Finite Group Databasebeta

← #27447148

Subgroups — #27447148

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×C4×#11719924C211yesmaximal, characteristic
3#20278587C211yesmaximal, characteristic
4#20276239C211yesmaximal, characteristic
5#20276197C211yesmaximal, characteristic, complemented
6#20276086C211yesmaximal, characteristic
7#20276053C211yesmaximal, characteristic, complemented
8#20275440C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275070C211yesmaximal, characteristic
11#20275070C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274969C211yesmaximal, characteristic, complemented
14#20274961C211yesmaximal, characteristic
15C53×#2351C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19#16810930—24no—
20C4×#11719924—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810558—21no—
35#16810547—21no—
36#16810525C2211yescharacteristic
37#16810525C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810494—21no—
41#16810482—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×C4×Q8×C53C2211yescharacteristic
53C53×#389—21no—
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#382—21no—
58C53×#382C2211yescharacteristic
59C4×C4⋊C4×C53—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C2×#11719924—22no—
64C2×#11719924—22no—
65C2×#11719924Q814yes—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133209—22no—
71#13133209—41no—
72C4×D212—22no—
73C4×D212—41no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86#13133150—22no—
87#13133150—41no—
88#13133148—22no—
89#13133148—22no—
90#13133148—22no—
91C4⋊#88322—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141D414yes—
95#13133137—22no—
96#13133137—22no—
97#13133137—41no—
98C4×#88322—22no—
99#13133124—21no—
100#13133124—21no—
101C22×Q8×C53C2311yescharacteristic
102C4×Q8×C53—22no—
103C4×Q8×C53—22no—
104C4×Q8×C53—22no—
105C4×Q8×C53—22no—
106C4×Q8×C53—41no—
107C4×Q8×C53—41no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C23×D53—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—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212C2×Q812yes—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—81no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—41no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53#4112yes—
173C42×C53—24no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—42no—
181C42×C53—42no—
182#11719924—28no—
183#11719921—42no—
184#11719921—42no—
185#11719921—81no—
186#11719921—81no—
187#11719921—81no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53C2×D412yes—
194#11719919—42no—
195#11719919—42no—
196#11719919—81no—
197#11719919—81no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322C2×D412yes—
209C2×#88322C2×Q812yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53C22⋊Q814yes—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
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—21no—
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—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53#13512yes—
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267D212—22no—
268D212—22no—
269D212—41no—
270D212—41no—
271C4×D53—84no—
272#88322—22no—
273#88322—22no—
274#88322—41no—
275#88322—41no—
276#26375—531nomaximal, Sylow
277C22×C53#52511yescharacteristic
278C22×C53D4211yescharacteristic
279C22×C53D4×Q811yescharacteristic
280C22×C53C2×#13511yescharacteristic
281C22×C53#57511yescharacteristic
282C22×C53#57711yescharacteristic
283C22×C53#57711yescharacteristic
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—81no—
288C2×D53—81no—
289C2×D53—81no—
290C2×D53—81no—
291C4×C53—24no—
292C4×C53—22no—
293C4×C53—41no—
294C4×C53—42no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53C2×C22⋊Q811yescharacteristic
302C4×C53#53911yescharacteristic
303C4×C53#54711yescharacteristic
304C4×C53#55411yescharacteristic
305Dic53—42no—
306Dic53—81no—
307Dic53—81no—
308C2×#479—531no—
309C2×C4×SD16—531no—
310#2454—531no—
311#2351D5311yescharacteristic, complemented
312#2092—531no—
313#2085—531no—
314#2065—531no—
315#2059—531no—
316#1941—531no—
317#1916—531no—
318#1880—531no—
319#1880—531no—
320#1865—531no—
321#1863—531no—
322#1862—531no—
323C2×C53#244911yescharacteristic
324C2×C53#310211yescharacteristic
325C2×C53#312611yescharacteristic
326C2×C53#327111yescharacteristic
327C2×C53#328211yescharacteristic
328C2×C53#338111yescharacteristic
329C2×C53#338811yescharacteristic
330D53—82no—
331C22×SD16—531no—
332C2×#125—531no—
333C2×C4×Q8C2×D5311yescharacteristic
334C2×C4×Q8C2×D5311yescharacteristic
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×#103—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×C4×C8—531no—
352#399—1061no—
353#396—1061no—
354#390—531no—
355#390—531no—
356#389—21no—
357#389—21no—
358#389—531no—
359#389—531no—
360#389C2×D5311yescharacteristic
361#389C2×D5311yescharacteristic
362#385—1061no—
363#382—21no—
364#382C2×D5311yescharacteristic
365#380—1061no—
366C4×C4⋊C4—21no—
367C4×C4⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#336—531no—
373#336—531no—
374#336—531no—
375#336—531no—
376C53#2637511yescharacteristic, complemented, Sylow
377C22×Q8C22×D5311yescharacteristic
378C22×D4—531no—
379C23×C4—1061no—
380C2×SD16—534no—
381C2×SD16—1062no—
382C2×SD16—1062no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387#125—1062no—
388#125—2121no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—41no—
394C4×Q8—41no—
395C4×D4—1062no—
396C4×D4—2121no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22⋊D5311yescharacteristic
410C2×C4⋊C4C22⋊D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42C22×D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424C2×C42C22×D5311yescharacteristic
425#107—1062no—
426#107—2121no—
427#106—1062no—
428#106—1062no—
429#106—1062no—
430C4⋊C8—1062no—
431#103—534no—
432#103—1062no—
433#103—1062no—
434#102—1062no—
435#102—1062no—
436#102—2121no—
437C4×C8—1062no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1061no—
443#95—1061no—
444#95C22⋊D5311yescharacteristic
445#95C22⋊D5311yescharacteristic
446C24—1061no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8#1171994512yes—
451C2×Q8D4×D5312yes—
452C2×Q8Q8×D5312yes—
453C2×D4—532no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—4241no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476SD16—1068no—
477C2×C8—532no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—532no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—4241no—
508C4⋊C4—4241no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4#1171994512yes—
512C22⋊C4—4241no—
513C22⋊C4—4241no—
514C22⋊C4—4241no—
515C22⋊C4—2122no—
516C22⋊C4—2122no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—42no—
526C23—1062no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23—2121no—
532C23—2121no—
533C23C22×C22⋊D5311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1313319514yes—
538D4—1062no—
539D4—1062no—
540D4—2121no—
541D4—2121no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—1062no—
572C2×C4—4244no—
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#1313323712yes—
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×Q8×D5311yescharacteristic
584C2×C4C2×#1171994511yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326511yescharacteristic
588C8—1062no—
589C8—1062no—
590C8—2121no—
591C8—2121no—
592C22—4241no—
593C22—4241no—
594C22—4241no—
595C22—4241no—
596C22—2122no—
597C22—2122no—
598C22—2122no—
599C22#1681113411yescharacteristic
600C22D4×C22⋊D5311yescharacteristic
601C22Q8×C22⋊D5311yescharacteristic
602C22C2×#1313323711yescharacteristic
603C22#1681142111yescharacteristic
604C22#1681143111yescharacteristic
605C22#1681143111yescharacteristic
606C4—22no—
607C4—24no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—81no—
615C4—81no—
616C4—4241no—
617C4—4241no—
618C4—2122no—
619C4C2×#1313319511yescharacteristic
620C4#1681122011yescharacteristic
621C4#1681127411yescharacteristic
622C4#1681132811yescharacteristic
623C2—4242no—
624C2#2027852411yescharacteristic
625C2#2028473311yescharacteristic
626C2#2028500611yescharacteristic
627C2#2028630111yescharacteristic
628C2#2028642311yescharacteristic
629C2#2028758611yescharacteristic
630C2#2028765411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2351C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2351D53
hypocenterC53#26375
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#16810930C2semidirect
5. #20276197C2semidirect
7. #20276053C2semidirect
8. #20275440C2semidirect
9. #20275274C2semidirect
12. #20274983C2semidirect
13. #20274969C2semidirect
15. C53×#2351C2semidirect
311. #2351D53semidirect
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×C4×#117199242
#202785872
#202762392
#202761972
#202760862
#202760532
#202754402
#202752742
#202750702
#202749832
#202749692
#202749612
C53×#23512
#2637553