Finite Group Databasebeta

← #27314880

Subgroups — #27314880

Order 13568.

1442 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810535C211yesmaximal, characteristic
2#20272985C212yesmaximal, complemented
3#20272973C212yesmaximal, complemented
4#20272973C212yesmaximal, complemented
5#20272971C212yesmaximal, complemented
6#20272960C211yesmaximal, characteristic
7#20272958C212yesmaximal
8#20272938C212yesmaximal
9C53×C22⋊#95C211yesmaximal, characteristic
10C23×Dic106C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719919C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15#16810535C418yescomplemented
16C2×#13133124C2212yes—
17C2×#13133124C2212yes—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C24×C4×C53C2211yescharacteristic
28C22×C22⋊C4×C53C2211yescharacteristic
29C22×C22⋊C4×C53C2211yescharacteristic
30C2×#95×C53C2212yes—
31C2×#95×C53C2212yes—
32C22×Dic106—41no—
33C22×Dic106—41no—
34C22×Dic106—42no—
35C22×Dic106—42no—
36C22×Dic106C2×C414yescomplemented
37C23×Dic53C2312yes—
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C2×C4⋊Dic53—41no—
42C2×C4⋊Dic53—42no—
43C2×#11719919—42no—
44C2×#11719919—42no—
45C2×#11719919—42no—
46C2×C22⋊Dic53—24no—
47C2×C22⋊Dic53—42no—
48C2×C22⋊Dic53—42no—
49C2×C22⋊Dic53—42no—
50C2×C22⋊Dic53—42no—
51C2×C22⋊Dic53C2×C418yes—
52#13133124—42no—
53#13133124—42no—
54#13133124—42no—
55#13133124—42no—
56#13133124—42no—
57#13133124—42no—
58#13133124—42no—
59#13133124—41no—
60#13133124—42no—
61#13133124C2×C414yescomplemented
62#13133124C2×C418yescomplemented
63C25×C53C2311yescharacteristic
64C23×C4×C53—42no—
65C23×C4×C53—42no—
66C23×C4×C53—41no—
67C23×C4×C53—41no—
68C23×C4×C53Q814yescomplemented
69C23×C4×C53C2312yes—
70C23×C4×C53C2311yescharacteristic
71C23×C4×C53C2311yescharacteristic
72C23×C4×C53C2312yes—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—41no—
75C2×C22⋊C4×C53—41no—
76C2×C22⋊C4×C53—41no—
77C2×C22⋊C4×C53—41no—
78C2×C22⋊C4×C53C2×C414yes—
79#95×C53—42no—
80#95×C53—42no—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—22no—
85C24×C53—22no—
86C24×C53—22no—
87C24×C53—22no—
88C24×C53—21no—
89C24×C53C22×C412yes—
90C24×C53C2×Q812yes—
91C24×C53#4112yes—
92C24×C53C2411yescharacteristic
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—48no—
115C22×C4×C53—44no—
116C22×C4×C53—44no—
117C22×C4×C53—48no—
118C22×C4×C53—44no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C2×Q814yes—
122C22×Dic53—22no—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53C22×C414yes—
149C22×Dic53C22×C414yes—
150C2×Dic106—44no—
151C2×Dic106—48no—
152C2×Dic106—48no—
153C2×Dic106—44no—
154C2×Dic106—44no—
155C22⋊C4×C53—44no—
156C22⋊C4×C53—44no—
157C22⋊Dic53—48no—
158C22⋊Dic53—48no—
159C4⋊Dic53—44no—
160#11719919—48no—
161C23×C53—24no—
162C23×C53—22no—
163C23×C53—24no—
164C23×C53—22no—
165C23×C53—24no—
166C23×C53—22no—
167C23×C53—24no—
168C23×C53—22no—
169C23×C53—22no—
170C23×C53—24no—
171C23×C53—24no—
172C23×C53—22no—
173C23×C53—42no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—42no—
177C23×C53—41no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—41no—
183C23×C53C4×Q814yes—
184C23×C53C23×C411yescharacteristic
185C23×C53C22×Q811yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C23×C53H(5,𝔽2)11yescharacteristic
188C23×C53H(5,𝔽2)12yes—
189C23×C53H(5,𝔽2)11yescharacteristic
190C23×C53H(5,𝔽2)12yes—
191C23×C53H(5,𝔽2)11yescharacteristic
192C23×C53H(5,𝔽2)12yes—
193C23×C532−1+412yes—
194C23×C532−1+411yescharacteristic
195C2×C4×C53—24no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—416no—
203C2×C4×C53—42no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—41no—
211C2×C4×C53—42no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×Dic53—24no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248Dic106—416no—
249#14635—531nomaximal, Sylow
250C22×C53—28no—
251C22×C53—24no—
252C22×C53—44no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—44no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—44no—
261C22×C53C2×C4×Q811yescharacteristic
262C22×C53#51812yes—
263C22×C53#51811yescharacteristic
264C22×C53#52012yes—
265C22×C53#52011yescharacteristic
266C22×C53#53812yes—
267C22×C53#53811yescharacteristic
268C22×C53#53912yes—
269C22×C53#53911yescharacteristic
270C22×C53#54312yes—
271C22×C53#54311yescharacteristic
272C22×C53#54412yes—
273C22×C53#54411yescharacteristic
274C22×C53#56012yes—
275C22×C53#56011yescharacteristic
276C22×C53H(3,𝔽4)11yescharacteristic
277C22×C53H(3,𝔽4)12yes—
278C22×C53H(3,𝔽4)11yescharacteristic
279C22×C53#56212yes—
280C22×C53#56212yes—
281C22×C53#56212yes—
282C22×C53#56312yes—
283C22×C53#56311yescharacteristic
284C4×C53—44no—
285C4×C53—42no—
286C4×C53—42no—
287C4×C53—82no—
288Dic53—44no—
289Dic53—44no—
290Dic53—82no—
291Dic53—82no—
292C2×#393—531no—
293#1446—532no—
294#1442—532no—
295#1442—532no—
296#1442—532no—
297#1439—532no—
298#1439—531no—
299C22⋊#95—532no—
300C22⋊#95D5311yescharacteristic
301C2×C53—44no—
302C2×C53#232912yes—
303C2×C53#232911yescharacteristic
304C2×C53#237511yescharacteristic
305C2×C53#237911yescharacteristic
306C2×C53#237912yes—
307C2×C53#282412yes—
308C2×C53#282411yescharacteristic
309C2×C53#283511yescharacteristic
310C2×C53#284011yescharacteristic
311C2×C53#284112yes—
312C2×C53#284111yescharacteristic
313C23×Q8—531no—
314C24×C4C2×D5311yescharacteristic
315C22×C4⋊C4—532no—
316C22×C4⋊C4—531no—
317C22×C22⋊C4—532no—
318C22×C22⋊C4—532no—
319C22×C22⋊C4C2×D5311yescharacteristic
320C22×C22⋊C4C2×D5311yescharacteristic
321#393—538no—
322C2×#95—531no—
323C2×#95—532no—
324C2×#95—532no—
325C2×#95—532no—
326C2×#95—532no—
327C2×#95—532no—
328C2×#95—532no—
329C2×#95—531no—
330C2×#95—532no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95C2×D5312yes—
334C2×#95C2×D5312yes—
335C53#1463511yescharacteristic, complemented, Sylow
336C25C22×D5311yescharacteristic
337C22×Q8—534no—
338C22×Q8—2122no—
339C22×Q8—2121no—
340C22×Q8—2121no—
341C22×Q8—2122no—
342C23×C4—42no—
343C23×C4—42no—
344C23×C4—41no—
345C23×C4—41no—
346C23×C4—532no—
347C23×C4—532no—
348C23×C4—532no—
349C23×C4—532no—
350C23×C4Dic10614yescomplemented
351C23×C4C22×D5312yes—
352C23×C4C22×D5312yes—
353C23×C4C22×D5311yescharacteristic
354C23×C4C22×D5311yescharacteristic
355C2×C4⋊C4—2122no—
356C2×C4⋊C4—2122no—
357C2×C4⋊C4—2122no—
358C2×C4⋊C4—2122no—
359C2×C4⋊C4—2121no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—538no—
366C2×C22⋊C4—1064no—
367C2×C22⋊C4—2122no—
368C2×C22⋊C4—2122no—
369C2×C22⋊C4—2122no—
370C2×C22⋊C4—2122no—
371C2×C22⋊C4C4×D5314yes—
372#95—42no—
373#95—42no—
374#95—534no—
375#95—538no—
376#95—2121no—
377#95—2122no—
378#95—2122no—
379#95—2122no—
380#95—2122no—
381#95—2122no—
382#95—2122no—
383#95—2122no—
384#95—2122no—
385C24—21no—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—21no—
393C24C2×Dic10612yes—
394C24C2×C4×D5312yes—
395C24#1171994512yes—
396C24C23×D5311yescharacteristic
397C2×Q8—2128no—
398C2×Q8—2124no—
399C2×Q8—2128no—
400C2×Q8—2124no—
401C2×Q8—2124no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—534no—
429C22×C4—534no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4C2×Dic10614yes—
457C22×C4C2×C4×D5312yes—
458C22×C4C2×C4×D5312yes—
459C4⋊C4—2128no—
460C4⋊C4—2124no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—2128no—
464C22⋊C4—2128no—
465C23—24no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23C4×Dic10614yes—
488C23C22×C4×D5311yescharacteristic
489C23C22×Dic10611yescharacteristic
490C23C2×#1171994511yescharacteristic
491C23#1313326212yes—
492C23#1313326211yescharacteristic
493C23#1313326211yescharacteristic
494C23#1313326212yes—
495C23#1313326211yescharacteristic
496C23#1313326312yes—
497C23#1313326412yes—
498C23#1313326511yescharacteristic
499Q8—21216no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—416no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C22—28no—
554C22—24no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22C2×C4×Dic10611yescharacteristic
565C22#1681109512yes—
566C22#1681109611yescharacteristic
567C22#1681110212yes—
568C22#1681110411yescharacteristic
569C22#1681120612yes—
570C22#1681121111yescharacteristic
571C22#1681121512yes—
572C22#1681122011yescharacteristic
573C22#1681124912yes—
574C22#1681124911yescharacteristic
575C22#1681125112yes—
576C22#1681125411yescharacteristic
577C22#1681136112yes—
578C22#1681136311yescharacteristic
579C22#1681136611yescharacteristic
580C22#1681136611yescharacteristic
581C22#1681136612yes—
582C22#1681136812yes—
583C22#1681136912yes—
584C22#1681137012yes—
585C22#1681137412yes—
586C22#1681137611yescharacteristic
587C4—44no—
588C4—42no—
589C4—42no—
590C4—82no—
591C4—4242no—
592C4—4242no—
593C4—2124no—
594C4—2124no—
595C2—44no—
596C2#2027744312yes—
597C2#2027744811yescharacteristic
598C2#2027785311yescharacteristic
599C2#2027788511yescharacteristic
600C2#2027788812yes—
601C2#2028279212yes—
602C2#2028279511yescharacteristic
603C2#2028290311yescharacteristic
604C2#2028296811yescharacteristic
605C2#2028297012yes—
606C2#2028297211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#14635
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. #20272985C2semidirect
3. #20272973C2semidirect
4. #20272973C2semidirect
5. #20272971C2semidirect
15. #16810535C4semidirect
36. C22×Dic106C2×C4semidirect
61. #13133124C2×C4semidirect
62. #13133124C2×C4semidirect
68. C23×C4×C53Q8semidirect
335. C53#14635semidirect
350. C23×C4Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168105352
#202729852
#202729732
#202729712
#202729602
#202729582
#202729382
C53×C22⋊#952
#1463553