Finite Group Databasebeta

← #27387128

Subgroups — #27387128

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810732C211yesmaximal, characteristic
2#20277173C211yesmaximal, characteristic, complemented
3C4×D4×Dic53C211yesmaximal, characteristic, complemented
4#20275341C211yesmaximal, characteristic, complemented
5#20275336C211yesmaximal, characteristic, complemented
6#20275297C211yesmaximal, characteristic, complemented
7#20275296C211yesmaximal, characteristic, complemented
8#20275063C211yesmaximal, characteristic
9#20275061C211yesmaximal, characteristic
10#20274964C211yesmaximal, characteristic, complemented
11#20274615C211yesmaximal, characteristic, complemented
12#20274542C211yesmaximal, characteristic, complemented
13#20274539C211yesmaximal, characteristic, complemented
14#20274462C211yesmaximal, characteristic, complemented
15C53×#1864C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133175—21no—
21C22×C4×Dic53—21no—
22#16810732C418yescomplemented
23C2×#13133151C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31#16810525C2211yescharacteristic
32#16810524C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810515—21no—
35#16810513—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810470—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×C4×D53—21no—
64C2×#11719922—22no—
65C2×#11719922—22no—
66C2×#11719922C2×C414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69D4×Dic53—22no—
70D4×Dic53—22no—
71D4×Dic53—22no—
72D4×Dic53C2×C414yes—
73#13133179—22no—
74#13133179C2×C414yescomplemented
75#13133175—44no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—44no—
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98#13133151—22no—
99#13133148C2×C414yescomplemented
100#13133139—22no—
101#13133139C2×C414yescomplemented
102#13133138—22no—
103#13133137—22no—
104#13133127—22no—
105#13133127C2×C414yescomplemented
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×D4×C53C2311yescharacteristic
112C23×C4×C53—21no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4×D4×C53—22no—
116C4×D4×C53—22no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C42×C53C2311yescharacteristic
123#106×C53—22no—
124#102×C53—22no—
125#102×C53C2×C414yes—
126#95×C53—21no—
127C24×C53—21no—
128C23×D53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C22×C412yes—
133C2×D4×C53#4112yes—
134C2×D4×C53#4112yes—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—24no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—24no—
148C22×Dic53—21no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212C22×C412yes—
163C2×C4×D53—24no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53C22×C412yes—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53C22×C412yes—
178C4⋊C4×C53#4112yes—
179C22⋊C4×C53—24no—
180C42×C53—21no—
181C42×C53—21no—
182C22⋊Dic53—24no—
183C22⋊Dic53—24no—
184#11719922—28no—
185#11719921—24no—
186#11719921—42no—
187#11719921—42no—
188#11719921—42no—
189#11719921—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—42no—
195C4⋊Dic53C22×C412yes—
196C4⋊Dic53#4112yes—
197C4×Dic53—22no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—22no—
201C4×Dic53—41no—
202C4×Dic53—41no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53C22×C412yes—
208C4×Dic53#4112yes—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C22×C412yes—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—22no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—22no—
233D4×C53#11714yescomplemented
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—48no—
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—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—48no—
268C2×Dic53—48no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53C4×D412yes—
277C2×Dic53C4×D412yes—
278C2×Dic53AGL(1,ℤ/8)12yes—
279D212—22no—
280D212—22no—
281D212—41no—
282D212—41no—
283C4×D53—48no—
284C8×C53—22no—
285C8×C53—41no—
286#88322—22no—
287#88322—41no—
288#21122—531nomaximal, Sylow
289C22×C53—24no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53C2×C4×D411yescharacteristic
297C22×C53#52511yescharacteristic
298C22×C53#52511yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C53#57511yescharacteristic
302C22×C53#57511yescharacteristic
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—81no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C4×C53—22no—
311C4×C53—42no—
312C4×C53—41no—
313C4×C53—42no—
314C4×C53—81no—
315C4×C53C2×#11711yescharacteristic
316C4×C53C4×#4111yescharacteristic
317C4×C53#52911yescharacteristic
318C4×C53#52911yescharacteristic
319Dic53—22no—
320Dic53—22no—
321Dic53—22no—
322Dic53—42no—
323Dic53—41no—
324Dic53—42no—
325Dic53—81no—
326C2×#442—531no—
327#2293—531no—
328C42×D4—531no—
329#1926—531no—
330#1925—531no—
331#1920—531no—
332#1919—531no—
333#1879—531no—
334#1879—531no—
335#1864D5311yescharacteristic, complemented
336#1862—531no—
337#1776—531no—
338#1762—531no—
339#1762—531no—
340#1745—531no—
341C2×C53—44no—
342C2×C53#229011yescharacteristic
343C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
345C2×C53#310911yescharacteristic
346C2×C53#311111yescharacteristic
347C2×C53#311411yescharacteristic
348C2×C53#311511yescharacteristic
349D53—82no—
350C22×D8—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—531no—
354C2×C4×D4C2×D5311yescharacteristic
355C2×#117—1061no—
356C22×C42—1061no—
357#442—538no—
358C2×#107—531no—
359C2×#106—531no—
360C2×#106C2×D5311yescharacteristic
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102C2×D5311yescharacteristic
366C2×#102C2×D5311yescharacteristic
367C2×#97—531no—
368C2×#97—531no—
369#390—531no—
370#390—531no—
371#389—531no—
372#389C2×D5311yescharacteristic
373#388—1061no—
374#388—1061no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—531no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4—531no—
379C4×C22⋊C4—1061no—
380C4×C22⋊C4—1061no—
381C4×C22⋊C4—1061no—
382C4×C22⋊C4—1061no—
383C43—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340C2×D5311yescharacteristic
390#336—531no—
391#336—531no—
392#336—531no—
393#336C2×D5311yescharacteristic
394C53#2112211yescharacteristic, complemented, Sylow
395C22×D4—531no—
396C22×D4C22×D5311yescharacteristic
397C23×C4—21no—
398C23×C4—1061no—
399C23×C4—1061no—
400C23×C4—1061no—
401C2×D8—534no—
402C2×D8—1062no—
403C2×D8—1062no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—534no—
411C4×D4—534no—
412C4×D4—1062no—
413C4×D4—1062no—
414C4×D4—1062no—
415C4×D4—1062no—
416#117—2124no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—1061no—
440C2×C42—1061no—
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42—2124no—
444C2×C42C22×D5311yescharacteristic
445#107—1062no—
446#106—22no—
447#106—534no—
448#102—22no—
449#102—534no—
450#102—1062no—
451#102—1062no—
452#102—1062no—
453#102C4×D5314yes—
454#97—534no—
455#97—1062no—
456#95—21no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462C24—21no—
463C24—1061no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—532no—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4C2×C4×D5312yes—
474C2×D4#1171994712yes—
475C2×D4#1171994912yes—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—1064no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4D4×D5311yescharacteristic
499C22×C4D4×D5311yescharacteristic
500C22×C4C23×D5311yescharacteristic
501D8—1068no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—532no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8C2×C4×D5312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—532no—
520C4⋊C4—532no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—2122no—
526C4⋊C4C2×C4×D5312yes—
527C4⋊C4#1171994712yes—
528C22⋊C4—24no—
529C22⋊C4—1064no—
530C22⋊C4—1064no—
531C22⋊C4—1064no—
532C22⋊C4—2122no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C42—21no—
537C42—21no—
538C42—532no—
539C42—532no—
540C42—1062no—
541C42—1062no—
542C42—1061no—
543C42—1061no—
544C42—2121no—
545C42—2121no—
546C42—2122no—
547C42—2122no—
548C42—2122no—
549C42—2122no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—1062no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23—2121no—
564C23C2×D4×D5311yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—1062no—
569D4—1062no—
570D4—2121no—
571D4—2121no—
572D4#1313317514yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—532no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1062no—
598C2×C4—1061no—
599C2×C4—1062no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—2128no—
603C2×C4—2128no—
604C2×C4—2128no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4#1313324412yes—
613C2×C4C22×C4×D5311yescharacteristic
614C2×C4#41×D5311yescharacteristic
615C2×C4#41×D5311yescharacteristic
616C2×C4C2×#1171994711yescharacteristic
617C2×C4C2×#1171994711yescharacteristic
618C2×C4C2×#1171994911yescharacteristic
619C8—22no—
620C8—41no—
621C8—1062no—
622C8—2121no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—4241no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—2122no—
635C22—2122no—
636C22—2122no—
637C22C4×D4×D5311yescharacteristic
638C22#1681112911yescharacteristic
639C22#1681113611yescharacteristic
640C22AGL(1,ℤ/8)×D5311yescharacteristic
641C22C2×#1313324411yescharacteristic
642C22#1681141911yescharacteristic
643C22#1681142011yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—81no—
649C4—1062no—
650C4—1062no—
651C4—1062no—
652C4—4241no—
653C4—2122no—
654C4—2122no—
655C4—2121no—
656C4C2×#1313317511yescharacteristic
657C4C4×#1171994711yescharacteristic
658C4#1681115011yescharacteristic
659C4#1681115911yescharacteristic
660C2—44no—
661C2—4242no—
662C2#2027715111yescharacteristic
663C2C4×#1313324411yescharacteristic
664C2#2028352611yescharacteristic
665C2#2028480811yescharacteristic
666C2#2028482911yescharacteristic
667C2#2028485411yescharacteristic
668C2#2028485911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1864C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
cosocleC22×C4×C53C24
hypercenter#1864D53
hypocenterC53#21122
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #20277173C2semidirect
3. C4×D4×Dic53C2semidirect
4. #20275341C2semidirect
5. #20275336C2semidirect
6. #20275297C2semidirect
7. #20275296C2semidirect
10. #20274964C2semidirect
11. #20274615C2semidirect
12. #20274542C2semidirect
13. #20274539C2semidirect
14. #20274462C2semidirect
15. C53×#1864C2semidirect
22. #16810732C4semidirect
74. #13133179C2×C4semidirect
99. #13133148C2×C4semidirect
101. #13133139C2×C4semidirect
105. #13133127C2×C4semidirect
233. D4×C53#117semidirect
335. #1864D53semidirect
394. C53#21122semidirect
572. D4#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×#168107322
#202771732
C4×D4×Dic532
#202753412
#202753362
#202752972
#202752962
#202750632
#202750612
#202749642
#202746152
#202745422
#202745392
#202744622
C53×#18642
#2112253