Finite Group Databasebeta

← #27430261

Subgroups — #27430261

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Dic53C211yesmaximal, characteristic, complemented
2#20276248C211yesmaximal, characteristic
3#20276243C211yesmaximal, characteristic
4#20276191C211yesmaximal, characteristic, complemented
5#20276186C211yesmaximal, characteristic, complemented
6#20275656C211yesmaximal, characteristic, complemented
7#20275481C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic
9#20275332C211yesmaximal, characteristic
10#20275304C211yesmaximal, characteristic, complemented
11#20275297C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C53×#2580C211yesmaximal, characteristic
16C2×Q8×Dic53—21no—
17C2×#13133150C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29#16810522—21no—
30#16810522—21no—
31#16810522—21no—
32#16810493C2211yescharacteristic
33#16810493C2211yescharacteristic
34#16810493C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4×Q8×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C53×#399—21no—
51C53×#393—21no—
52C53×#390C2211yescharacteristic
53C53×#390C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#385—21no—
57C53×#385—21no—
58C53×#382C2211yescharacteristic
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—
81#13133148—41no—
82#13133148—41no—
83C4⋊#88322—22no—
84#13133138—22no—
85#13133138—41no—
86#13133138—41no—
87#13133138Q814yescomplemented
88#13133128—22no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92C22×Q8×C53D411yescharacteristic
93C22×D4×C53C2311yescharacteristic
94C23×C4×C53—21no—
95C4×Q8×C53—22no—
96C4×Q8×C53—22no—
97C4×D4×C53—22no—
98C4×D4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
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×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119C24×C53—21no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53C2×Q812yes—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—42no—
133C22×C4×C53—81no—
134C22×C4×C53—81no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—81no—
156C4⋊C4×C53—81no—
157C22⋊C4×C53—42no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—81no—
161C22⋊C4×C53—81no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53#4112yes—
167C42×C53#4112yes—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C4⋊Dic53—42no—
178C4⋊Dic53—81no—
179C4⋊Dic53—81no—
180C4⋊Dic53—81no—
181C4⋊Dic53C2×Q812yes—
182C4×Dic53—42no—
183C4×Dic53—42no—
184C4×Dic53—42no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322C2×Q812yes—
196C2×#88322#4112yes—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22×D411yescharacteristic
205Q8×C53—28no—
206D4×C53—22no—
207D4×C53—22no—
208D4×C53—41no—
209D4×C53—41no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
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—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—84no—
234C2×C4×C53#12312yes—
235C2×C4×C53#12312yes—
236C2×C4×C53AGL(1,ℤ/8)12yes—
237C2×C4×C53#13712yes—
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256#88322—22no—
257#88322—22no—
258#88322—41no—
259#88322—41no—
260#25068—531nomaximal, Sylow
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53C2×#12311yescharacteristic
269C22×C53D4×Q811yescharacteristic
270C22×C53#55011yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57611yescharacteristic
274C22×C53#57711yescharacteristic
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#53311yescharacteristic
285C4×C53#54211yescharacteristic
286C4×C53#55211yescharacteristic
287C4×C53#55411yescharacteristic
288Dic53—42no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2580D5311yescharacteristic
293Q8×C4⋊C4—531no—
294#2094—531no—
295#2093—531no—
296#2084—531no—
297#2083—531no—
298#1985—531no—
299#1949—531no—
300#1926—531no—
301#1925—531no—
302#1921—531no—
303#1920—531no—
304#1847—531no—
305#1802—531no—
306#1762—531no—
307C2×C53—82no—
308C2×C53#242311yescharacteristic
309C2×C53#309211yescharacteristic
310C2×C53#309611yescharacteristic
311C2×C53#338411yescharacteristic
312C2×C53#338511yescharacteristic
313C2×C53#339911yescharacteristic
314C2×C53#340011yescharacteristic
315C2×C4×Q8—1061no—
316C2×C4×Q8C2×D5311yescharacteristic
317C2×C4×D4C2×D5311yescharacteristic
318C2×#107—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#106—531no—
322C2×#106—531no—
323C2×#106—531no—
324C2×C4⋊C8—531no—
325C2×#102—531no—
326C2×#102—531no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#97—531no—
330#399—21no—
331#393—21no—
332#390C2×D5311yescharacteristic
333#390C2×D5311yescharacteristic
334#389—1061no—
335#389—1061no—
336#389—1061no—
337#389C2×D5311yescharacteristic
338#389C2×D5311yescharacteristic
339#385—21no—
340#385—21no—
341#384—531no—
342#384—531no—
343#384—531no—
344#382C2×D5311yescharacteristic
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#336—531no—
355#336—531no—
356#336—531no—
357#336—531no—
358C53#2506811yescharacteristic, complemented, Sylow
359C22×Q8C22⋊D5311yescharacteristic
360C22×D4C22×D5311yescharacteristic
361C23×C4—21no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—531no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—2124no—
369C4×D4—22no—
370C4×D4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4C22⋊D5311yescharacteristic
384C2×C4⋊C4C22⋊D5311yescharacteristic
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400#107—1062no—
401#107—2121no—
402#107—2121no—
403#106—1062no—
404#106—2121no—
405#106—2121no—
406C4⋊C8—1062no—
407#102—534no—
408#102—1062no—
409#102—2121no—
410#102—2121no—
411#97—1062no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4Q8×D5312yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—532no—
447C2×C8—532no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—532no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—4241no—
478C4⋊C4—4241no—
479C4⋊C4—4241no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—2122no—
493C42—2122no—
494C42—2122no—
495C42#1171994712yes—
496C42#1171994712yes—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C22×C22⋊D5311yescharacteristic
505Q8—28no—
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4—1062no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4#1313319812yes—
547C2×C4#1313319812yes—
548C2×C4#1313324212yes—
549C2×C4#1313324712yes—
550C2×C4C2×Q8×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C2×C4#1313326311yescharacteristic
555C2×C4#1313326511yescharacteristic
556C8—1062no—
557C8—1062no—
558C8—2121no—
559C8—2121no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C2×#1313319811yescharacteristic
568C22Q8×C22⋊D5311yescharacteristic
569C22#1681129211yescharacteristic
570C22C2×#1313324211yescharacteristic
571C22C2×#1313324711yescharacteristic
572C22#1681142611yescharacteristic
573C22#1681143411yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—2122no—
587C4#1681117511yescharacteristic
588C4#1681124611yescharacteristic
589C4#1681130711yescharacteristic
590C4#1681133111yescharacteristic
591C2—82no—
592C2#2027823611yescharacteristic
593C2#2028463511yescharacteristic
594C2#2028466711yescharacteristic
595C2#2028761411yescharacteristic
596C2#2028762511yescharacteristic
597C2#2028775311yescharacteristic
598C2#2028776811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2580C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2580D53
hypocenterC53#25068
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Dic53C2semidirect
4. #20276191C2semidirect
5. #20276186C2semidirect
6. #20275656C2semidirect
7. #20275481C2semidirect
10. #20275304C2semidirect
11. #20275297C2semidirect
12. #20274884C2semidirect
87. #13133138Q8semidirect
358. C53#25068semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Dic532
#202762482
#202762432
#202761912
#202761862
#202756562
#202754812
#202753392
#202753322
#202753042
#202752972
#202748842
#202747152
#202745402
C53×#25802
#2506853