
540C22⋊C4—1064no—
541C22⋊C4—2122no—
542C22⋊C4—2122no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—532no—
549C42—532no—
550C42—1062no—
551C42—1062no—
552C42—2122no—
553C42—2122no—
554C42—2122no—
555C42—2122no—
556C23—532no—
557C23—532no—
558C23—532no—
559C23—1061no—
560C23—1061no—
561C23—1061no—
562C23—1061no—
563C23—1062no—
564C23—2121no—
565C23—2121no—
566C23—2121no—
567C23—2121no—
568C23—2121no—
569C23—2121no—
570C23C2×D4×D5311yescharacteristic
571D4—1062no—
572D4—1062no—
573D4—2128no—
574D4—2121no—
575D4—2121no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—532no—
594C2×C4—532no—
595C2×C4—532no—
596C2×C4—1068no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1062no—
604C2×C4—1062no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—4244no—
610C2×C4—2124no—
611C2×C4—2128no—
612C2×C4—2128no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4#1313324412yes—
621C2×C4C22×C4×D5311yescharacteristic
622C2×C4#41×D5311yescharacteristic
623C2×C4#41×D5311yescharacteristic
624C2×C4#41×D5311yescharacteristic
625C2×C4#41×D5311yescharacteristic
626C2×C4C2×#1171994911yescharacteristic
627C8—22no—
628C8—41no—
629C8—1062no—
630C8—2121no—
631C22—534no—
632C22—1062no—
633C22—1062no—
634C22—1062no—
635C22—1062no—
636C22—1062no—
637C22—1062no—
638C22—4241no—
639C22—4241no—
640C22—4241no—
641C22—4241no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22C4×D4×D5311yescharacteristic
646C22#123×D5311yescharacteristic
647C22#1681113611yescharacteristic
648C22AGL(1,ℤ/8)×D5311yescharacteristic
649C22C2×#1313324411yescharacteristic
650C22#137×D5311yescharacteristic
651C22#1681142011yescharacteristic
652C4—22no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—81no—
658C4—1062no—
659C4—1062no—
660C4—1062no—
661C4—4241no—
662C4—2122no—
663C4—2122no—
664C4#1681109111yescharacteristic
665C4C4×#1171994911yescharacteristic
666C4#1681115611yescharacteristic
667C4#1681115811yescharacteristic
668C2—1064no—
669C2—4242no—
670C2#2027723611yescharacteristic
671C2#440×D5311yescharacteristic
672C2C4×#1313324411yescharacteristic
673C2#2028463911yescharacteristic
674C2#2028465411yescharacteristic
675C2#2028480811yescharacteristic
676C2#2028489111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1920C2
socleC23×C53C22×D4
prime radicalC2×#13133192C22
cosocleC22×C4×C53C24
hypercenter#1920D53
hypocenterC53#21580
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
3. C4×C4⋊C4×D53C2semidirect
4. #20275336C2semidirect
5. #20275335C2semidirect
6. #20275297C2semidirect
8. #20274904C2semidirect
9. #20274804C2semidirect
10. #20274615C2semidirect
11. #20274611C2semidirect
12. #20274539C2semidirect
13. #20274539C2semidirect
14. #20274476C2semidirect
15. C53×#1920C2semidirect
23. #16810943C4semidirect
83. C4⋊C4×D53C2×C4semidirect
101. #13133148C2×C4semidirect
103. #13133139C2×C4semidirect
105. #13133137C2×C4semidirect
107. #13133129C2×C4semidirect
124. #106×C53C2×C4semidirect
336. #1920D53semidirect
398. C53#21580semidirect
450. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109432
#202772232
C4×C4⋊C4×D532
#202753362
#202753352
#202752972
#202750312
#202749042
#202748042
#202746152
#202746112
#202745392
#202744762
C53×#19202
#2158053