Finite Group Databasebeta

← #27418845

Subgroups — #27418845

Order 13568.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810815C211yesmaximal, characteristic
2C2×C4×#11719924C211yesmaximal, characteristic, complemented
3#20279633C211yesmaximal, characteristic, complemented
4#20275951C212yesmaximal, complemented
5#20275792C212yesmaximal, complemented
6#20275652C211yesmaximal, characteristic, complemented
7#20275522C211yesmaximal, characteristic, complemented
8#20275063C212yesmaximal, complemented
9#20274976C212yesmaximal, complemented
10#20274720C211yesmaximal, characteristic, complemented
11C2×C53×#559C211yesmaximal, characteristic, complemented
12C22×#11719924C2211yescharacteristic
13C22×#11719922C2212yes—
14C2×#13133193—21no—
15C2×C4×D212C2211yescharacteristic
16#16810815—24no—
17C4×#11719924—24no—
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×#13133138C2212yes—
25C2×#13133137C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27#16810545—22no—
28#16810525C2212yes—
29#16810521C2212yes—
30#16810493C2211yescharacteristic
31#16810480—21no—
32#16810385C2212yes—
33#16810378C2212yes—
34#16810378C2212yes—
35C53×#559—24no—
36C53×#559—24no—
37C2×#127×C53C2211yescharacteristic
38C2×#124×C53—21no—
39C2×#124×C53C2212yes—
40C2×#123×C53—22no—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C2×C4×D4×C53C2212yes—
45C22×D212C2311yescharacteristic
46C22×C4×D53—21no—
47C2×#11719924—22no—
48C2×#11719924—22no—
49C2×#11719924—22no—
50C2×#11719922—24no—
51C2×#11719922—24no—
52C2×#11719922—42no—
53C22×#88322C2311yescharacteristic
54C22×#88322C2312yes—
55C22×#88322C2311yescharacteristic
56#13133193—44no—
57C4×D212—22no—
58C4×D212—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2312yes—
63C2×#11719919—22no—
64C2×#11719921—22no—
65C2×#11719921—21no—
66C2×#11719921—22no—
67#13133148—22no—
68C4⋊#88322—22no—
69#13133141—22no—
70#13133141—42no—
71#13133138—24no—
72#13133138—24no—
73#13133138—42no—
74#13133137—22no—
75C4×#88322—22no—
76#13133124—22no—
77C22×Q8×C53D411yescharacteristic
78C22×Q8×C53C2311yescharacteristic
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53C2312yes—
82C23×C4×C53—22no—
83#127×C53—22no—
84#127×C53D414yes—
85#124×C53—28no—
86#124×C53—24no—
87#124×C53—24no—
88#123×C53—216no—
89C4×Q8×C53—22no—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92C4×Q8×C53D414yes—
93C4×D4×C53—28no—
94C4×D4×C53—24no—
95C4×D4×C53—24no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D412yes—
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53—22no—
103C2×C42×C53—21no—
104C2×C42×C53C2312yes—
105C2×C42×C53C2311yescharacteristic
106C24×C53—22no—
107C23×D53—21no—
108C2×Q8×C53—22no—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—24no—
118C2×D4×C53—24no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—81no—
127C2×D4×C53—81no—
128C2×D4×C53C2×D414yes—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—28no—
132C22×C4×C53—82no—
133C22×C4×C53—82no—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—22no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212#4112yes—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53C2×D412yes—
155C22⋊C4×C53—28no—
156C22⋊C4×C53—28no—
157C22⋊C4×C53—28no—
158C42×C53—24no—
159C42×C53—22no—
160C42×C53—22no—
161C42×C53C2×D412yes—
162C42×C53C2×D412yes—
163#11719924—28no—
164#11719922—48no—
165#11719921—44no—
166#11719921—42no—
167#11719921—82no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—42no—
173C4⋊Dic53—81no—
174#11719919—44no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—21no—
181C2×#88322—22no—
182C2×#88322—21no—
183C2×#88322#4112yes—
184C23×C53—24no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53C22×D411yescharacteristic
192C22×D53—22no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199Q8×C53—22no—
200Q8×C53—22no—
201Q8×C53—41no—
202Q8×C53—41no—
203Q8×C53—44no—
204D4×C53—24no—
205D4×C53—24no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—48no—
209D4×C53—84no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—88no—
225C2×C4×C53#12012yes—
226C2×C4×C53#12012yes—
227C2×C4×C53#13512yes—
228C2×C4×C53AGL(1,ℤ/8)12yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53H(5,𝔽2)11yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×Dic53—41no—
236C2×Dic53—42no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241D212—22no—
242D212—22no—
243D212—41no—
244D212—41no—
245C4×D53—84no—
246#88322—22no—
247#88322—41no—
248#88322—42no—
249#24186—531nomaximal, Sylow
250C22×C53—44no—
251C22×C53—44no—
252C22×C53—44no—
253C22×C53—82no—
254C22×C53—82no—
255C22×C53—82no—
256C22×C53—82no—
257C22×C53C2×#12011yescharacteristic
258C22×C53#54811yescharacteristic
259C22×C53#55011yescharacteristic
260C22×C53C2×#13511yescharacteristic
261C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C53#57611yescharacteristic
263C22×C53#57711yescharacteristic
264C2×D53—42no—
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—81no—
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C4×C53—22no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53—42no—
275C4×C53—42no—
276C4×C53#44914yes—
277C4×C53#52511yescharacteristic
278C4×C53#53711yescharacteristic
279C4×C53#54611yescharacteristic
280C4×C53#54711yescharacteristic
281Dic53—82no—
282Dic53—81no—
283C2×#559D5311yescharacteristic, complemented
284C2×#459—531no—
285C2×C4×SD16—531no—
286#2538—531no—
287#2042—532no—
288#2012—532no—
289#1984—531no—
290#1957—531no—
291#1879—532no—
292#1864—532no—
293#1803—531no—
294C2×C53—84no—
295C2×C53#240711yescharacteristic
296C2×C53C2×#44911yescharacteristic
297C2×C53#301211yescharacteristic
298C2×C53#336111yescharacteristic
299C2×C53#337311yescharacteristic
300C2×C53#339411yescharacteristic
301C2×C53#339911yescharacteristic
302D53—82no—
303C22×SD16—531no—
304C22×D8—532no—
305#559—24no—
306#559—24no—
307C2×#127C2×D5311yescharacteristic
308C2×#124—21no—
309C2×#124C2×D5312yes—
310C2×#123—22no—
311C2×C22⋊Q8—1061no—
312C2×C4×Q8C2×D5311yescharacteristic
313C2×C4×Q8C2×D5311yescharacteristic
314C2×C4×D4—21no—
315C2×C4×D4—531no—
316C2×C4×D4C2×D5312yes—
317#459—1064no—
318C4×SD16—1064no—
319C2×#106—531no—
320C2×C4⋊C8—531no—
321C2×#103—531no—
322C2×#103—532no—
323C2×#102—532no—
324C2×#102—532no—
325C2×#102—531no—
326C2×#102—532no—
327C2×C4×C8—531no—
328#396—1062no—
329#390—532no—
330#389—532no—
331#384—531no—
332#380—1061no—
333#340—532no—
334#336—532no—
335#336—532no—
336C53#2418611yescharacteristic, complemented, Sylow
337C22×Q8C22⋊D5311yescharacteristic
338C22×Q8C22×D5311yescharacteristic
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—531no—
342C22×D4C22×D5312yes—
343C23×C4—22no—
344C23×C4—1061no—
345C2×SD16—1062no—
346C2×SD16—1062no—
347C2×SD16—1062no—
348C2×D8—1064no—
349C2×D8—1064no—
350C2×D8—2122no—
351C22×C8—531no—
352C22×C8—532no—
353C22×C8—531no—
354#127—22no—
355#127C22⋊D5314yes—
356#124—28no—
357#124—24no—
358#124—24no—
359#123—216no—
360C22⋊Q8—2124no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8C22⋊D5314yes—
365C4×D4—28no—
366C4×D4—24no—
367C4×D4—24no—
368C4×D4—1062no—
369C4×D4—1062no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22⋊D5311yescharacteristic
377C2×C4⋊C4C22⋊D5312yes—
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1062no—
385C2×C42—21no—
386C2×C42C22×D5312yes—
387C2×C42C22×D5311yescharacteristic
388#106—1062no—
389C4⋊C8—1062no—
390#103—1062no—
391#103—2122no—
392#102—1064no—
393#102—1064no—
394#102—1062no—
395#102—2122no—
396C4×C8—1062no—
397#95—1062no—
398C24—22no—
399C24—1061no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×C22⋊D5312yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—532no—
421C2×D4—1061no—
422C2×D4—1061no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4C2×C22⋊D5314yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—28no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—2122no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4C2×C22⋊D5312yes—
438C22×C4C2×C22⋊D5312yes—
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C23×D5311yescharacteristic
442SD16—1068no—
443D8—2128no—
444C2×C8—532no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1062no—
449C2×C8—1062no—
450C2×C8—1061no—
451C2×C8—1062no—
452C2×C8—1061no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4—4241no—
462C4⋊C4—2124no—
463C4⋊C4—2122no—
464C4⋊C4C2×C22⋊D5312yes—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—4242no—
469C22⋊C4—2124no—
470C22⋊C4—2122no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42C2×C22⋊D5312yes—
475C42C2×C22⋊D5312yes—
476C23—24no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—1062no—
484C23—2121no—
485C23—2121no—
486C23—2121no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23C22×C22⋊D5311yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—41no—
494Q8—44no—
495Q8—41no—
496D4—24no—
497D4—24no—
498D4—48no—
499D4—42no—
500D4—42no—
501D4—84no—
502D4—1062no—
503D4—1062no—
504D4—2121no—
505D4—2121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—88no—
521C2×C4—4244no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2122no—
526C2×C4—2121no—
527C2×C4—2122no—
528C2×C4#1313318612yes—
529C2×C4#1313318612yes—
530C2×C4#1313323712yes—
531C2×C4#1313324212yes—
532C2×C4C22×C22⋊D5311yescharacteristic
533C2×C4C22×C22⋊D5311yescharacteristic
534C2×C4#41×D5311yescharacteristic
535C2×C4#41×D5311yescharacteristic
536C2×C4#1313326311yescharacteristic
537C2×C4#1313326411yescharacteristic
538C8—1062no—
539C8—2121no—
540C8—2122no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—4241no—
549C22—4241no—
550C22—4241no—
551C22—4241no—
552C22—2122no—
553C22—2122no—
554C22—2122no—
555C22C2×#1313318611yescharacteristic
556C22#1681128011yescharacteristic
557C22#1681129211yescharacteristic
558C22C2×#1313323711yescharacteristic
559C22C2×#1313324211yescharacteristic
560C22#1681142611yescharacteristic
561C22#1681143411yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—4242no—
568C4—4241no—
569C4#1681076814yes—
570C4#1681113011yescharacteristic
571C4#1681120111yescharacteristic
572C4#1681126111yescharacteristic
573C4#1681127211yescharacteristic
574C2—84no—
575C2—4242no—
576C2#2027805411yescharacteristic
577C2C2×#1681076811yescharacteristic
578C2#2028401911yescharacteristic
579C2#2028737111yescharacteristic
580C2#2028750311yescharacteristic
581C2#2028770711yescharacteristic
582C2#2028775311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#559C2
socleC23×C53C22×D4
prime radicalC2×#16810815C2
hypercenterC2×#559D53
hypocenterC53#24186
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719924C2semidirect
3. #20279633C2semidirect
4. #20275951C2semidirect
5. #20275792C2semidirect
6. #20275652C2semidirect
7. #20275522C2semidirect
8. #20275063C2semidirect
9. #20274976C2semidirect
10. #20274720C2semidirect
11. C2×C53×#559C2semidirect
283. C2×#559D53semidirect
336. C53#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#168108152
C2×C4×#117199242
#202796332
#202759512
#202757922
#202756522
#202755222
#202750632
#202749762
#202747202
C2×C53×#5592
#2418653