Finite Group Databasebeta

← #27449669

Subgroups — #27449669

Order 13568.

900 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic, complemented
2C2×C4×#11719924C211yesmaximal, characteristic, complemented
3#20277729C211yesmaximal, characteristic
4#20276239C211yesmaximal, characteristic, complemented
5#20276030C211yesmaximal, characteristic, complemented
6#20275897C211yesmaximal, characteristic
7#20275716C211yesmaximal, characteristic
8#20275481C211yesmaximal, characteristic, complemented
9#20275445C211yesmaximal, characteristic, complemented
10#20275353C211yesmaximal, characteristic, complemented
11#20275323C211yesmaximal, characteristic, complemented
12#20274983C211yesmaximal, characteristic, complemented
13#20274961C211yesmaximal, characteristic, complemented
14#20274907C211yesmaximal, characteristic, complemented
15C53×#2421C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133177—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21#16810956—24no—
22C4×#11719924—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×C22⋊#88322C2211yescharacteristic
36C2×C22⋊#88322C2211yescharacteristic
37C2×C4×#88322C2211yescharacteristic
38#16810525C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810515—21no—
41#16810500—21no—
42#16810497—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#397—21no—
54C53×#391C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#387—21no—
58C53×#382C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C23×Dic53—21no—
64C22×C4×D53—21no—
65C2×#11719924—22no—
66C2×#11719924—22no—
67C2×#11719924—22no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133191—22no—
73#13133191—22no—
74C4×D212—22no—
75C4×D212—41no—
76#13133177—44no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53—21no—
89#13133150—22no—
90#13133150—22no—
91#13133150—22no—
92#13133148—22no—
93#13133148—41no—
94C4⋊#88322—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141D414yescomplemented
98#13133137—22no—
99#13133137—22no—
100#13133137—41no—
101C22⋊#88322—22no—
102C22⋊#88322—22no—
103C4×#88322—22no—
104#13133124—21no—
105#13133124—21no—
106C22×Q8×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C22⋊Q8×C53—22no—
109C22⋊Q8×C53—41no—
110C4×Q8×C53—22no—
111C4×Q8×C53—41no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53D411yescharacteristic
132#95×C53D411yescharacteristic
133C24×C53C2×D411yescharacteristic
134C23×D53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53C2×D412yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C2×C22⋊D53—24no—
158C2×C22⋊D53—42no—
159C2×C22⋊D53—42no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—42no—
164C22×Dic53—42no—
165C22×Dic53—81no—
166C2×D212—21no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212—21no—
170C2×D212—21no—
171C2×D212#4112yes—
172C2×C4×D53—42no—
173C2×C4×D53—42no—
174C2×C4×D53—81no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53—42no—
183C4⋊C4×C53—42no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—42no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—42no—
193#11719924—28no—
194#11719921—24no—
195#11719921—42no—
196#11719921—42no—
197#11719921—81no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53C2×D412yes—
204C4⋊Dic53#4112yes—
205#11719919—42no—
206C4×Dic53—42no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322C2×D412yes—
218C2×#88322#4112yes—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53#13512yes—
226C23×C53C22×D411yescharacteristic
227C22×D53—22no—
228C22×D53—41no—
229C22×D53—41no—
230C22×D53—41no—
231C22×D53—41no—
232C22×D53—41no—
233C22×D53—41no—
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
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—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53AGL(1,ℤ/8)12yes—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53H(5,𝔽2)11yescharacteristic
274C2×C4×C532−1+411yescharacteristic
275C22⋊D53—48no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—84no—
286D212—22no—
287D212—22no—
288D212—41no—
289D212—41no—
290C4×D53—84no—
291#88322—22no—
292#88322—22no—
293#88322—41no—
294#88322—41no—
295#26555—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53#52511yescharacteristic
304C22×C53D4211yescharacteristic
305C22×C53#54711yescharacteristic
306C22×C53C2×#13511yescharacteristic
307C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C53#57711yescharacteristic
309C22×C53#57811yescharacteristic
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—81no—
314C2×D53—81no—
315C2×D53—81no—
316C2×D53—81no—
317C4×C53—22no—
318C4×C53—41no—
319C4×C53—41no—
320C4×C53—41no—
321C4×C53—42no—
322C4×C53—41no—
323C4×C53—81no—
324C4×C53—81no—
325C4×C53—81no—
326C4×C53#52511yescharacteristic
327C4×C53#53911yescharacteristic
328C4×C53#54711yescharacteristic
329C4×C53#55911yescharacteristic
330Dic53—42no—
331Dic53—42no—
332Dic53—81no—
333C2×#484—531no—
334C2×C4×SD16—531no—
335#2421D5311yescharacteristic, complemented
336#2361—531no—
337#2092—531no—
338#2055—531no—
339#2033—531no—
340#2000—531no—
341#1949—531no—
342#1942—531no—
343#1928—531no—
344#1923—531no—
345#1865—531no—
346#1862—531no—
347#1853—531no—
348C2×C53—42no—
349C2×C53#245511yescharacteristic
350C2×C53#310511yescharacteristic
351C2×C53#312011yescharacteristic
352C2×C53#326511yescharacteristic
353C2×C53#328511yescharacteristic
354C2×C53#333711yescharacteristic
355C2×C53#335311yescharacteristic
356D53—82no—
357C22×SD16—531no—
358C2×C22⋊Q8C2×D5311yescharacteristic
359C2×#121—531no—
360C2×C4×Q8C2×D5311yescharacteristic
361C2×C4×D4—531no—
362C2×C4×D4—1061no—
363C22×C4⋊C4—531no—
364#484—1064no—
365C4×SD16—1064no—
366C2×#107—531no—
367C2×#107—531no—
368C2×#107—531no—
369C2×#106—531no—
370C2×#106—531no—
371C2×C4⋊C8—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×#102—531no—
378C2×C22⋊C8—531no—
379C2×C22⋊C8—531no—
380C2×C4×C8—531no—
381#400—21no—
382#397—21no—
383#391C2×D5311yescharacteristic
384#390—531no—
385#389—531no—
386#389C2×D5311yescharacteristic
387#388—21no—
388#388—1061no—
389#387—21no—
390#386—1061no—
391#385—1061no—
392#382C2×D5311yescharacteristic
393#380C2×D5311yescharacteristic
394C4×C4⋊C4—531no—
395C4×C22⋊C4C2×D5311yescharacteristic
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#336—531no—
401#336—531no—
402C53#2655511yescharacteristic, complemented, Sylow
403C22×Q8C22×D5311yescharacteristic
404C22×D4—531no—
405C22×D4—1061no—
406C23×C4—1061no—
407C23×C4—1061no—
408C23×C4C22×D5311yescharacteristic
409C2×SD16—1062no—
410C2×SD16—1062no—
411C2×SD16—1062no—
412C22×C8—531no—
413C22×C8—531no—
414C22×C8—531no—
415C22×C8—531no—
416C22⋊Q8—22no—
417C22⋊Q8—41no—
418#121—1062no—
419#121—1062no—
420C4×Q8—22no—
421C4×Q8—41no—
422C4×D4—1062no—
423C4×D4—2121no—
424C4×D4—2124no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—2121no—
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4C22⋊D5311yescharacteristic
445C2×C22⋊C4C22⋊D5311yescharacteristic
446C2×C42—21no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449C2×C42C22×D5311yescharacteristic
450#107—1062no—
451#107—1062no—
452#107—1062no—
453#106—1062no—
454#106—2121no—
455C4⋊C8—1062no—
456#103—534no—
457#103—1062no—
458#103—1062no—
459#102—1062no—
460#102—1062no—
461#102—2121no—
462C22⋊C8—1062no—
463C22⋊C8—1062no—
464C4×C8—1062no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—1061no—
472#95—1061no—
473#95C22⋊D5311yescharacteristic
474#95C22⋊D5311yescharacteristic
475C24—1061no—
476C24C2×C22⋊D5311yescharacteristic
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8D4×D5312yes—
483C2×D4—532no—
484C2×D4—1064no—
485C2×D4—1061no—
486C2×D4—1061no—
487C2×D4—1061no—
488C2×D4—1061no—
489C2×D4—1061no—
490C2×D4—2122no—
491C2×D4—2122no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—4241no—
506C22×C4—4241no—
507C22×C4—2122no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4—2122no—
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C23×D5311yescharacteristic
517SD16—1068no—
518C2×C8—532no—
519C2×C8—532no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—532no—
540C4⋊C4—532no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—1064no—
552C22⋊C4—4241no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—42no—
560C42—2122no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—1062no—
568C23—2121no—
569C23—2121no—
570C23—2121no—
571C23—2121no—
572C23—2121no—
573C23—2121no—
574C23#1313323712yes—
575C23C22×C22⋊D5311yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580D4—1062no—
581D4—1062no—
582D4—2128no—
583D4—2121no—
584D4—2121no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—4244no—
618C2×C4—4244no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4#1313324212yes—
627C2×C4C2×D4×D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4#41×D5311yescharacteristic
630C2×C4#41×D5311yescharacteristic
631C2×C4#1313326211yescharacteristic
632C2×C4#1313326411yescharacteristic
633C8—1062no—
634C8—1062no—
635C8—2121no—
636C8—2121no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—4241no—
645C22—4241no—
646C22—4241no—
647C22—4241no—
648C22—2122no—
649C22—2122no—
650C22—2122no—
651C22#1681113011yescharacteristic
652C22D4×C22⋊D5311yescharacteristic
653C22#1681127211yescharacteristic
654C22C2×#1313323711yescharacteristic
655C22C2×#1313324211yescharacteristic
656C22#1681143111yescharacteristic
657C22#1681143811yescharacteristic
658C4—22no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—4241no—
668C4—2122no—
669C4—2122no—
670C4#1681113111yescharacteristic
671C4#1681121611yescharacteristic
672C4#1681127111yescharacteristic
673C4#1681135711yescharacteristic
674C2—42no—
675C2—4242no—
676C2#2027859411yescharacteristic
677C2#2028475811yescharacteristic
678C2#2028492311yescharacteristic
679C2#2028621111yescharacteristic
680C2#2028645211yescharacteristic
681C2#2028709411yescharacteristic
682C2#2028729311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2421C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2421D53
hypocenterC53#26555
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2421

Complements

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

NH ≅ G/Ntype
1. C2×#16810956C2semidirect
2. C2×C4×#11719924C2semidirect
4. #20276239C2semidirect
5. #20276030C2semidirect
8. #20275481C2semidirect
9. #20275445C2semidirect
10. #20275353C2semidirect
11. #20275323C2semidirect
12. #20274983C2semidirect
13. #20274961C2semidirect
14. #20274907C2semidirect
15. C53×#2421C2semidirect
97. #13133141D4semidirect
335. #2421D53semidirect
402. C53#26555semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×C4×#117199242
#202777292
#202762392
#202760302
#202758972
#202757162
#202754812
#202754452
#202753532
#202753232
#202749832
#202749612
#202749072
C53×#24212
#2655553