Finite Group Databasebeta

← #27465182

Subgroups — #27465182

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277322C211yesmaximal, characteristic, complemented
2#20277307C211yesmaximal, characteristic
3#20276256C211yesmaximal, characteristic, complemented
4#20276246C211yesmaximal, characteristic, complemented
5#20276136C211yesmaximal, characteristic, complemented
6#20276131C211yesmaximal, characteristic, complemented
7#20275352C211yesmaximal, characteristic, complemented
8#20275351C211yesmaximal, characteristic, complemented
9#20275345C211yesmaximal, characteristic, complemented
10#20275344C211yesmaximal, characteristic, complemented
11#20274910C211yesmaximal, characteristic, complemented
12#20274907C211yesmaximal, characteristic, complemented
13#20274784C211yesmaximal, characteristic, complemented
14#20274655C211yesmaximal, characteristic
15C53×#1788C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133149C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810530C2211yescharacteristic
29#16810526C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810517—21no—
32#16810511—21no—
33#16810508—21no—
34#16810506—21no—
35#16810503—21no—
36#16810503—21no—
37#16810492C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810483—21no—
40#16810483—21no—
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×#41×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133176—22no—
63#13133176—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—41no—
66C2×C4⋊Dic53—41no—
67C2×C4⋊Dic53D411yescharacteristic
68C2×C4⋊Dic53D411yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D411yescharacteristic
81C2×C22⋊Dic53D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133149—41no—
87#13133149—41no—
88#13133148—41no—
89#13133148—41no—
90#13133141—22no—
91#13133141—41no—
92#13133138—22no—
93#13133138—41no—
94C22⋊#88322—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C2×#41×C53—22no—
105C2×#41×C53—22no—
106C2×#41×C53—22no—
107C2×#41×C53—22no—
108C2×#41×C53—41no—
109C2×#41×C53—41no—
110C22×Q8×C53C2311yescharacteristic
111C22×D4×C53—21no—
112C22×D4×C53C2311yescharacteristic
113C23×C4×C53—21no—
114C23×C4×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120#103×C53—22no—
121#103×C53—41no—
122#102×C53—22no—
123#102×C53—41no—
124C22⋊C8×C53—22no—
125#95×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128#41×C53—44no—
129#41×C53—44no—
130#41×C53—44no—
131#41×C53—44no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53#4112yes—
138C2×D4×C53—24no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—42no—
145C2×D4×C53—42no—
146C2×D4×C53#4112yes—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—24no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53#4112yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—81no—
167C22×Dic53—81no—
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C22⋊C4×C53—81no—
183C22⋊Dic53—22no—
184C22⋊Dic53—22no—
185C22⋊Dic53—81no—
186C22⋊Dic53—81no—
187C22⋊Dic53—81no—
188C22⋊Dic53—81no—
189C22⋊Dic53—81no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201#11719919—81no—
202#11719919—81no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—81no—
208C4×Dic53—81no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—22no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53#12312yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—22no—
231Q8×C53—22no—
232Q8×C53—41no—
233Q8×C53—41no—
234D4×C53—22no—
235D4×C53—22no—
236D4×C53—48no—
237D4×C53—41no—
238D4×C53—41no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—48no—
247C2×C4×C53—41no—
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#12312yes—
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—84no—
287C8×C53—41no—
288C8×C53—41no—
289#88322—41no—
290#88322—41no—
291#27597—531nomaximal, Sylow
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53C2×#12311yescharacteristic
307C22×C53#52511yescharacteristic
308C22×C53#53511yescharacteristic
309C22×C53#57711yescharacteristic
310C22×C53#57711yescharacteristic
311C22×C53#57711yescharacteristic
312C22×C53#57711yescharacteristic
313C4×C53—22no—
314C4×C53—42no—
315C4×C53—81no—
316C4×C53—81no—
317C4×C53#55111yescharacteristic
318C4×C53#55111yescharacteristic
319C4×C53#55111yescharacteristic
320C4×C53#55111yescharacteristic
321Dic53—41no—
322Dic53—41no—
323Dic53—41no—
324Dic53—41no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329#2314—531no—
330#2311—531no—
331#2096—531no—
332#2094—531no—
333#2074—531no—
334#2073—531no—
335#1927—531no—
336#1927—531no—
337#1926—531no—
338#1926—531no—
339#1853—531no—
340#1853—531no—
341#1818—531no—
342#1788—531no—
343#1788D5311yescharacteristic
344C2×C53—42no—
345C2×C53—82no—
346C2×C53#248211yescharacteristic
347C2×C53#309911yescharacteristic
348C2×C53#309911yescharacteristic
349C2×C53#312511yescharacteristic
350C2×C53#312511yescharacteristic
351C2×C53#322111yescharacteristic
352C2×C53#322211yescharacteristic
353C22×#41C2×D5311yescharacteristic
354C2×#117—531no—
355C22×C4⋊C4—531no—
356C2×#106—531no—
357C2×#106—531no—
358C2×#106—531no—
359C2×#106—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103C2×D5311yescharacteristic
363C2×#103C2×D5311yescharacteristic
364C2×#102—531no—
365C2×#102—531no—
366C2×#102C2×D5311yescharacteristic
367C2×#102C2×D5311yescharacteristic
368C2×C22⋊C8—531no—
369C2×C22⋊C8C2×D5311yescharacteristic
370#391—531no—
371#391—531no—
372#390—531no—
373#390—531no—
374#388—1061no—
375#388—1061no—
376#387—1061no—
377#387—1061no—
378#386—1061no—
379#386—1061no—
380#384—531no—
381#382—531no—
382#380—531no—
383#380—1061no—
384#380—1061no—
385#380C2×D5311yescharacteristic
386C4×C4⋊C4—531no—
387C4×C4⋊C4—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336—531no—
393#336—531no—
394#336—531no—
395#336—531no—
396C53#2759711yescharacteristic, complemented, Sylow
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—41no—
402C2×#41—41no—
403C22×Q8C22×D5311yescharacteristic
404C22×D4—21no—
405C22×D4C22×D5311yescharacteristic
406C23×C4—21no—
407C23×C4—1061no—
408C23×C4C22×D5311yescharacteristic
409C22×C8—531no—
410C22×C8—531no—
411C22×C8C22×D5311yescharacteristic
412C22×C8C22×D5311yescharacteristic
413#117—1062no—
414#117—1062no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—2121no—
425C2×C4⋊C4—2121no—
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—531no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—1061no—
439C2×C42—1061no—
440#106—2121no—
441#106—2121no—
442#106—2121no—
443#106—2121no—
444#103—22no—
445#103—41no—
446#103—1062no—
447#103—2121no—
448#102—22no—
449#102—41no—
450#102—1062no—
451#102—2121no—
452C22⋊C8—22no—
453C22⋊C8—1062no—
454#95—21no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464C24—21no—
465C24D4×D5311yescharacteristic
466#41—44no—
467#41—44no—
468#41—44no—
469#41—44no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8#1171994712yes—
476C2×D4—24no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4#1171994712yes—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—4241no—
504C22×C4—4241no—
505C22×C4—2122no—
506C22×C4#1171994512yes—
507C22×C4D4×D5311yescharacteristic
508C22×C4#1171994712yes—
509C22×C4C23×D5311yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—1062no—
527C4⋊C4—1062no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—4241no—
537C4⋊C4—4241no—
538C4⋊C4—2122no—
539C22⋊C4—81no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C22⋊C4—4241no—
543C22⋊C4—4241no—
544C22⋊C4—4241no—
545C22⋊C4—4241no—
546C22⋊C4—4241no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C42—1061no—
551C42—4241no—
552C42—4241no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23#1313319912yes—
567C23C2×D4×D5311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—41no—
576D4—41no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
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—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—4244no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4#1313319912yes—
619C2×C4C2×#1171994711yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4C2×#1171994711yescharacteristic
622C2×C4C2×#1171994511yescharacteristic
623C2×C4#1313326211yescharacteristic
624C2×C4#1313326211yescharacteristic
625C8—41no—
626C8—41no—
627C8—2121no—
628C8—2121no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×#1313319911yescharacteristic
644C22#1681112911yescharacteristic
645C22#1681118811yescharacteristic
646C22#1681143011yescharacteristic
647C22#1681143011yescharacteristic
648C22#1681143211yescharacteristic
649C22#1681143311yescharacteristic
650C4—22no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—4241no—
655C4—4241no—
656C4—4241no—
657C4—4241no—
658C4—2121no—
659C4—2121no—
660C4—2121no—
661C4—2121no—
662C4#1681129911yescharacteristic
663C4#1681129911yescharacteristic
664C4#1681130411yescharacteristic
665C4#1681130411yescharacteristic
666C2—42no—
667C2—82no—
668C2#2027898611yescharacteristic
669C2#2028470211yescharacteristic
670C2#2028470311yescharacteristic
671C2#2028498511yescharacteristic
672C2#2028499311yescharacteristic
673C2#2028577411yescharacteristic
674C2#2028578511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1788C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#1788D53
hypocenterC53#27597
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
1. #20277322C2semidirect
3. #20276256C2semidirect
4. #20276246C2semidirect
5. #20276136C2semidirect
6. #20276131C2semidirect
7. #20275352C2semidirect
8. #20275351C2semidirect
9. #20275345C2semidirect
10. #20275344C2semidirect
11. #20274910C2semidirect
12. #20274907C2semidirect
13. #20274784C2semidirect
396. C53#27597semidirect

Maximal subgroups

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

subgroupindex
#202773222
#202773072
#202762562
#202762462
#202761362
#202761312
#202753522
#202753512
#202753452
#202753442
#202749102
#202749072
#202747842
#202746552
C53×#17882
#2759753