Finite Group Databasebeta

← #27446469

Subgroups — #27446469

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Dic53C211yesmaximal, characteristic, complemented
2#20276253C211yesmaximal, characteristic, complemented
3#20276243C211yesmaximal, characteristic
4#20276123C211yesmaximal, characteristic, complemented
5#20276118C211yesmaximal, characteristic
6#20275656C211yesmaximal, characteristic, complemented
7#20275648C211yesmaximal, characteristic
8#20275532C211yesmaximal, characteristic
9#20275473C211yesmaximal, characteristic, complemented
10#20275353C211yesmaximal, characteristic, complemented
11#20275346C211yesmaximal, characteristic, complemented
12#20275339C211yesmaximal, characteristic
13#20275332C211yesmaximal, characteristic
14#20274613C211yesmaximal, characteristic, complemented
15C53×#2592C211yesmaximal, characteristic
16C2×Q8×Dic53—21no—
17C2×#13133150C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810522—21no—
32#16810522—21no—
33#16810522—21no—
34#16810493C2211yescharacteristic
35#16810493C2211yescharacteristic
36#16810493C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C53×#397—21no—
51C53×#393—21no—
52C53×#391C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#388—21no—
57C53×#388—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—44no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77#13133150—22no—
78#13133150—41no—
79#13133150—41no—
80#13133148—22no—
81C4⋊#88322—22no—
82#13133141—22no—
83#13133141—22no—
84#13133141—41no—
85#13133141—41no—
86#13133138—22no—
87#13133138—22no—
88#13133138—41no—
89#13133138—41no—
90#13133128—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C22×Q8×C53D411yescharacteristic
95C22×Q8×C53C2311yescharacteristic
96C22×D4×C53C2311yescharacteristic
97#124×C53—22no—
98#124×C53—22no—
99C4×Q8×C53—22no—
100C4×Q8×C53—22no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121C24×C53—21no—
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53—22no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—81no—
163C22⋊C4×C53—81no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—81no—
169C42×C53#4112yes—
170C42×C53#4112yes—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—42no—
179C4⋊Dic53—42no—
180C4⋊Dic53—42no—
181C4⋊Dic53—81no—
182C4⋊Dic53—81no—
183C4⋊Dic53—81no—
184C4⋊Dic53#4112yes—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322#4112yes—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208Q8×C53—28no—
209Q8×C53—41no—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213D4×C53—41no—
214D4×C53—41no—
215D4×C53—41no—
216D4×C53—41no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53#12312yes—
241C2×C4×C53#12312yes—
242C2×C4×C53AGL(1,ℤ/8)12yes—
243C2×C4×C53#13712yes—
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—22no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262#88322—22no—
263#88322—41no—
264#88322—41no—
265#88322—41no—
266#26330—531nomaximal, Sylow
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53C2×#12311yescharacteristic
275C22×C53#54811yescharacteristic
276C22×C53#55011yescharacteristic
277C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C53C2×#13711yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57811yescharacteristic
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#52911yescharacteristic
291C4×C53#54211yescharacteristic
292C4×C53#55611yescharacteristic
293C4×C53#55911yescharacteristic
294Dic53—42no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298#2592D5311yescharacteristic
299Q8×C4⋊C4—531no—
300#2095—531no—
301#2093—531no—
302#2072—531no—
303#2071—531no—
304#1985—531no—
305#1983—531no—
306#1959—531no—
307#1947—531no—
308#1928—531no—
309#1927—531no—
310#1926—531no—
311#1925—531no—
312#1776—531no—
313C2×C53—161no—
314C2×C53#244811yescharacteristic
315C2×C53#309211yescharacteristic
316C2×C53#309811yescharacteristic
317C2×C53#336711yescharacteristic
318C2×C53#337511yescharacteristic
319C2×C53#339811yescharacteristic
320C2×C53#339911yescharacteristic
321C2×#124C2×D5311yescharacteristic
322C2×C4×Q8—1061no—
323C2×C4×Q8C2×D5311yescharacteristic
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#106—531no—
328C2×C4⋊C8—531no—
329C2×#103—531no—
330C2×#103—531no—
331C2×#103—531no—
332C2×#103—531no—
333C2×#102—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#97—531no—
338#397—21no—
339#393—21no—
340#391C2×D5311yescharacteristic
341#391C2×D5311yescharacteristic
342#390C2×D5311yescharacteristic
343#390C2×D5311yescharacteristic
344#389—1061no—
345#389—1061no—
346#389—1061no—
347#388—21no—
348#388—21no—
349#384—531no—
350#384—531no—
351#384—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4C2×D5311yescharacteristic
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#336—531no—
361#336—531no—
362#336—531no—
363#336—531no—
364C53#2633011yescharacteristic, complemented, Sylow
365C22×Q8C22⋊D5311yescharacteristic
366C22×Q8C22×D5311yescharacteristic
367C22×D4C22×D5311yescharacteristic
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—531no—
372#124—22no—
373#124—22no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—2124no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4C22⋊D5311yescharacteristic
388C2×C4⋊C4C22⋊D5311yescharacteristic
389C2×C4⋊C4C22⋊D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—21no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42—1061no—
402C2×C42C22×D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404C2×C42C22×D5311yescharacteristic
405#107—1062no—
406#107—2121no—
407#107—2121no—
408#106—1062no—
409C4⋊C8—1062no—
410#103—1062no—
411#103—1062no—
412#103—2121no—
413#103—2121no—
414#102—1062no—
415#102—1062no—
416#102—2121no—
417#102—2121no—
418#97—1062no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C2×C8—532no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—532no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—4241no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—2122no—
502C42—2122no—
503C42—2122no—
504C42#1171994712yes—
505C42#1171994712yes—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C22×C22⋊D5311yescharacteristic
514Q8—28no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—1062no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4#1313319812yes—
559C2×C4#1313319812yes—
560C2×C4#1313324212yes—
561C2×C4#1313324712yes—
562C2×C4#41×D5311yescharacteristic
563C2×C4#41×D5311yescharacteristic
564C2×C4C2×#1171994711yescharacteristic
565C2×C4C2×#1171994711yescharacteristic
566C2×C4#1313326311yescharacteristic
567C2×C4#1313326411yescharacteristic
568C8—1062no—
569C8—2121no—
570C8—2121no—
571C8—2121no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1313319811yescharacteristic
580C22#1681128011yescharacteristic
581C22#1681129211yescharacteristic
582C22C2×#1313324211yescharacteristic
583C22C2×#1313324711yescharacteristic
584C22#1681142611yescharacteristic
585C22#1681143811yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—2122no—
599C4#1681115211yescharacteristic
600C4#1681124211yescharacteristic
601C4#1681134211yescharacteristic
602C4#1681135511yescharacteristic
603C2—161no—
604C2#2027850911yescharacteristic
605C2#2028463511yescharacteristic
606C2#2028468911yescharacteristic
607C2#2028742911yescharacteristic
608C2#2028752511yescharacteristic
609C2#2028774611yescharacteristic
610C2#2028775311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2592C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2592D53
hypocenterC53#26330
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Dic53C2semidirect
2. #20276253C2semidirect
4. #20276123C2semidirect
6. #20275656C2semidirect
9. #20275473C2semidirect
10. #20275353C2semidirect
11. #20275346C2semidirect
14. #20274613C2semidirect
364. C53#26330semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Dic532
#202762532
#202762432
#202761232
#202761182
#202756562
#202756482
#202755322
#202754732
#202753532
#202753462
#202753392
#202753322
#202746132
C53×#25922
#2633053