Finite Group Databasebeta

← #27275974

Subgroups — #27275974

Order 13568.

852 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275635C211yesmaximal, characteristic, complemented
2#20274540C211yesmaximal, characteristic, complemented
3#20273326C211yesmaximal, characteristic, complemented
4#20273303C211yesmaximal, characteristic, complemented
5#20273240C211yesmaximal, characteristic, complemented
6#20273158C211yesmaximal, characteristic, complemented
7C53×#3519C211yesmaximal, characteristic
8#16810680C2211yescharacteristic, complemented
9C2×#13133144—22no—
10C2×#13133138—21no—
11C4×#11719917C2211yescharacteristic, complemented
12C2×#13133128C2211yescharacteristic, complemented
13C4×C4⋊Dic53C2211yescharacteristic, complemented
14#16810385—21no—
15#16810365—21no—
16#16810363—21no—
17#16810363—22no—
18#16810360—21no—
19#16810357—21no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C53×#555C411yescharacteristic, complemented
23C53×#554C411yescharacteristic, complemented
24C53×#553—21no—
25C53×#553—21no—
26C53×#550C411yescharacteristic, complemented
27C53×#546—22no—
28C53×#546—21no—
29D42×C53—21no—
30D42×C53C411yescharacteristic, complemented
31C53×#540—21no—
32C53×#538—22no—
33C53×#532C2211yescharacteristic
34C53×#529C2211yescharacteristic
35C53×#525—21no—
36C2×#121×C53—21no—
37C53×#518—21no—
38C2×C4×D4×C53C2211yescharacteristic
39C2×#11719917—21no—
40C22×#88322—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44#13133144—44no—
45#13133138—24no—
46C22⋊#88322—21no—
47#13133128—21no—
48#13133128—21no—
49#13133128—21no—
50#13133128—21no—
51C4×#88322—21no—
52#13133124—21no—
53C2×#41×C53—21no—
54C2×#41×C53D412yes—
55C22×D4×C53—22no—
56C22×D4×C53—21no—
57C22×D4×C53—21no—
58C22×D4×C53D411yescharacteristic
59C23×C4×C53—21no—
60C4⋊Q8×C53C2×C411yescharacteristic
61#127×C53—21no—
62#127×C53C2×C411yescharacteristic
63#126×C53—22no—
64#126×C53—21no—
65#125×C53—21no—
66#125×C53D411yescharacteristic
67#124×C53—22no—
68#124×C53D411yescharacteristic
69#124×C53D412yes—
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—21no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53—22no—
76C22⋊Q8×C53D411yescharacteristic
77#121×C53—22no—
78#121×C53—22no—
79#121×C53—22no—
80#121×C53—22no—
81#121×C53—21no—
82#121×C53—21no—
83#121×C53—21no—
84#121×C53—21no—
85#121×C53—22no—
86#121×C53—21no—
87#121×C53—21no—
88#121×C53—21no—
89#121×C53D411yescharacteristic
90#120×C53—22no—
91#120×C53—22no—
92#120×C53—22no—
93C4×Q8×C53C2×C411yescharacteristic
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53C2×C411yescharacteristic
105C4×D4×C53C2×C411yescharacteristic
106C4×D4×C53C2×C411yescharacteristic
107C4×D4×C53D411yescharacteristic
108C4×D4×C53D411yescharacteristic
109#117×C53—21no—
110#117×C53—21no—
111#117×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53—22no—
115C2×C22⋊C4×C53—22no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C24×C53—22no—
119C24×C53—21no—
120#41×C53—42no—
121#41×C53—42no—
122#41×C53—82no—
123C2×Q8×C53C22⋊C411yescharacteristic
124C2×Q8×C53C22⋊C412yes—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—21no—
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—22no—
132C2×D4×C53—22no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—41no—
137C2×D4×C53—42no—
138C2×D4×C53—42no—
139C2×D4×C53—82no—
140C2×D4×C53—82no—
141C2×D4×C53—82no—
142C2×D4×C53C22⋊C411yescharacteristic
143C2×D4×C53C22⋊C411yescharacteristic
144C2×D4×C53C22⋊C411yescharacteristic
145C2×D4×C53C22⋊C412yes—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—22no—
154C22×C4×C53—21no—
155C22×C4×C53—82no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×Dic53—41no—
160C4⋊C4×C53—22no—
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—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C22⋊C411yescharacteristic
170C4⋊C4×C53C22⋊C411yescharacteristic
171C4⋊C4×C53C22⋊C411yescharacteristic
172C4⋊C4×C53C22⋊C411yescharacteristic
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53—21no—
184C22⋊C4×C53—21no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—21no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—22no—
189C42×C53—22no—
190C42×C53—21no—
191C42×C53C22×C411yescharacteristic
192C42×C53C2×D411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C4⋊Dic53—22no—
196C4⋊Dic53—41no—
197C4×Dic53—42no—
198#11719917—22no—
199#11719917—41no—
200C2×#88322—22no—
201C2×#88322—41no—
202C2×#88322—41no—
203C2×#88322—41no—
204C2×#88322—41no—
205C23×C53—21no—
206C23×C53—22no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—42no—
210C23×C53—44no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—81no—
214C23×C53—82no—
215C23×C53—82no—
216C23×C53#12011yescharacteristic
217Q8×C53—42no—
218Q8×C53—41no—
219D4×C53—21no—
220D4×C53—21no—
221D4×C53—21no—
222D4×C53—42no—
223D4×C53—41no—
224D4×C53—42no—
225D4×C53—42no—
226D4×C53—44no—
227D4×C53—41no—
228D4×C53—42no—
229D4×C53—41no—
230D4×C53—42no—
231D4×C53—41no—
232D4×C53—41no—
233D4×C53—82no—
234D4×C53—82no—
235D4×C53—82no—
236D4×C53—82no—
237D4×C53#10412yescomplemented
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—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—41no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—81no—
254C2×C4×C53—81no—
255C2×C4×C53—82no—
256C2×C4×C53—81no—
257C2×C4×C53—82no—
258C2×C4×C53—82no—
259C2×C4×C53C2×C22⋊C411yescharacteristic
260C2×C4×C53C2×C22⋊C411yescharacteristic
261C2×C4×C53C2×C22⋊C411yescharacteristic
262C2×C4×C53#12011yescharacteristic
263C2×C4×C53#12011yescharacteristic
264C2×C4×C53#12011yescharacteristic
265C2×Dic53—42no—
266C2×Dic53—81no—
267C2×Dic53—81no—
268#88322—42no—
269#88322—42no—
270#88322—81no—
271#8416—531nomaximal, Sylow
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—22no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—42no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—42no—
286C22×C53—82no—
287C22×C53—82no—
288C22×C53—164no—
289C22×C53#37911yescharacteristic
290C22×C53#45311yescharacteristic
291C22×C53#45511yescharacteristic
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—21no—
295C4×C53—21no—
296C4×C53—21no—
297C4×C53—41no—
298C4×C53—42no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53C2×#10411yescharacteristic
304C4×C53#42111yescharacteristic
305Dic53—82no—
306#3519D5311yescharacteristic
307#1980—531no—
308#1762—531no—
309#1530—531no—
310#1524—531no—
311#1509—531no—
312#1490—531no—
313C2×C53—21no—
314C2×C53—21no—
315C2×C53—42no—
316C2×C53—81no—
317C2×C53—82no—
318C2×C53—81no—
319C2×C53#178911yescharacteristic
320C2×C53#179211yescharacteristic
321C2×C53#179511yescharacteristic
322#555Dic5311yescharacteristic, complemented
323#554Dic5311yescharacteristic, complemented
324#553—21no—
325#553—21no—
326#550Dic5311yescharacteristic, complemented
327#546—22no—
328#546—21no—
329D42—21no—
330D42Dic5311yescharacteristic, complemented
331#540—21no—
332#538—22no—
333#532C2×D5311yescharacteristic
334#529C2×D5311yescharacteristic
335#525—21no—
336C2×#121—21no—
337#518—21no—
338C2×C4×D4C2×D5311yescharacteristic
339#432—531no—
340C2×#104—1062no—
341C2×#102—1061no—
342C4×M16—531no—
343C2×#97—531no—
344C4×C4⋊C4—531no—
345#340—1061no—
346#330—1061no—
347#329—1061no—
348#329—1062no—
349#328—1061no—
350#327—1061no—
351D4⋊C8—1061no—
352D4⋊C8—1061no—
353C53#841611yescharacteristic, complemented, Sylow
354C2×#41—21no—
355C2×#41C22⋊D5312yes—
356C22×D4—22no—
357C22×D4—21no—
358C22×D4—21no—
359C22×D4C22⋊D5311yescharacteristic
360C23×C4—21no—
361C2×M16—1061no—
362C22×C8—1061no—
363C4⋊Q8C2×Dic5311yescharacteristic
364#127—21no—
365#127C2×Dic5311yescharacteristic
366#126—22no—
367#126—21no—
368#125—21no—
369#125C22⋊D5311yescharacteristic
370#124—22no—
371#124C22⋊D5311yescharacteristic
372#124C22⋊D5312yes—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—21no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8C22⋊D5311yescharacteristic
380#121—22no—
381#121—22no—
382#121—22no—
383#121—22no—
384#121—21no—
385#121—22no—
386#121—21no—
387#121—21no—
388#121—21no—
389#121—21no—
390#121—21no—
391#121—21no—
392#121C22⋊D5311yescharacteristic
393#120—22no—
394#120—22no—
395#120—22no—
396C4×Q8C2×Dic5311yescharacteristic
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4C2×Dic5311yescharacteristic
408C4×D4C2×Dic5311yescharacteristic
409C4×D4C2×Dic5311yescharacteristic
410C4×D4C22⋊D5311yescharacteristic
411C4×D4C22⋊D5311yescharacteristic
412#117—21no—
413#117—21no—
414#117—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22⋊D5311yescharacteristic
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—21no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423C4⋊C8—1061no—
424#104—2124no—
425#102—1064no—
426C22⋊C8—1061no—
427#97—1061no—
428#97—1061no—
429#97—1061no—
430#97—1061no—
431C4×C8—1061no—
432#95—1061no—
433C24—22no—
434C24—21no—
435#41—42no—
436#41—42no—
437#41—82no—
438C2×Q8C22⋊Dic5311yescharacteristic
439C2×Q8C22⋊Dic5312yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—82no—
457C2×D4C22⋊Dic5311yescharacteristic
458C2×D4C22⋊Dic5311yescharacteristic
459C2×D4C22⋊Dic5311yescharacteristic
460C2×D4C22⋊Dic5312yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—82no—
471C22×C4—2121no—
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475M16—1062no—
476M16—2121no—
477C2×C8—1062no—
478C2×C8—2121no—
479C2×C8—2121no—
480C2×C8—2121no—
481C2×C8—2121no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—1062no—
492C4⋊C4—2121no—
493C4⋊C4C22⋊Dic5311yescharacteristic
494C4⋊C4C22⋊Dic5311yescharacteristic
495C4⋊C4C22⋊Dic5311yescharacteristic
496C4⋊C4C22⋊Dic5311yescharacteristic
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C42—22no—
514C42—21no—
515C42—2122no—
516C42C22×Dic5311yescharacteristic
517C42C2×C22⋊D5311yescharacteristic
518C42C2×C22⋊D5311yescharacteristic
519C42C2×C22⋊D5311yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—42no—
525C23—44no—
526C23—41no—
527C23—41no—
528C23—81no—
529C23—82no—
530C23—82no—
531C23#1313318611yescharacteristic
532Q8—42no—
533Q8—41no—
534D4—21no—
535D4—21no—
536D4—21no—
537D4—42no—
538D4—41no—
539D4—42no—
540D4—42no—
541D4—44no—
542D4—41no—
543D4—42no—
544D4—41no—
545D4—42no—
546D4—41no—
547D4—41no—
548D4—82no—
549D4—82no—
550D4—82no—
551D4—82no—
552D4#1313314412yescomplemented
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—4241no—
575C2×C4—4241no—
576C2×C4—2122no—
577C2×C4C2×C22⋊Dic5311yescharacteristic
578C2×C4C2×C22⋊Dic5311yescharacteristic
579C2×C4C2×C22⋊Dic5311yescharacteristic
580C2×C4#1313318611yescharacteristic
581C2×C4#1313318611yescharacteristic
582C2×C4#1313318611yescharacteristic
583C8—4241no—
584C8—2122no—
585C8—2122no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22—82no—
601C22—82no—
602C22—164no—
603C22#1681047911yescharacteristic
604C22#1681079111yescharacteristic
605C22#1681080111yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—4242no—
618C4C2×#1313314411yescharacteristic
619C4#1681064011yescharacteristic
620C2—21no—
621C2—21no—
622C2—42no—
623C2—81no—
624C2—81no—
625C2—82no—
626C2#2027465911yescharacteristic
627C2#2027466911yescharacteristic
628C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3519C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3519D53
hypocenterC53#8416
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3519

Complements

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

NH ≅ G/Ntype
1. #20275635C2semidirect
2. #20274540C2semidirect
3. #20273326C2semidirect
4. #20273303C2semidirect
5. #20273240C2semidirect
6. #20273158C2semidirect
8. #16810680C22semidirect
11. C4×#11719917C22semidirect
12. C2×#13133128C22semidirect
13. C4×C4⋊Dic53C22semidirect
22. C53×#555C4semidirect
23. C53×#554C4semidirect
26. C53×#550C4semidirect
30. D42×C53C4semidirect
237. D4×C53#104semidirect
322. #555Dic53semidirect
323. #554Dic53semidirect
326. #550Dic53semidirect
330. D42Dic53semidirect
353. C53#8416semidirect
552. D4#13133144semidirect

Maximal subgroups

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

subgroupindex
#202756352
#202745402
#202733262
#202733032
#202732402
#202731582
C53×#35192
#841653