Finite Group Databasebeta

← #27434387

Subgroups — #27434387

Order 13568.

860 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810915C211yesmaximal, characteristic, complemented
2C2×#16810695C211yesmaximal, characteristic
3#20279624C211yesmaximal, characteristic
4#20276146C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275918C211yesmaximal, characteristic, complemented
7#20275911C211yesmaximal, characteristic, complemented
8#20275689C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275146C211yesmaximal, characteristic, complemented
11#20275136C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic, complemented
14#20274548C211yesmaximal, characteristic, complemented
15C53×#2547C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133196C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21#16810915—24no—
22#16810695—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35#16810554—21no—
36#16810537—21no—
37#16810511—21no—
38#16810511—21no—
39#16810489C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×#124×C53—21no—
52C2×C4×D4×C53C2211yescharacteristic
53C53×#397—21no—
54C53×#397—21no—
55C53×#391—21no—
56C53×#380C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917—41no—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322—41no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133196—22no—
73#13133196—22no—
74#13133196—41no—
75#13133196—41no—
76C4×Dic106—22no—
77C4×Dic106—22no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89#13133150—22no—
90#13133150—22no—
91#13133150—41no—
92#13133148—41no—
93C4⋊#88322D414yes—
94#13133140—22no—
95#13133140—22no—
96#13133140—22no—
97#13133140—22no—
98C22⋊#88322—22no—
99C22⋊#88322—22no—
100#13133128—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53—21no—
106C22×D4×C53D411yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C23×C4×C53C2311yescharacteristic
109C4⋊Q8×C53—22no—
110C4⋊Q8×C53—22no—
111#124×C53—44no—
112C4×D4×C53—22no—
113C4×D4×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53D411yescharacteristic
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C42×C53—21no—
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53—21no—
133C24×C53C2×D411yescharacteristic
134C24×C53C2×D411yescharacteristic
135C2×Q8×C53—42no—
136C2×Q8×C53—42no—
137C2×Q8×C53—42no—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—41no—
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×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2411yescharacteristic
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106#4112yes—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
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—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—42no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—42no—
188C42×C53—42no—
189C42×C53C2×D412yes—
190C42×C53#4112yes—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C22⋊Dic53—42no—
194C22⋊Dic53—42no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201#11719919—42no—
202#11719919—42no—
203#11719919—42no—
204C4×Dic53—42no—
205#11719917—44no—
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—
215C2×#88322—44no—
216C2×#88322C2×D412yes—
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—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53AGL(1,ℤ/8)12yes—
233C23×C53C22×D411yescharacteristic
234Q8×C53—84no—
235D4×C53—44no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—41no—
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—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#12112yes—
267C2×C4×C53#12112yes—
268C2×C4×C53C22×D411yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53H(5,𝔽2)11yescharacteristic
273C2×C4×C532−1+411yescharacteristic
274C2×Dic53—22no—
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—
286Dic106—22no—
287Dic106—22no—
288Dic106—41no—
289Dic106—41no—
290#88322—22no—
291#88322—22no—
292#88322—22no—
293#88322—41no—
294#25352—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53C2×#12111yescharacteristic
310C22×C53#54611yescharacteristic
311C22×C53#54711yescharacteristic
312C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C53#57511yescharacteristic
314C22×C53#57811yescharacteristic
315C22×C53#57811yescharacteristic
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—41no—
319C4×C53—41no—
320C4×C53—81no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53—81no—
324C4×C53#52911yescharacteristic
325C4×C53#54211yescharacteristic
326C4×C53#54611yescharacteristic
327C4×C53#54811yescharacteristic
328Dic53—42no—
329Dic53—81no—
330Dic53—81no—
331Dic53—81no—
332C2×#477—531no—
333C2×#435—531no—
334#2547D5311yescharacteristic
335#2536—531no—
336#2076—531no—
337#2075—531no—
338#2036—531no—
339#2035—531no—
340#1993—531no—
341#1949—531no—
342#1894—531no—
343#1892—531no—
344#1809—531no—
345#1808—531no—
346#1763—531no—
347C2×C53—42no—
348C2×C53—81no—
349C2×C53#242711yescharacteristic
350C2×C53#305511yescharacteristic
351C2×C53#305911yescharacteristic
352C2×C53#330811yescharacteristic
353C2×C53#331311yescharacteristic
354C2×C53#334311yescharacteristic
355C2×C53#334911yescharacteristic
356C22×M16—531no—
357C23×C8—531no—
358C2×C4⋊Q8C2×D5311yescharacteristic
359C2×#124—21no—
360C2×C22⋊Q8—531no—
361C2×C22⋊Q8—531no—
362C2×C4×Q8—531no—
363C2×C4×D4C2×D5311yescharacteristic
364#477—1064no—
365#435—1064no—
366C2×#107—531no—
367C2×#107—531no—
368C2×#107—531no—
369C2×#106—531no—
370C2×C4⋊C8—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×C22⋊C8—531no—
376C2×C22⋊C8—531no—
377C2×#97—531no—
378#398—1061no—
379#397—21no—
380#397—21no—
381#394—1061no—
382#391—21no—
383#388—1061no—
384#388—1061no—
385#382—531no—
386#380—531no—
387#380—531no—
388#380C2×D5311yescharacteristic
389#380C2×D5311yescharacteristic
390#380C2×D5311yescharacteristic
391#380C2×D5311yescharacteristic
392C4×C4⋊C4C2×D5311yescharacteristic
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#340—531no—
401C53#2535211yescharacteristic, complemented, Sylow
402C22×Q8—21no—
403C22×Q8—531no—
404C22×D4C22⋊D5311yescharacteristic
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C2×M16—1062no—
408C2×M16—1062no—
409C2×M16—2121no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8—531no—
413C22×C8—531no—
414C22×C8—1062no—
415C22×C8—1062no—
416C22×C8—2121no—
417C4⋊Q8—22no—
418C4⋊Q8—22no—
419#124—44no—
420C22⋊Q8—1062no—
421C22⋊Q8—1062no—
422C22⋊Q8—2121no—
423C22⋊Q8—2121no—
424C4×Q8—1062no—
425C4×Q8—1062no—
426C4×D4—22no—
427C4×D4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4C22⋊D5311yescharacteristic
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4C22⋊D5311yescharacteristic
449C2×C22⋊C4C22⋊D5311yescharacteristic
450C2×C42—21no—
451C2×C42—1061no—
452C2×C42C22×D5311yescharacteristic
453#107—1062no—
454#107—1062no—
455#107—2121no—
456#106—2121no—
457C4⋊C8—534no—
458#103—1062no—
459#103—1062no—
460#103—1062no—
461#103—1062no—
462C22⋊C8—1062no—
463C22⋊C8—1062no—
464#97—1062no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474C24C2×C22⋊D5311yescharacteristic
475C24C2×C22⋊D5311yescharacteristic
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—532no—
480C2×Q8—1061no—
481C2×Q8—1061no—
482C2×Q8—1061no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—41no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
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—21no—
503C22×C4—21no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C23×D5311yescharacteristic
513M16—2124no—
514C2×C8—532no—
515C2×C8—532no—
516C2×C8—532no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—2124no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—2122no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C22⋊C4—2122no—
556C42—42no—
557C42—2122no—
558C42D4×D5312yes—
559C42#1171994712yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1313324312yes—
574C23C22×C22⋊D5311yescharacteristic
575Q8—84no—
576Q8—1062no—
577Q8—1062no—
578Q8—2121no—
579Q8—2121no—
580D4—44no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
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—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—1062no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4#1313318912yes—
624C2×C4#1313318912yes—
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4#41×D5311yescharacteristic
627C2×C4#41×D5311yescharacteristic
628C2×C4C2×#1171994711yescharacteristic
629C2×C4#1313326311yescharacteristic
630C2×C4#1313326411yescharacteristic
631C8—1062no—
632C8—1062no—
633C8—1062no—
634C8—2121no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1313318911yescharacteristic
650C22#1681126111yescharacteristic
651C22#1681127211yescharacteristic
652C22C2×#1313324311yescharacteristic
653C22#1681142111yescharacteristic
654C22#1681143811yescharacteristic
655C22#1681143811yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—4241no—
665C4—4241no—
666C4—4241no—
667C4—2122no—
668C4#1681115211yescharacteristic
669C4#1681124211yescharacteristic
670C4#1681126611yescharacteristic
671C4#1681128111yescharacteristic
672C2—42no—
673C2—81no—
674C2#2027829511yescharacteristic
675C2#2028433011yescharacteristic
676C2#2028436711yescharacteristic
677C2#2028670011yescharacteristic
678C2#2028677511yescharacteristic
679C2#2028714911yescharacteristic
680C2#2028723911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2547C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2547D53
hypocenterC53#25352
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#16810915C2semidirect
4. #20276146C2semidirect
5. #20276141C2semidirect
6. #20275918C2semidirect
7. #20275911C2semidirect
8. #20275689C2semidirect
9. #20275483C2semidirect
10. #20275146C2semidirect
11. #20275136C2semidirect
12. #20274746C2semidirect
13. #20274742C2semidirect
14. #20274548C2semidirect
401. C53#25352semidirect

Maximal subgroups

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

subgroupindex
C2×#168109152
C2×#168106952
#202796242
#202761462
#202761412
#202759182
#202759112
#202756892
#202754832
#202751462
#202751362
#202747462
#202747422
#202745482
C53×#25472
#2535253