Finite Group Databasebeta

← #27432281

Subgroups — #27432281

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic
2#20279194C211yesmaximal, characteristic, complemented
3#20277735C211yesmaximal, characteristic
4#20276194C211yesmaximal, characteristic, complemented
5#20276121C211yesmaximal, characteristic, complemented
6#20276092C211yesmaximal, characteristic, complemented
7#20276028C211yesmaximal, characteristic, complemented
8#20275475C211yesmaximal, characteristic, complemented
9#20275352C211yesmaximal, characteristic, complemented
10#20275051C211yesmaximal, characteristic, complemented
11#20275023C211yesmaximal, characteristic, complemented
12#20274893C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic, complemented
15C2×C53×#465C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810939—24no—
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810553—21no—
31#16810553—21no—
32#16810530C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810526C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810516—21no—
37#16810503—21no—
38#16810498—21no—
39#16810495—21no—
40#16810495—21no—
41#16810492C2211yescharacteristic
42#16810492C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×#121×C53C2211yescharacteristic
57C53×#465—24no—
58C2×#106×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C23×Dic53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66D4×Dic53—22no—
67D4×Dic53—41no—
68D4×Dic53—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53D411yescharacteristic
88C2×C22⋊Dic53D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133148—22no—
93#13133148—22no—
94#13133146—22no—
95#13133146—41no—
96#13133141—22no—
97#13133141—41no—
98#13133138—22no—
99#13133138—22no—
100C22⋊#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×Q8×C53C2311yescharacteristic
110C22×D4×C53—21no—
111C22×D4×C53C2311yescharacteristic
112C23×C4×C53C2311yescharacteristic
113C2×SD16×C53—22no—
114C2×SD16×C53—22no—
115C2×SD16×C53—41no—
116C22×C8×C53—22no—
117C22×C8×C53—22no—
118C22×C8×C53—41no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53—22no—
123#121×C53—22no—
124#121×C53D414yes—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C22⋊C4×C53—21no—
129C2×C22⋊C4×C53—21no—
130#106×C53—22no—
131#103×C53—22no—
132#102×C53—22no—
133C24×C53—21no—
134C24×C53C2×D411yescharacteristic
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×D4×C53—24no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—42no—
148C2×D4×C53—42no—
149C2×D4×C53C2×D412yes—
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×D412yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C22×Dic53—42no—
169C22×Dic53—42no—
170C22×Dic53—81no—
171C22×Dic53—81no—
172C22×Dic53—81no—
173C22×Dic53—81no—
174C22×Dic53C2×D411yescharacteristic
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178SD16×C53—44no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—44no—
185C2×C8×C53#4112yes—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—42no—
190C4⋊C4×C53C2×D412yes—
191C22⋊C4×C53—24no—
192C22⋊C4×C53—42no—
193C22⋊Dic53—22no—
194C22⋊Dic53—42no—
195C22⋊Dic53—42no—
196C22⋊Dic53—81no—
197C22⋊Dic53SD1614yescomplemented
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—42no—
206C4⋊Dic53—42no—
207C4⋊Dic53—42no—
208C4⋊Dic53#4112yes—
209#11719919—81no—
210#11719919—81no—
211#11719919—81no—
212#11719919—81no—
213C4×Dic53—21no—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—21no—
217C4×Dic53—42no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—22no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53#12112yes—
236C23×C53C2×SD1612yes—
237C23×C53#13712yes—
238C23×C53C22×D411yescharacteristic
239Q8×C53—41no—
240Q8×C53—41no—
241Q8×C53—41no—
242Q8×C53—41no—
243D4×C53—22no—
244D4×C53—22no—
245D4×C53—48no—
246D4×C53—41no—
247D4×C53—41no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
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#12112yes—
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×C4×C532−1+411yescharacteristic
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—22no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—84no—
293C2×Dic53—84no—
294C2×Dic53—84no—
295C2×Dic53C2×SD1612yes—
296C2×Dic53C2×SD1612yes—
297C8×C53—22no—
298C8×C53—41no—
299#88322—41no—
300#88322—41no—
301#25202—531nomaximal, Sylow
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—42no—
311C22×C53—81no—
312C22×C53—81no—
313C22×C53—81no—
314C22×C53—81no—
315C22×C53#46114yes—
316C22×C53C2×#12111yescharacteristic
317C22×C53#54611yescharacteristic
318C22×C53#54711yescharacteristic
319C22×C53C22×SD1611yescharacteristic
320C22×C53C2×#13711yescharacteristic
321C22×C53#57611yescharacteristic
322C22×C53#57711yescharacteristic
323C4×C53—22no—
324C4×C53—42no—
325C4×C53—81no—
326C4×C53—81no—
327C4×C53#52911yescharacteristic
328C4×C53#53911yescharacteristic
329C4×C53#54611yescharacteristic
330C4×C53#54711yescharacteristic
331Dic53—22no—
332Dic53—22no—
333Dic53—41no—
334Dic53—42no—
335Dic53—41no—
336Dic53—81no—
337Dic53—81no—
338Dic53—81no—
339C2×#481—531no—
340C2×#465D5311yescharacteristic
341#2502—531no—
342#2361—531no—
343#2084—531no—
344#2071—531no—
345#2066—531no—
346#2055—531no—
347#1948—531no—
348#1927—531no—
349#1877—531no—
350#1872—531no—
351#1850—531no—
352#1802—531no—
353C2×#340—531no—
354C2×C53—24no—
355C2×C53—82no—
356C2×C53#242511yescharacteristic
357C2×C53C2×#46111yescharacteristic
358C2×C53#304511yescharacteristic
359C2×C53#329511yescharacteristic
360C2×C53#331611yescharacteristic
361C2×C53#333511yescharacteristic
362C2×C53#334611yescharacteristic
363C22×SD16C2×D5311yescharacteristic
364C23×C8C2×D5311yescharacteristic
365C2×C22⋊Q8C2×D5311yescharacteristic
366C2×#121C2×D5311yescharacteristic
367C2×C4×D4—531no—
368C2×C4×D4—1061no—
369C22×C4⋊C4—531no—
370C22×C4⋊C4—531no—
371#481—1064no—
372#465—24no—
373C2×#106—531no—
374C2×#106—531no—
375C2×#106C2×D5311yescharacteristic
376C2×C4⋊C8—531no—
377C2×C4⋊C8—531no—
378C2×#103—531no—
379C2×#103—531no—
380C2×#103C2×D5311yescharacteristic
381C2×#102—531no—
382C2×#102—531no—
383C2×#102C2×D5311yescharacteristic
384C2×C22⋊C8—531no—
385#398—1061no—
386#398—1061no—
387#391—531no—
388#391—531no—
389#390—531no—
390#389—531no—
391#388—1061no—
392#386—1061no—
393#385—1061no—
394#385—1061no—
395#385—1061no—
396#384—531no—
397#384—531no—
398C4×C4⋊C4—531no—
399C4×C22⋊C4—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#340—531no—
404#340—531no—
405#340—531no—
406#336—531no—
407#336—531no—
408C53#2520211yescharacteristic, complemented, Sylow
409C22×Q8C22×D5311yescharacteristic
410C22×D4—21no—
411C22×D4C22×D5311yescharacteristic
412C23×C4—1061no—
413C23×C4—1061no—
414C23×C4—1061no—
415C23×C4C22×D5311yescharacteristic
416C2×SD16—22no—
417C2×SD16—22no—
418C2×SD16—41no—
419C22×C8—22no—
420C22×C8—22no—
421C22×C8—41no—
422C22×C8—531no—
423C22×C8—531no—
424C22×C8C22×D5311yescharacteristic
425C22×C8C22×D5311yescharacteristic
426C22⋊Q8—22no—
427C22⋊Q8—22no—
428#121—22no—
429#121C22⋊D5314yes—
430C4×D4—1062no—
431C4×D4—2121no—
432C4×D4—2124no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—1062no—
439C2×C4⋊C4—1062no—
440C2×C4⋊C4—1062no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—2121no—
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—531no—
454C2×C22⋊C4—531no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C42—531no—
459C2×C42—531no—
460C2×C42—1061no—
461#106—22no—
462#106—1062no—
463#106—1062no—
464C4⋊C8—1062no—
465C4⋊C8—2121no—
466#103—22no—
467#103—1062no—
468#103—2121no—
469#102—22no—
470#102—1062no—
471#102—1062no—
472C22⋊C8—1062no—
473#95—531no—
474#95—531no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479#95—1061no—
480#95—1061no—
481C24—21no—
482C24D4×D5311yescharacteristic
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C22⋊D5312yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—531no—
506C22×C4—531no—
507C22×C4—531no—
508C22×C4—531no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—4241no—
514C22×C4—4241no—
515C22×C4—4241no—
516C22×C4—4241no—
517C22×C4—2122no—
518C22×C4—2122no—
519C22×C4—2122no—
520C22×C4—2122no—
521C22×C4—2122no—
522C22×C4D4×D5311yescharacteristic
523C22×C4#1171994712yes—
524C22×C4C2×C22⋊D5312yes—
525C22×C4C23×D5311yescharacteristic
526SD16—44no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—44no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C2×C8—1061no—
539C2×C8#1171994712yes—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—532no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—4241no—
553C4⋊C4—4241no—
554C4⋊C4—4241no—
555C4⋊C4—4241no—
556C4⋊C4—2122no—
557C4⋊C4—2122no—
558C4⋊C4—2122no—
559C4⋊C4C2×C22⋊D5312yes—
560C22⋊C4—24no—
561C22⋊C4—42no—
562C22⋊C4—534no—
563C22⋊C4—1062no—
564C22⋊C4—4241no—
565C22⋊C4—2122no—
566C22⋊C4—2122no—
567C42—1061no—
568C42—1061no—
569C42—1061no—
570C42—1061no—
571C42—2122no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23#1313318912yes—
584C23SD16×D5312yes—
585C23#1313324612yes—
586C23C2×D4×D5311yescharacteristic
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—41no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4—41no—
595D4—41no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
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—532no—
612C2×C4—532no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1062no—
624C2×C4—4244no—
625C2×C4—4244no—
626C2×C4—4244no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4#1313318912yes—
639C2×C4C22×C22⋊D5311yescharacteristic
640C2×C4#41×D5311yescharacteristic
641C2×C4C2×#1171994711yescharacteristic
642C2×C4C2×#1171994711yescharacteristic
643C2×C4#1313326211yescharacteristic
644C2×C4#1313326411yescharacteristic
645C8—22no—
646C8—41no—
647C8—2121no—
648C8—2121no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#1681082614yes—
663C22C2×#1313318911yescharacteristic
664C22#1681126211yescharacteristic
665C22#1681127311yescharacteristic
666C22C2×SD16×D5311yescharacteristic
667C22C2×#1313324611yescharacteristic
668C22#1681142511yescharacteristic
669C22#1681143011yescharacteristic
670C4—22no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—1062no—
675C4—1062no—
676C4—4241no—
677C4—4241no—
678C4—4241no—
679C4—2122no—
680C4—2121no—
681C4—2121no—
682C4#1681115411yescharacteristic
683C4#1681121811yescharacteristic
684C4#1681126711yescharacteristic
685C4#1681127211yescharacteristic
686C2—24no—
687C2—82no—
688C2#2027827211yescharacteristic
689C2C2×#1681082611yescharacteristic
690C2#2028425211yescharacteristic
691C2#2028653411yescharacteristic
692C2#2028683211yescharacteristic
693C2#2028706711yescharacteristic
694C2#2028720611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#465C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#465D53
hypocenterC53#25202
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
2. #20279194C2semidirect
4. #20276194C2semidirect
5. #20276121C2semidirect
6. #20276092C2semidirect
7. #20276028C2semidirect
8. #20275475C2semidirect
9. #20275352C2semidirect
10. #20275051C2semidirect
11. #20275023C2semidirect
12. #20274893C2semidirect
13. #20274713C2semidirect
14. C2×#16810386C2semidirect
197. C22⋊Dic53SD16semidirect
408. C53#25202semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
#202791942
#202777352
#202761942
#202761212
#202760922
#202760282
#202754752
#202753522
#202750512
#202750232
#202748932
#202747132
C2×#168103862
C2×C53×#4652
#2520253