Finite Group Databasebeta

← #27452177

Subgroups — #27452177

Order 13568.

826 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#20279078C211yesmaximal, characteristic
2#20276030C211yesmaximal, characteristic, complemented
3#20275908C211yesmaximal, characteristic, complemented
4#20275856C211yesmaximal, characteristic, complemented
5#20275716C211yesmaximal, characteristic
6#20275667C211yesmaximal, characteristic, complemented
7#20275397C211yesmaximal, characteristic, complemented
8#20275342C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic, complemented
10#20275297C211yesmaximal, characteristic, complemented
11#20275070C211yesmaximal, characteristic, complemented
12#20275032C211yesmaximal, characteristic
13#20274805C211yesmaximal, characteristic, complemented
14C4×#13133141C211yesmaximal, characteristic, complemented
15C53×#2360C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133203—21no—
19C2×#13133191C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810539—21no—
32#16810536—21no—
33#16810525C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810515—21no—
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#387—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C22⋊C4×C53—21no—
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C2×#11719924—22no—
63C2×#11719924—41no—
64C2×#11719924—41no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133217—22no—
70#13133217—41no—
71#13133203—44no—
72#13133191—22no—
73#13133191—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85#13133148—22no—
86C4⋊#88322—41no—
87#13133141—22no—
88#13133141—22no—
89#13133141—41no—
90#13133137—22no—
91#13133137—22no—
92#13133137—41no—
93C22⋊#88322—41no—
94C22⋊#88322—41no—
95C4×#88322—41no—
96#13133124—21no—
97#13133124—21no—
98C22×Q8×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53—22no—
103C4×Q8×C53—22no—
104C4×Q8×C53—41no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53—41no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53D411yescharacteristic
126#95×C53D411yescharacteristic
127C24×C53C2×D411yescharacteristic
128C23×D53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53#4112yes—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—81no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C2×C22⋊D53—42no—
155C2×C22⋊D53—42no—
156C2×C22⋊D53—81no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×Dic106—42no—
167C2×Dic106—42no—
168C2×Dic106—81no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—81no—
177C4⋊C4×C53#4112yes—
178C4⋊C4×C53#4112yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—41no—
181C22⋊C4×C53SD1614yescomplemented
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—42no—
187C42×C53—81no—
188C22⋊Dic53—42no—
189#11719924—44no—
190#11719921—42no—
191#11719921—42no—
192#11719921—42no—
193#11719921—42no—
194#11719921—81no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53#4112yes—
202C4×Dic53—42no—
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—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53C2×SD1612yes—
222C23×C53C22×D411yescharacteristic
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230Q8×C53—22no—
231Q8×C53—22no—
232Q8×C53—41no—
233Q8×C53—41no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
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—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—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—84no—
264C2×C4×C53C2×SD1612yes—
265C2×C4×C53C2×SD1612yes—
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×C4×C53H(5,𝔽2)11yescharacteristic
272C22⋊D53—84no—
273C2×Dic53—22no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282D212—41no—
283D212—41no—
284D212—41no—
285D212—41no—
286Dic106—84no—
287#88322—41no—
288#88322—41no—
289#88322—41no—
290#88322—41no—
291#26724—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#52511yescharacteristic
300C22×C53#54611yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53C22×SD1611yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57611yescharacteristic
305C22×C53#57811yescharacteristic
306C2×D53—81no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C2×D53—81no—
311C2×D53—81no—
312C2×D53—81no—
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—42no—
316C4×C53—41no—
317C4×C53—42no—
318C4×C53—41no—
319C4×C53—42no—
320C4×C53—41no—
321C4×C53—81no—
322C4×C53#52911yescharacteristic
323C4×C53#53811yescharacteristic
324C4×C53#55511yescharacteristic
325C4×C53#55911yescharacteristic
326Dic53—42no—
327Dic53—81no—
328Dic53—81no—
329#2491—531no—
330#2360D5311yescharacteristic, complemented
331#2055—531no—
332#2035—531no—
333#2024—531no—
334#2000—531no—
335#1988—531no—
336#1934—531no—
337#1926—531no—
338#1923—531no—
339#1920—531no—
340#1880—531no—
341#1874—531no—
342#1824—531no—
343C4×#103—531no—
344C2×C53—42no—
345C2×C53#245911yescharacteristic
346C2×C53#310311yescharacteristic
347C2×C53#312411yescharacteristic
348C2×C53#329511yescharacteristic
349C2×C53#329811yescharacteristic
350C2×C53#331911yescharacteristic
351C2×C53#332711yescharacteristic
352D53—161no—
353C22×SD16—531no—
354C2×C4⋊Q8—531no—
355C2×#124—1061no—
356C2×C22⋊Q8C2×D5311yescharacteristic
357C2×#121—531no—
358C2×C4×Q8C2×D5311yescharacteristic
359C22×C4⋊C4C2×D5311yescharacteristic
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×#102—531no—
368C2×C22⋊C8—531no—
369C2×C22⋊C8—531no—
370C2×C4×C8—531no—
371#394—1061no—
372#393—1061no—
373#391C2×D5311yescharacteristic
374#390—531no—
375#390—531no—
376#389—21no—
377#389C2×D5311yescharacteristic
378#388—1061no—
379#387—21no—
380#385—21no—
381#380—531no—
382C4×C4⋊C4—531no—
383C4×C4⋊C4C2×D5311yescharacteristic
384C4×C22⋊C4—21no—
385C4×C22⋊C4C2×D5311yescharacteristic
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#336—531no—
395#336—531no—
396C53#2672411yescharacteristic, complemented, Sylow
397C22×Q8—1061no—
398C22×Q8C22×D5311yescharacteristic
399C22×D4—531no—
400C22×D4—1061no—
401C23×C4—21no—
402C23×C4C22×D5311yescharacteristic
403C2×SD16—1062no—
404C2×SD16—2121no—
405C2×SD16—2121no—
406C22×C8—531no—
407C22×C8—531no—
408C22×C8—531no—
409C22×C8—531no—
410C4⋊Q8—1062no—
411C4⋊Q8—2121no—
412#124—2124no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415#121—1062no—
416#121—2121no—
417C4×Q8—22no—
418C4×Q8—41no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
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—1061no—
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4C22⋊D5311yescharacteristic
440C2×C22⋊C4C22⋊D5311yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445C2×C42C22×D5311yescharacteristic
446#106—1062no—
447C4⋊C8—2121no—
448#103—1062no—
449#103—1062no—
450#103—2121no—
451#102—1062no—
452#102—1062no—
453#102—2121no—
454C22⋊C8—2121no—
455C22⋊C8—2121no—
456C4×C8—2121no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—1061no—
462#95—1061no—
463#95C22⋊D5311yescharacteristic
464#95C22⋊D5311yescharacteristic
465C24—1061no—
466C24C2×C22⋊D5311yescharacteristic
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—4241no—
473C2×Q8—2122no—
474C2×Q8—2122no—
475C2×Q8#1171994512yes—
476C2×D4—1061no—
477C2×D4—1061no—
478C2×D4—1061no—
479C2×D4—1061no—
480C2×D4—1061no—
481C2×D4—1061no—
482C2×D4—4241no—
483C2×D4—2122no—
484C2×D4—2122no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—81no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507SD16—2124no—
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—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—532no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—2122no—
535C4⋊C4#1171994512yes—
536C4⋊C4#1171994712yes—
537C22⋊C4—24no—
538C22⋊C4—41no—
539C22⋊C4—4241no—
540C22⋊C4—2122no—
541C22⋊C4—2122no—
542C22⋊C4—2122no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C22⋊C4#1171992414yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—81no—
552C42—2122no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23—2121no—
564C23—2121no—
565C23—2121no—
566C23C2×#1171992412yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—4244no—
573D4—4244no—
574D4—2121no—
575D4—2121no—
576D4—2121no—
577D4—2121no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—1062no—
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×C4C2×#1171992412yes—
618C2×C4C2×#1171992412yes—
619C2×C4#41×D5311yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4C2×#1171994511yescharacteristic
622C2×C4C2×#1171994511yescharacteristic
623C2×C4#1313326211yescharacteristic
624C2×C4#1313326311yescharacteristic
625C8—2121no—
626C8—2121no—
627C8—2121no—
628C8—2121no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—4241no—
637C22—4241no—
638C22—4241no—
639C22—4241no—
640C22—4241no—
641C22—4241no—
642C22—4241no—
643C22#1681113411yescharacteristic
644C22#1681126111yescharacteristic
645C22#1681126711yescharacteristic
646C22C22×#1171992411yescharacteristic
647C22#1681142111yescharacteristic
648C22#1681142611yescharacteristic
649C22#1681143811yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—81no—
659C4—4241no—
660C4—4241no—
661C4—2122no—
662C4#1681115311yescharacteristic
663C4#1681120611yescharacteristic
664C4#1681133811yescharacteristic
665C4#1681135611yescharacteristic
666C2—42no—
667C2—8481no—
668C2#2027864611yescharacteristic
669C2#2028474611yescharacteristic
670C2#2028497711yescharacteristic
671C2#2028653311yescharacteristic
672C2#2028658011yescharacteristic
673C2#2028687111yescharacteristic
674C2#2028697711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2360D53
hypocenterC53#26724
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #20276030C2semidirect
3. #20275908C2semidirect
4. #20275856C2semidirect
6. #20275667C2semidirect
7. #20275397C2semidirect
8. #20275342C2semidirect
9. #20275323C2semidirect
10. #20275297C2semidirect
11. #20275070C2semidirect
13. #20274805C2semidirect
14. C4×#13133141C2semidirect
15. C53×#2360C2semidirect
181. C22⋊C4×C53SD16semidirect
330. #2360D53semidirect
396. C53#26724semidirect
545. C22⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
#202790782
#202760302
#202759082
#202758562
#202757162
#202756672
#202753972
#202753422
#202753232
#202752972
#202750702
#202750322
#202748052
C4×#131331412
C53×#23602
#2672453