Finite Group Databasebeta

← #27275895

Subgroups — #27275895

Order 13568.

1364 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275033C212yesmaximal, complemented
2C2×#16810357C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic, complemented
4#20272960C211yesmaximal, characteristic, complemented
5C2×C53×#534C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic, complemented
7C22×C22⋊Dic53—21no—
8C2×#13133138—22no—
9C2×C22⋊#88322C2212yescomplemented
10C2×#13133124—21no—
11C2×#13133124C2211yescharacteristic, complemented
12#16810385—22no—
13#16810357—24no—
14#16810347—22no—
15C23×D4×C53—21no—
16C53×#534—24no—
17C53×#534—22no—
18C53×#534C414yescomplemented
19C2×#123×C53—22no—
20C2×#121×C53—21no—
21C2×#121×C53C2212yes—
22C2×#120×C53—22no—
23C22×C22⋊C4×C53C2211yescharacteristic
24C23×Dic53—21no—
25C23×Dic53—21no—
26C22×#88322—22no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—42no—
29C2×C22⋊Dic53—42no—
30C2×C22⋊Dic53—42no—
31C2×C22⋊Dic53—42no—
32#13133138—44no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133124—22no—
36#13133124—42no—
37C25×C53—21no—
38C25×C53D411yescharacteristic
39C22×D4×C53—22no—
40C22×D4×C53—24no—
41C22×D4×C53—24no—
42C22×D4×C53—22no—
43C22×D4×C53—21no—
44C22×D4×C53—42no—
45C22×D4×C53—42no—
46C22×D4×C53—42no—
47C22×D4×C53—42no—
48C22×D4×C53D412yes—
49C23×C4×C53—21no—
50C23×C4×C53D411yescharacteristic
51C23×C4×C53C2311yescharacteristic
52#123×C53—28no—
53#123×C53—28no—
54#121×C53—24no—
55#121×C53—24no—
56#121×C53—22no—
57#121×C53—22no—
58#121×C53C2×C414yes—
59#121×C53D414yes—
60#120×C53—28no—
61#120×C53—28no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D412yes—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53C2×C412yes—
69C2×C22⋊C4×C53D412yes—
70C24×C53—21no—
71C24×C53—21no—
72C24×C53—21no—
73C24×C53—21no—
74C24×C53—24no—
75C24×C53—22no—
76C24×C53—41no—
77C24×C53—41no—
78C24×C53—41no—
79C24×C53—41no—
80C24×C53—42no—
81C24×C53—42no—
82C24×C53—81no—
83C24×C53—81no—
84C24×C53—81no—
85C24×C53—81no—
86C24×C53C22⋊C412yes—
87C24×C53D812yes—
88C24×C53SD1612yes—
89C24×C53C2×D411yescharacteristic
90C2×D4×C53—28no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—22no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—48no—
98C2×D4×C53—48no—
99C2×D4×C53—48no—
100C2×D4×C53—44no—
101C2×D4×C53—44no—
102C2×D4×C53—48no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C2×D4×C53—42no—
106C2×D4×C53—42no—
107C2×D4×C53—81no—
108C2×D4×C53—81no—
109C2×D4×C53C22⋊C414yescomplemented
110C22×C4×C53—24no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C22⋊C412yes—
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D411yescharacteristic
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—41no—
128C22×Dic53—41no—
129C22×Dic53—41no—
130C22×Dic53—41no—
131C22×Dic53—41no—
132C22×Dic53—41no—
133C22×Dic53—41no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—41no—
137C22×Dic53—41no—
138C22×Dic53—41no—
139C4⋊C4×C53—24no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53C22⋊C414yescomplemented
142C22⋊C4×C53—28no—
143C22⋊C4×C53—24no—
144C22⋊C4×C53—24no—
145C22⋊C4×C53—24no—
146C22⋊C4×C53—24no—
147C22⋊Dic53—48no—
148C4⋊Dic53—24no—
149C4⋊Dic53—42no—
150C2×#88322—42no—
151C2×#88322—42no—
152C2×#88322—42no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53—22no—
162C23×C53—22no—
163C23×C53—24no—
164C23×C53—42no—
165C23×C53—42no—
166C23×C53—42no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—48no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—81no—
182C23×C53—81no—
183C23×C53—81no—
184C23×C53—81no—
185C23×C53—81no—
186C23×C53—81no—
187C23×C53—84no—
188C23×C53—84no—
189C23×C53—84no—
190C23×C53—84no—
191C23×C53—84no—
192C23×C53—84no—
193C23×C53#10214yescomplemented
194C23×C53C2×C22⋊C411yescharacteristic
195C23×C53#12011yescharacteristic
196C23×C53#12011yescharacteristic
197C23×C53C2×D811yescharacteristic
198C23×C53C2×SD1611yescharacteristic
199C23×C53AGL(1,ℤ/8)11yescharacteristic
200C23×C53AGL(1,ℤ/8)11yescharacteristic
201D4×C53—44no—
202D4×C53—416no—
203D4×C53—44no—
204D4×C53—44no—
205D4×C53—44no—
206D4×C53—44no—
207D4×C53—82no—
208D4×C53—84no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—48no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—81no—
223C2×C4×C53C2×C22⋊C412yes—
224C2×C4×C53#12012yes—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—81no—
232C2×Dic53—81no—
233C2×Dic53—81no—
234C2×Dic53—81no—
235C2×Dic53—81no—
236C2×Dic53—81no—
237C2×Dic53—81no—
238C2×Dic53—81no—
239#88322—82no—
240#8403—531nomaximal, Sylow
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—24no—
254C22×C53—44no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—44no—
262C22×C53—44no—
263C22×C53—44no—
264C22×C53—82no—
265C22×C53—82no—
266C22×C53—82no—
267C22×C53—816no—
268C22×C53—82no—
269C22×C53—82no—
270C22×C53—82no—
271C22×C53—82no—
272C22×C53#37911yescharacteristic
273C22×C53C2×#10211yescharacteristic
274C22×C53#41811yescharacteristic
275C22×C53#44711yescharacteristic
276C22×C53#44711yescharacteristic
277C22×C53#45011yescharacteristic
278C22×C53#45011yescharacteristic
279C22×C53#45311yescharacteristic
280C22×C53#45511yescharacteristic
281C22×C53B(4,𝔽2)11yescharacteristic
282C22×C53#45811yescharacteristic
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—44no—
287Dic53—82no—
288Dic53—82no—
289C2×#534D5311yescharacteristic
290#1874—532no—
291C2×#327—532no—
292C2×#323—531no—
293#1439—531no—
294C2×C53—21no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—44no—
299C2×C53—84no—
300C2×C53#159511yescharacteristic
301C2×C53#160111yescharacteristic
302C2×C53#160711yescharacteristic
303C2×C53#161311yescharacteristic
304C2×C53#178611yescharacteristic
305C2×C53#179011yescharacteristic
306C2×C53#179211yescharacteristic
307C23×D4—21no—
308#534—24no—
309#534—22no—
310#534Dic5314yescomplemented
311C2×#123—22no—
312C2×#121—21no—
313C2×#121C2×D5312yes—
314C2×#120—22no—
315C22×C4⋊C4—531no—
316C22×C22⋊C4—1061no—
317C22×C22⋊C4C2×D5311yescharacteristic
318C2×#102—1062no—
319C2×C22⋊C8—532no—
320C2×#95—531no—
321C2×#95—1061no—
322#340—1062no—
323#327—1064no—
324#323—1062no—
325C53#840311yescharacteristic, complemented, Sylow
326C25—21no—
327C25C22⋊D5311yescharacteristic
328C22×D4—22no—
329C22×D4—24no—
330C22×D4—22no—
331C22×D4—24no—
332C22×D4—21no—
333C22×D4—42no—
334C22×D4—42no—
335C22×D4—42no—
336C22×D4—42no—
337C22×D4C22⋊D5312yes—
338C23×C4—21no—
339C23×C4—1061no—
340C23×C4—1061no—
341C23×C4C22⋊D5311yescharacteristic
342C23×C4C22×D5311yescharacteristic
343C22×C8—1062no—
344#123—28no—
345#123—28no—
346#121—24no—
347#121—22no—
348#121—22no—
349#121—24no—
350#121C2×Dic5314yes—
351#121C22⋊D5314yes—
352#120—28no—
353#120—28no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4—2122no—
357C2×C4⋊C4C22⋊D5312yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—2122no—
363C2×C22⋊C4—2122no—
364C2×C22⋊C4—2122no—
365C2×C22⋊C4C2×Dic5312yes—
366C2×C22⋊C4C22⋊D5312yes—
367#102—2124no—
368C22⋊C8—1062no—
369C22⋊C8—1062no—
370#95—1062no—
371#95—2122no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—24no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—42no—
383C24—42no—
384C24—81no—
385C24—81no—
386C24—81no—
387C24—81no—
388C24#1171992212yes—
389C24#1171992312yes—
390C24C22⋊Dic5312yes—
391C24C2×C22⋊D5311yescharacteristic
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4C22⋊Dic5314yescomplemented
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—1062no—
423C22×C4—1062no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4—2121no—
427C22×C4—2121no—
428C22×C4—2121no—
429C22×C4—2121no—
430C22×C4—2121no—
431C22×C4—2121no—
432C22×C4—2121no—
433C22×C4—2121no—
434C22×C4—2121no—
435C22×C4—2121no—
436C22×C4C22⋊Dic5312yes—
437C22×C4C22×Dic5311yescharacteristic
438C22×C4C2×C22⋊D5312yes—
439C22×C4C2×C22⋊D5312yes—
440C22×C4C2×C22⋊D5311yescharacteristic
441C2×C8—2122no—
442C2×C8—2122no—
443C2×C8—2122no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—1064no—
447C4⋊C4—2122no—
448C4⋊C4C22⋊Dic5314yescomplemented
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—2128no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—48no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23#1313313814yescomplemented
496C23C2×C22⋊Dic5311yescharacteristic
497C23#1313318611yescharacteristic
498C23#1313318611yescharacteristic
499C23C2×#1171992211yescharacteristic
500C23C2×#1171992311yescharacteristic
501C23#1313324311yescharacteristic
502C23#1313324311yescharacteristic
503D4—44no—
504D4—416no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—82no—
510D4—84no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—4241no—
526C2×C4—4241no—
527C2×C4—4241no—
528C2×C4—4241no—
529C2×C4—4241no—
530C2×C4—4241no—
531C2×C4—4241no—
532C2×C4—4241no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4C2×C22⋊Dic5312yes—
540C2×C4#1313318612yes—
541C8—4242no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—24no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—816no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22#1681047911yescharacteristic
574C22C2×#1313313811yescharacteristic
575C22#1681062811yescharacteristic
576C22#1681075711yescharacteristic
577C22#1681075711yescharacteristic
578C22#1681077411yescharacteristic
579C22#1681077411yescharacteristic
580C22#1681079111yescharacteristic
581C22#1681080111yescharacteristic
582C22#1681081011yescharacteristic
583C22#1681081311yescharacteristic
584C4—22no—
585C4—22no—
586C4—44no—
587C4—42no—
588C4—4242no—
589C4—4242no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—84no—
596C2#2027363311yescharacteristic
597C2#2027366711yescharacteristic
598C2#2027370111yescharacteristic
599C2#2027373511yescharacteristic
600C2#2027464611yescharacteristic
601C2#2027466311yescharacteristic
602C2#2027466911yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#534C2
socleC23×C53#120
cosocleC23×C4×C53C23
hypercenterC2×#534D53
hypocenterC53#8403
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #20275033C2semidirect
2. C2×#16810357C2semidirect
3. C2×#16810347C2semidirect
4. #20272960C2semidirect
6. C22×C4⋊Dic53C22semidirect
9. C2×C22⋊#88322C22semidirect
11. C2×#13133124C22semidirect
18. C53×#534C4semidirect
109. C2×D4×C53C22⋊C4semidirect
141. C4⋊C4×C53C22⋊C4semidirect
193. C23×C53#102semidirect
310. #534Dic53semidirect
325. C53#8403semidirect
411. C2×D4C22⋊Dic53semidirect
448. C4⋊C4C22⋊Dic53semidirect
495. C23#13133138semidirect

Maximal subgroups

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

subgroupindex
#202750332
C2×#168103572
C2×#168103472
#202729602
C2×C53×#5342
#840353