Finite Group Databasebeta

← #27429338

Subgroups — #27429338

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277672C211yesmaximal, characteristic, complemented
2#20276283C211yesmaximal, characteristic, complemented
3#20276278C211yesmaximal, characteristic, complemented
4#20276128C211yesmaximal, characteristic
5#20276118C211yesmaximal, characteristic
6#20275652C211yesmaximal, characteristic, complemented
7#20275481C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic
9#20275332C211yesmaximal, characteristic
10#20275297C211yesmaximal, characteristic, complemented
11#20275283C211yesmaximal, characteristic, complemented
12#20274845C211yesmaximal, characteristic, complemented
13#20274725C211yesmaximal, characteristic
14C4×#13133138C211yesmaximal, characteristic
15C53×#2580C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133138C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24#16810530—21no—
25#16810509—21no—
26#16810509—21no—
27#16810493C2211yescharacteristic
28#16810489C2211yescharacteristic
29#16810489C2211yescharacteristic
30C4⋊C4×Dic53—21no—
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C4⋊Dic53C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
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
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—21no—
72#13133150—22no—
73#13133148—22no—
74C4⋊#88322—22no—
75#13133138—22no—
76#13133138—22no—
77#13133138—22no—
78#13133138—22no—
79C4×#88322—22no—
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87C22×Q8×C53D411yescharacteristic
88C22×D4×C53C2311yescharacteristic
89C23×C4×C53—21no—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114C24×C53—21no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—41no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53#4112yes—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—42no—
128C22×C4×C53—81no—
129C22×C4×C53—81no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—81no—
151C4⋊C4×C53—81no—
152C22⋊C4×C53—42no—
153C22⋊C4×C53—42no—
154C22⋊C4×C53—42no—
155C22⋊C4×C53—81no—
156C22⋊C4×C53—81no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53#4112yes—
162C42×C53#4112yes—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—42no—
171C4⋊Dic53—81no—
172C4⋊Dic53#4112yes—
173C4×Dic53—42no—
174C4×Dic53—81no—
175C4×Dic53—81no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C23×C53—22no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53C22×D411yescharacteristic
196Q8×C53—44no—
197D4×C53—22no—
198D4×C53—22no—
199D4×C53—41no—
200D4×C53—41no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
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—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—84no—
226C2×C4×C53#12312yes—
227C2×C4×C53#12312yes—
228C2×C4×C53#13512yes—
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247#88322—41no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#24999—531nomaximal, Sylow
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—81no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53C2×#12311yescharacteristic
260C22×C53#54811yescharacteristic
261C22×C53#54811yescharacteristic
262C22×C53C2×#13511yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57611yescharacteristic
265C22×C53#57711yescharacteristic
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#52911yescharacteristic
276C4×C53#54111yescharacteristic
277C4×C53#55211yescharacteristic
278C4×C53#55611yescharacteristic
279Dic53—42no—
280Dic53—81no—
281Dic53—81no—
282Dic53—81no—
283#2580D5311yescharacteristic
284#2355—531no—
285#2101—531no—
286#2100—531no—
287#2073—531no—
288#2071—531no—
289#1984—531no—
290#1949—531no—
291#1926—531no—
292#1925—531no—
293#1920—531no—
294#1918—531no—
295#1835—531no—
296#1804—531no—
297C4×#102—531no—
298C2×C53—82no—
299C2×C53#242111yescharacteristic
300C2×C53#309111yescharacteristic
301C2×C53#309611yescharacteristic
302C2×C53#336011yescharacteristic
303C2×C53#336111yescharacteristic
304C2×C53#337011yescharacteristic
305C2×C53#337111yescharacteristic
306C2×C4×Q8C2×D5311yescharacteristic
307C2×C4×D4C2×D5311yescharacteristic
308C2×#107—531no—
309C2×#106—531no—
310C2×C4⋊C8—531no—
311C2×#102—531no—
312C2×#102—531no—
313C2×#102—531no—
314C2×#102—531no—
315C2×C4×C8—531no—
316#399—21no—
317#393—21no—
318#391—1061no—
319#390C2×D5311yescharacteristic
320#390C2×D5311yescharacteristic
321#389C2×D5311yescharacteristic
322#389C2×D5311yescharacteristic
323#387—1061no—
324#387—1061no—
325#385—21no—
326#385—21no—
327#384—531no—
328#382—531no—
329#382—531no—
330#382C2×D5311yescharacteristic
331C4×C4⋊C4—531no—
332C4×C4⋊C4—531no—
333C4×C4⋊C4—531no—
334C4×C4⋊C4—1061no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#336—531no—
346#336—531no—
347#336—531no—
348#336—531no—
349C53#2499911yescharacteristic, complemented, Sylow
350C22×Q8C22⋊D5311yescharacteristic
351C22×D4C22×D5311yescharacteristic
352C23×C4—21no—
353C22×C8—531no—
354C22×C8—531no—
355C22×C8—531no—
356C22×C8—531no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22⋊D5311yescharacteristic
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22⋊D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42—1061no—
381C2×C42—1061no—
382C2×C42—1061no—
383C2×C42C22×D5311yescharacteristic
384C2×C42C22×D5311yescharacteristic
385C2×C42C22×D5311yescharacteristic
386#107—1062no—
387#106—1062no—
388C4⋊C8—1062no—
389#102—1062no—
390#102—1062no—
391#102—1062no—
392#102—1062no—
393C4×C8—1062no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405C24—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—41no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4#1171994912yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C2×C22⋊D5311yescharacteristic
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C23×D5311yescharacteristic
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—532no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—4241no—
464C4⋊C4—2122no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—4241no—
475C42—4241no—
476C42—2122no—
477C42#1171994712yes—
478C42#1171994712yes—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C22×C22⋊D5311yescharacteristic
487Q8—44no—
488D4—22no—
489D4—22no—
490D4—41no—
491D4—41no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—84no—
517C2×C4—1062no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4#1313319812yes—
530C2×C4#1313319812yes—
531C2×C4#1313323712yes—
532C2×C4#41×D5311yescharacteristic
533C2×C4C2×#1171994711yescharacteristic
534C2×C4C2×#1171994711yescharacteristic
535C2×C4C2×#1171994911yescharacteristic
536C2×C4#1313326411yescharacteristic
537C2×C4#1313326511yescharacteristic
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22C2×#1313319811yescharacteristic
550C22#1681128011yescharacteristic
551C22#1681128411yescharacteristic
552C22C2×#1313323711yescharacteristic
553C22#1681142111yescharacteristic
554C22#1681142611yescharacteristic
555C22#1681143411yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4—2122no—
569C4#1681115911yescharacteristic
570C4#1681123311yescharacteristic
571C4#1681131511yescharacteristic
572C4#1681134411yescharacteristic
573C2—82no—
574C2#2027821411yescharacteristic
575C2#2028462411yescharacteristic
576C2#2028466711yescharacteristic
577C2#2028736411yescharacteristic
578C2#2028737111yescharacteristic
579C2#2028746811yescharacteristic
580C2#2028747711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2580C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2580D53
hypocenterC53#24999
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. #20277672C2semidirect
2. #20276283C2semidirect
3. #20276278C2semidirect
6. #20275652C2semidirect
7. #20275481C2semidirect
10. #20275297C2semidirect
11. #20275283C2semidirect
12. #20274845C2semidirect
349. C53#24999semidirect

Maximal subgroups

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

subgroupindex
#202776722
#202762832
#202762782
#202761282
#202761182
#202756522
#202754812
#202753392
#202753322
#202752972
#202752832
#202748452
#202747252
C4×#131331382
C53×#25802
#2499953