Finite Group Databasebeta

← #27447639

Subgroups — #27447639

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277729C211yesmaximal, characteristic
2#20276272C211yesmaximal, characteristic, complemented
3#20276206C211yesmaximal, characteristic, complemented
4#20276100C211yesmaximal, characteristic, complemented
5#20275848C211yesmaximal, characteristic, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275283C211yesmaximal, characteristic, complemented
8#20275141C211yesmaximal, characteristic
9#20275032C211yesmaximal, characteristic
10#20274961C211yesmaximal, characteristic, complemented
11#20274902C211yesmaximal, characteristic, complemented
12#20274895C211yesmaximal, characteristic, complemented
13#20274712C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#2440C211yesmaximal, characteristic, complemented
16C23×#88322C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133177—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810525C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810515—21no—
33#16810500—21no—
34#16810497—21no—
35C4×C4⋊Dic53C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×C22⋊Q8×C53—21no—
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#398—21no—
52C53×#395—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#385—21no—
56C53×#384C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×C4×D53—21no—
63C22×#88322—22no—
64C22×#88322—41no—
65C22×#88322—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133191—22no—
71#13133191Q814yescomplemented
72C4×D212—22no—
73C4×D212—41no—
74#13133177—44no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87#13133150—22no—
88#13133150—41no—
89C4⋊#88322—41no—
90C4⋊#88322—41no—
91#13133137—22no—
92#13133137—41no—
93#13133137—41no—
94#13133137—41no—
95C22⋊#88322—22no—
96#13133124—21no—
97#13133124—21no—
98C22×Q8×C53—21no—
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C4⋊Q8×C53—22no—
102C4⋊Q8×C53—41no—
103C22⋊Q8×C53—44no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53—41no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C2×Q8×C53—42no—
129C2×Q8×C53—42no—
130C2×Q8×C53—81no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—81no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×Q812yes—
148C22×C4×C53#4112yes—
149C22×C4×C53C2411yescharacteristic
150C2×C22⋊D53—24no—
151C2×C22⋊D53—42no—
152C2×C22⋊D53—42no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212C2×Q812yes—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—81no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—42no—
180C4⋊C4×C53—81no—
181C4⋊C4×C53#4112yes—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53D814yes—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189#11719921—24no—
190#11719921—42no—
191#11719921—42no—
192#11719921—81no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—42no—
198C4⋊Dic53C2×Q812yes—
199C4⋊Dic53#4112yes—
200#11719919—42no—
201C4×Dic53—42no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—44no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C22⋊Q812yes—
220C23×C53C2×D812yes—
221C23×C53#13712yes—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230Q8×C53—84no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—84no—
260C2×C4×C53C22⋊Q812yes—
261C2×C4×C53C2×D812yes—
262C2×C4×C53C2×D812yes—
263C2×C4×C53C22×Q811yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53H(5,𝔽2)11yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C22⋊D53—48no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—84no—
280D212—22no—
281D212—22no—
282D212—41no—
283D212—41no—
284C4×D53—84no—
285#88322—41no—
286#88322—41no—
287#88322—41no—
288#88322—41no—
289#26408—531nomaximal, Sylow
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53#47414yes—
297C22×C53C2×C22⋊Q811yescharacteristic
298C22×C53#54611yescharacteristic
299C22×C53#54611yescharacteristic
300C22×C53C22×D811yescharacteristic
301C22×C53C2×#13711yescharacteristic
302C22×C53#57711yescharacteristic
303C22×C53#57711yescharacteristic
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C2×D53—81no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—42no—
315C4×C53—41no—
316C4×C53—41no—
317C4×C53—81no—
318C4×C53—81no—
319C4×C53—81no—
320C4×C53#52911yescharacteristic
321C4×C53#53811yescharacteristic
322C4×C53#55411yescharacteristic
323C4×C53#55411yescharacteristic
324Dic53—42no—
325Dic53—42no—
326Dic53—81no—
327#2440D5311yescharacteristic, complemented
328#2361—531no—
329#2099—531no—
330#2086—531no—
331#2067—531no—
332#2023—531no—
333#1988—531no—
334#1918—531no—
335#1893—531no—
336#1874—531no—
337#1862—531no—
338#1852—531no—
339#1850—531no—
340#1802—531no—
341C2×#340—531no—
342C2×C53—24no—
343C2×C53#245011yescharacteristic
344C2×C53C2×#47411yescharacteristic
345C2×C53#308411yescharacteristic
346C2×C53#329411yescharacteristic
347C2×C53#329411yescharacteristic
348C2×C53#331611yescharacteristic
349C2×C53#332011yescharacteristic
350D53—82no—
351C23×C8—531no—
352C2×C4⋊Q8C2×D5311yescharacteristic
353C2×C22⋊Q8—21no—
354C2×#121—531no—
355C2×C4×D4—531no—
356C2×C4×D4—1061no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C2×#107—531no—
360C2×#107—531no—
361C2×C4⋊C8—531no—
362C2×C4⋊C8—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×C22⋊C8—531no—
368#398—21no—
369#395—21no—
370#390—531no—
371#389—531no—
372#389C2×D5311yescharacteristic
373#389C2×D5311yescharacteristic
374#388—1061no—
375#386—1061no—
376#385—21no—
377#385—1061no—
378#384C2×D5311yescharacteristic
379#380C2×D5311yescharacteristic
380C4×C4⋊C4—531no—
381C4×C22⋊C4C2×D5311yescharacteristic
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336—531no—
393#336—531no—
394C53#2640811yescharacteristic, complemented, Sylow
395C22×Q8—21no—
396C22×D4—531no—
397C22×D4—1061no—
398C23×C4—21no—
399C23×C4—1061no—
400C23×C4—1061no—
401C23×C4C22×D5311yescharacteristic
402C22×C8—531no—
403C22×C8—531no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—1062no—
407C22×C8—2121no—
408C22×C8—2121no—
409C4⋊Q8—22no—
410C4⋊Q8—41no—
411C22⋊Q8—44no—
412#121—534no—
413#121—1062no—
414C4×D4—1062no—
415C4×D4—2121no—
416C4×D4—2124no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—41no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—2121no—
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4C22⋊D5311yescharacteristic
442C2×C22⋊C4C22⋊D5311yescharacteristic
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445C2×C42C22×D5311yescharacteristic
446#107—1062no—
447#107—2121no—
448C4⋊C8—2121no—
449C4⋊C8—2121no—
450#102—1062no—
451#102—2121no—
452#102—2121no—
453#102—2121no—
454C22⋊C8—1062no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—1061no—
459#95—1061no—
460#95C22⋊D5311yescharacteristic
461#95C22⋊D5311yescharacteristic
462C24—1061no—
463C24C2×C22⋊D5311yescharacteristic
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—81no—
467C2×D4—532no—
468C2×D4—1064no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—2122no—
475C2×D4—2122no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—4241no—
491C22×C4—4241no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4Q8×D5312yes—
497C22×C4#1171994912yes—
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C23×D5311yescharacteristic
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—2124no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—532no—
531C4⋊C4—532no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C4⋊C4#1171994712yes—
539C22⋊C4—22no—
540C22⋊C4—42no—
541C22⋊C4—1064no—
542C22⋊C4—4241no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C22⋊C4#1171992214yes—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—2122no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—1062no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23#1313319312yes—
563C23C2×#1171992212yes—
564C23#1313324812yes—
565C23C22×C22⋊D5311yescharacteristic
566Q8—84no—
567D4—1062no—
568D4—1062no—
569D4—2128no—
570D4—2121no—
571D4—2121no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—4244no—
604C2×C4—4244no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4#1313319312yes—
613C2×C4C2×#1171992212yes—
614C2×C4C2×#1171992212yes—
615C2×C4C2×Q8×D5311yescharacteristic
616C2×C4#41×D5311yescharacteristic
617C2×C4C2×#1171994711yescharacteristic
618C2×C4C2×#1171994911yescharacteristic
619C2×C4#1313326211yescharacteristic
620C2×C4#1313326311yescharacteristic
621C8—2121no—
622C8—2121no—
623C8—2121no—
624C8—2121no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—4241no—
635C22—2122no—
636C22—2122no—
637C22—2122no—
638C22#1681090014yes—
639C22C2×#1313319311yescharacteristic
640C22#1681126111yescharacteristic
641C22#1681126711yescharacteristic
642C22C22×#1171992211yescharacteristic
643C22C2×#1313324811yescharacteristic
644C22#1681143111yescharacteristic
645C22#1681143411yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—4241no—
656C4—2122no—
657C4—2122no—
658C4#1681115911yescharacteristic
659C4#1681121311yescharacteristic
660C4#1681132711yescharacteristic
661C4#1681133111yescharacteristic
662C2—24no—
663C2—4242no—
664C2#2027853511yescharacteristic
665C2C2×#1681090011yescharacteristic
666C2#2028457311yescharacteristic
667C2#2028652211yescharacteristic
668C2#2028652811yescharacteristic
669C2#2028682011yescharacteristic
670C2#2028688611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2440C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2440D53
hypocenterC53#26408
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #20276272C2semidirect
3. #20276206C2semidirect
4. #20276100C2semidirect
5. #20275848C2semidirect
6. #20275668C2semidirect
7. #20275283C2semidirect
10. #20274961C2semidirect
11. #20274902C2semidirect
12. #20274895C2semidirect
13. #20274712C2semidirect
14. C2×#16810385C2semidirect
15. C53×#2440C2semidirect
71. #13133191Q8semidirect
327. #2440D53semidirect
394. C53#26408semidirect

Maximal subgroups

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

subgroupindex
#202777292
#202762722
#202762062
#202761002
#202758482
#202756682
#202752832
#202751412
#202750322
#202749612
#202749022
#202748952
#202747122
C2×#168103852
C53×#24402
#2640853