Finite Group Databasebeta

← #27447152

Subgroups — #27447152

Order 13568.

820 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810930C211yesmaximal, characteristic, complemented
2C2×SD16×Dic53C211yesmaximal, characteristic
3#20278589C211yesmaximal, characteristic
4#20277633C211yesmaximal, characteristic, complemented
5#20276234C211yesmaximal, characteristic
6#20276201C211yesmaximal, characteristic, complemented
7#20276089C211yesmaximal, characteristic
8#20276054C211yesmaximal, characteristic, complemented
9#20275442C211yesmaximal, characteristic, complemented
10#20275275C211yesmaximal, characteristic, complemented
11#20275072C211yesmaximal, characteristic
12#20274981C211yesmaximal, characteristic, complemented
13#20274971C211yesmaximal, characteristic, complemented
14#20274960C211yesmaximal, characteristic
15C53×#1880C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20#16810930—24no—
21SD16×Dic53—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810559—21no—
34#16810546—21no—
35#16810526C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810521—21no—
38#16810521C2211yescharacteristic
39#16810519—21no—
40#16810519C2211yescharacteristic
41#16810519C2211yescharacteristic
42#16810495—21no—
43#16810488—21no—
44#16810488C2211yescharacteristic
45#16810481—21no—
46C4×#11719919—21no—
47C4×#11719919C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×SD16×C53C2211yescharacteristic
55C2×C4×Q8×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C53×#390C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133209—22no—
65#13133209—41no—
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69Q8×Dic53—41no—
70D4×Dic53—22no—
71D4×Dic53—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919D411yescharacteristic
85C2×#11719919D411yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—41no—
94#13133150—22no—
95#13133149—22no—
96#13133148—22no—
97#13133148—22no—
98C4⋊#88322—22no—
99#13133141—22no—
100#13133141D414yes—
101#13133138—22no—
102#13133138—22no—
103C8×Dic53—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C22×D4×C53C2311yescharacteristic
114C2×SD16×C53—22no—
115C2×SD16×C53—22no—
116C2×SD16×C53Q814yes—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C4×Q8×C53—22no—
120C4×Q8×C53—41no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53—21no—
126C2×C42×C53—21no—
127C2×C42×C53C2311yescharacteristic
128#103×C53—22no—
129#102×C53—41no—
130C24×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53C2×D412yes—
135C2×Q8×C53C2×Q812yes—
136C2×Q8×C53#4112yes—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53C2×Q812yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160SD16×C53—28no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C2×Q812yes—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C22⋊C4×C53—81no—
173C22⋊C4×C53—81no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—81no—
180C4⋊Dic53—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53—81no—
189C4⋊Dic53C2×D412yes—
190C4⋊Dic53#4112yes—
191#11719919—22no—
192#11719919—41no—
193#11719919—42no—
194#11719919—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—81no—
198C4×Dic53—24no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322C2×D412yes—
210C23×C53—22no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53C22×D411yescharacteristic
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53C22⋊Q814yes—
222D4×C53—22no—
223D4×C53—22no—
224D4×C53—41no—
225D4×C53—41no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—22no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—84no—
271C2×Dic53#13512yes—
272C8×C53—22no—
273C8×C53—41no—
274#88322—22no—
275#88322—41no—
276#26375—531nomaximal, Sylow
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53#52511yescharacteristic
285C22×C53D4211yescharacteristic
286C22×C53D4×Q811yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53#57511yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57711yescharacteristic
291C4×C53—24no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53C2×C22⋊Q811yescharacteristic
297C4×C53#53911yescharacteristic
298C4×C53#54711yescharacteristic
299C4×C53#55411yescharacteristic
300Dic53—22no—
301Dic53—41no—
302Dic53—42no—
303Dic53—41no—
304Dic53—41no—
305Dic53—42no—
306Dic53—81no—
307Dic53—81no—
308C2×#479—531no—
309C2×C4×SD16—531no—
310#2454—531no—
311#2351—531no—
312#2092—531no—
313#2085—531no—
314#2065—531no—
315#2059—531no—
316#1941—531no—
317#1916—531no—
318#1880—531no—
319#1880D5311yescharacteristic
320#1865—531no—
321#1863—531no—
322#1862—531no—
323C2×C53—82no—
324C2×C53#244911yescharacteristic
325C2×C53#310211yescharacteristic
326C2×C53#312611yescharacteristic
327C2×C53#327111yescharacteristic
328C2×C53#328211yescharacteristic
329C2×C53#338111yescharacteristic
330C2×C53#338811yescharacteristic
331C22×SD16C2×D5311yescharacteristic
332C2×#125—531no—
333C2×C4×Q8—531no—
334C2×C4×Q8—531no—
335C2×C4×Q8C2×D5311yescharacteristic
336C2×C4×D4—531no—
337#479—1064no—
338C4×SD16—1064no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×#106—531no—
344C2×C4⋊C8—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103C2×D5311yescharacteristic
348C2×#102—531no—
349C2×#102—531no—
350C2×#102C2×D5311yescharacteristic
351C2×C4×C8—531no—
352#399—1061no—
353#396—1061no—
354#390—531no—
355#390C2×D5311yescharacteristic
356#389—531no—
357#389—531no—
358#389—531no—
359#389—531no—
360#389—1061no—
361#389—1061no—
362#385—1061no—
363#382—531no—
364#382—1061no—
365#380—1061no—
366C4×C4⋊C4—531no—
367C4×C4⋊C4—1061no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#336—531no—
373#336—531no—
374#336—531no—
375#336C2×D5311yescharacteristic
376C53#2637511yescharacteristic, complemented, Sylow
377C22×Q8C22×D5311yescharacteristic
378C22×D4C22×D5311yescharacteristic
379C23×C4—1061no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16Dic10614yes—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387#125—1062no—
388#125—2121no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—1062no—
394C4×Q8—2121no—
395C4×D4—1062no—
396C4×D4—2121no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C42—21no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—1061no—
423C2×C42—1061no—
424C2×C42C22×D5311yescharacteristic
425#107—1062no—
426#107—2121no—
427#106—1062no—
428#106—1062no—
429#106—1062no—
430C4⋊C8—1062no—
431#103—22no—
432#103—534no—
433#103—1062no—
434#102—41no—
435#102—1062no—
436#102—1062no—
437C4×C8—1062no—
438#95—531no—
439#95—531no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446C24—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8C2×Dic10612yes—
451C2×Q8D4×D5312yes—
452C2×Q8#1171994712yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4C2×Dic10612yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—4241no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4D4×D5311yescharacteristic
474C22×C4D4×D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476SD16—28no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—532no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8C2×Dic10612yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—532no—
495C4⋊C4—532no—
496C4⋊C4—1062no—
497C4⋊C4—1064no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—4241no—
505C4⋊C4—4241no—
506C4⋊C4—2121no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—4241no—
515C22⋊C4—2122no—
516C22⋊C4—2122no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—1064no—
521C42—1061no—
522C42—1061no—
523C42—1061no—
524C42—1061no—
525C42—2122no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×D4×D5311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1313319414yes—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—532no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1062no—
571C2×C4—4244no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C22×Dic10611yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326411yescharacteristic
588C8—22no—
589C8—41no—
590C8—1062no—
591C8—2121no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1681113311yescharacteristic
600C22#1681125511yescharacteristic
601C22D4×Dic10611yescharacteristic
602C22#135×D5311yescharacteristic
603C22#1681142211yescharacteristic
604C22#1681142911yescharacteristic
605C22#1681143311yescharacteristic
606C4—24no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—1062no—
612C4—4241no—
613C4—4241no—
614C4—2121no—
615C4—2122no—
616C4—2121no—
617C4—2121no—
618C4—2122no—
619C4C2×#1313319411yescharacteristic
620C4#1681121811yescharacteristic
621C4#1681127111yescharacteristic
622C4#1681133211yescharacteristic
623C2—82no—
624C2#2027852811yescharacteristic
625C2#2028473211yescharacteristic
626C2#2028500411yescharacteristic
627C2#2028630311yescharacteristic
628C2#2028642511yescharacteristic
629C2#2028759111yescharacteristic
630C2#2028766111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1880C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1880D53
hypocenterC53#26375
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#16810930C2semidirect
4. #20277633C2semidirect
6. #20276201C2semidirect
8. #20276054C2semidirect
9. #20275442C2semidirect
10. #20275275C2semidirect
12. #20274981C2semidirect
13. #20274971C2semidirect
376. C53#26375semidirect

Maximal subgroups

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

subgroupindex
C2×#168109302
C2×SD16×Dic532
#202785892
#202776332
#202762342
#202762012
#202760892
#202760542
#202754422
#202752752
#202750722
#202749812
#202749712
#202749602
C53×#18802
#2637553