Finite Group Databasebeta

← #27543190

Subgroups — #27543190

Order 13568.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811228C211yesmaximal, characteristic
2#20277511C214yesmaximal, complemented
3#20277478C214yesmaximal, complemented
4#20277454C211yesmaximal, characteristic, complemented
5#20277438C211yesmaximal, characteristic, complemented
6#20277404C212yesmaximal, complemented
7#20277353C212yesmaximal, complemented
8#20277277C211yesmaximal, characteristic, complemented
9#20277270C211yesmaximal, characteristic, complemented
10#20277236C214yesmaximal, complemented
11#20277227C214yesmaximal, complemented
12#20277221C212yesmaximal, complemented
13C4×#13133179C212yesmaximal, complemented
14#117×Dic53C211yesmaximal, characteristic, complemented
15C53×#2314C211yesmaximal, characteristic, complemented
16#16811228C4116yescomplemented
17C2×#13133214C2211yescharacteristic
18C2×#13133209C2211yescharacteristic
19C2×#13133202C2214yes—
20C2×#13133191C2212yes—
21C2×#13133190C2214yes—
22C2×C4×D212C2212yes—
23C2×#13133179C2212yes—
24C2×#13133179C2212yes—
25C2×#13133179C2214yes—
26C2×#13133177C2214yes—
27C2×#13133176C2211yescharacteristic
28C22×C4×Dic53C2211yescharacteristic
29C2×C42×D53C2214yes—
30#16810524C2214yes—
31#16810524C2212yes—
32#16810524C2212yes—
33#16810523C2214yes—
34#16810519C2212yes—
35#16810519C2212yes—
36#16810518C2214yes—
37#16810515C2214yes—
38#16810514C2214yes—
39#16810514C2214yes—
40#16810512C2214yes—
41#16810505C2214yes—
42#16810504C2214yes—
43#16810501C2214yes—
44#16810501C2214yes—
45#16810500C2214yes—
46#16810499C2214yes—
47#16810497C2214yes—
48#16810485C2214yes—
49#16810482C2214yes—
50C4⋊C4×Dic53C2212yes—
51C4⋊C4×Dic53C2212yes—
52C4⋊C4×Dic53C2212yes—
53C4⋊C4×Dic53C2212yes—
54#16810474C2214yes—
55C22⋊C4×Dic53C2212yes—
56C22⋊C4×Dic53C2212yes—
57C4×#11719921C2214yes—
58#16810470C2214yes—
59#16810470C2214yes—
60#16810470C2214yes—
61#16810469C2211yescharacteristic
62#16810469C2211yescharacteristic
63C42×Dic53C2211yescharacteristic
64C42×Dic53C2211yescharacteristic
65C2×#117×C53C2211yescharacteristic
66C53×#388C2214yes—
67C53×#387C2214yes—
68C53×#384C2211yescharacteristic
69C53×#382C2211yescharacteristic
70C53×#380C2212yes—
71C4×C4⋊C4×C53C2212yes—
72C22×D212C2312yes—
73C22×D212C2314yes—
74C22×C22⋊D53C2314yes—
75C23×Dic53C2311yescharacteristic
76C23×Dic53C2311yescharacteristic
77C22×C4×D53C2314yes—
78C22×C4×D53C2314yes—
79C22×C4×D53C2314yes—
80#13133214C2×C418yescomplemented
81#13133209C2×C418yescomplemented
82#13133202C2×C4132yescomplemented
83#13133191C2×C4116yescomplemented
84#13133190C2×C4132yescomplemented
85C4×D212C2×C4116yescomplemented
86#13133179—28no—
87#13133179—28no—
88#13133179—48no—
89#13133177—48no—
90#13133176—24no—
91C2×C4⋊Dic53C2312yes—
92C2×C4⋊Dic53C2312yes—
93C2×#11719919C2314yes—
94C2×#11719919C2314yes—
95C2×#11719919C2314yes—
96C2×C22⋊Dic53C2312yes—
97C2×#11719921C2314yes—
98C2×#11719921C2314yes—
99C2×#11719921C2314yes—
100C2×#11719921C2314yes—
101C2×#11719921C2314yes—
102C2×#11719921C2314yes—
103C2×#11719921C2314yes—
104C2×#11719921C2314yes—
105C2×C4×Dic53—24no—
106C2×C4×Dic53C2311yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C2×C4×Dic53C2312yes—
110C2×C4×Dic53C2314yes—
111C2×C4×Dic53C2311yescharacteristic
112C2×C4×Dic53C2312yes—
113C2×C4×Dic53C2314yes—
114C2×C4×Dic53C2312yes—
115C2×C4×Dic53C2312yes—
116C2×C4×Dic53C2311yescharacteristic
117C2×C4×Dic53C2311yescharacteristic
118C42×D53—48no—
119#13133124C2314yes—
120#13133124C2314yes—
121#13133124C2314yes—
122#13133124C2314yes—
123#13133124C2312yes—
124#13133124C2312yes—
125#13133124C2312yes—
126#13133124C2312yes—
127C23×C4×C53C2311yescharacteristic
128#117×C53C2×C418yescomplemented
129C2×C4⋊C4×C53C2312yes—
130C2×C4⋊C4×C53C2312yes—
131C2×C4⋊C4×C53C2314yes—
132C2×C4⋊C4×C53C2312yes—
133C2×C22⋊C4×C53C2314yes—
134C2×C22⋊C4×C53C2312yes—
135C2×C42×C53C2314yes—
136C2×C42×C53C2311yescharacteristic
137C2×C42×C53C2311yescharacteristic
138#95×C53C2314yes—
139#95×C53C2314yes—
140#95×C53C2314yes—
141C24×C53C2411yescharacteristic
142C23×D53C2414yes—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53C22×C414yes—
146C22×C4×C53C2414yes—
147C22×C4×C53C2414yes—
148C22×C4×C53C2411yescharacteristic
149C22×C4×C53C2412yes—
150C22×C4×C53C2412yes—
151C22×C4×C53C2411yescharacteristic
152C2×C22⋊D53—44no—
153C2×C22⋊D53—44no—
154C2×C22⋊D53C22×C4116yes—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—24no—
158C22×Dic53—42no—
159C22×Dic53C2411yescharacteristic
160C22×Dic53C2412yes—
161C22×Dic53C2411yescharacteristic
162C22×Dic53C2414yes—
163C22×Dic53C2411yescharacteristic
164C22×Dic53C2412yes—
165C22×Dic53C2411yescharacteristic
166C2×D212—24no—
167C2×D212—24no—
168C2×D212—44no—
169C2×D212—44no—
170C2×D212C22×C418yes—
171C2×D212C22×C4116yes—
172C2×C4×D53—28no—
173C2×C4×D53—28no—
174C2×C4×D53—28no—
175C2×C4×D53—28no—
176C2×C4×D53—44no—
177C2×C4×D53—44no—
178C2×C4×D53—44no—
179C2×C4×D53—44no—
180C2×C4×D53C22×C4116yes—
181C4⋊C4×C53—24no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—44no—
184C4⋊C4×C53C22×C418yes—
185C22⋊C4×C53—44no—
186C22⋊C4×C53C22×C418yes—
187C42×C53—44no—
188C42×C53C22×C414yes—
189C42×C53C22×C414yes—
190C22⋊Dic53—24no—
191#11719921—28no—
192#11719921—28no—
193#11719921—44no—
194#11719921—44no—
195#11719921—44no—
196#11719921—44no—
197#11719921C22×C4116yes—
198#11719921C22×C4116yes—
199C4⋊Dic53—24no—
200C4⋊Dic53C22×C418yes—
201#11719919—44no—
202#11719919—44no—
203#11719919C22×C4116yes—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—24no—
209C4×Dic53—28no—
210C4×Dic53—24no—
211C4×Dic53—44no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C4×Dic53#4114yes—
215C4×Dic53#4114yes—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53C23×C412yes—
221C23×C53C2511yescharacteristic
222C22×D53—24no—
223C22×D53—24no—
224C22×D53—24no—
225C22×D53—24no—
226C22×D53—24no—
227C22×D53—24no—
228C22×D53C23×C418yes—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—28no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53C23×C414yes—
243C2×C4×C53C23×C414yes—
244C2×C4×C53C23×C412yes—
245C2×C4×C53C23×C412yes—
246C2×C4×C53C2×#4112yes—
247C2×C4×C53C2×#4112yes—
248C22⋊D53—416no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—44no—
262C2×Dic53—44no—
263C2×Dic53C23×C418yes—
264C2×Dic53C2×#4114yes—
265C2×Dic53C2×#4112yes—
266C2×Dic53C2×#4112yes—
267C2×Dic53H(5,𝔽2)14yes—
268D212—216no—
269D212—416no—
270C4×D53—416no—
271C4×D53—416no—
272C4×D53—416no—
273C4×#540—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—24no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—42no—
279C22×C53C24×C411yescharacteristic
280C22×C53C22×#4111yescharacteristic
281C22×C53C22×#4111yescharacteristic
282C22×C53C2×H(5,𝔽2)12yes—
283C22×C53#58512yes—
284C2×D53—28no—
285C2×D53—28no—
286C2×D53—28no—
287C2×D53—44no—
288C2×D53—44no—
289C2×D53—44no—
290C2×D53—44no—
291C4×C53—22no—
292C4×C53—24no—
293C4×C53—24no—
294C4×C53—44no—
295C4×C53—44no—
296C4×C53C4×#4114yes—
297Dic53—28no—
298Dic53—22no—
299Dic53—24no—
300Dic53—41no—
301Dic53—41no—
302Dic53—42no—
303Dic53#54014yescomplemented
304C2×#540—531no—
305#2335—534no—
306#2333—534no—
307#2330—531no—
308#2328—531no—
309#2324—532no—
310#2319—532no—
311#2314D5311yescharacteristic, complemented
312C4×#127—531no—
313C4×#125—531no—
314C4×#123—534no—
315C4×#121—534no—
316C4×#121—532no—
317C42×D4—532no—
318C4×#117—531no—
319C2×C53—42no—
320C2×C53C2×C4×#4111yescharacteristic
321C2×C53C4×H(5,𝔽2)12yes—
322C2×C53C2×#54011yescharacteristic
323C2×C53#345511yescharacteristic
324C2×C53#346012yes—
325D53—48no—
326#540—5316no—
327C2×#127—531no—
328C2×#125—531no—
329C2×#123—534no—
330C2×#121—532no—
331C2×#121—534no—
332C2×C4×D4—532no—
333C2×C4×D4—532no—
334C2×C4×D4—534no—
335C2×C4×D4—534no—
336C2×C4×D4—532no—
337C2×#117—531no—
338C2×#117C2×D5311yescharacteristic
339C22×C42—531no—
340C22×C42—534no—
341#390—532no—
342#390—534no—
343#390—532no—
344#390—534no—
345#389—532no—
346#389—532no—
347#389—534no—
348#388—534no—
349#388—534no—
350#388—534no—
351#388—534no—
352#388C2×D5314yes—
353#387—534no—
354#387—534no—
355#387C2×D5314yes—
356#386—534no—
357#386—534no—
358#386—534no—
359#386—534no—
360#385—534no—
361#384C2×D5311yescharacteristic
362#382C2×D5311yescharacteristic
363#381—534no—
364#380—534no—
365#380C2×D5312yes—
366C4×C4⋊C4—532no—
367C4×C4⋊C4—532no—
368C4×C4⋊C4—532no—
369C4×C4⋊C4—532no—
370C4×C4⋊C4—534no—
371C4×C4⋊C4C2×D5312yes—
372C4×C22⋊C4—532no—
373C4×C22⋊C4—532no—
374C4×C22⋊C4—534no—
375C4×C22⋊C4—534no—
376C4×C22⋊C4—534no—
377C4×C22⋊C4—534no—
378#376—531no—
379#376—531no—
380C43—531no—
381C43—531no—
382C53C4×#54011yescharacteristic, complemented, Sylow
383C22×D4—534no—
384C22×D4—532no—
385C22×D4—534no—
386C23×C4—534no—
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—534no—
390C23×C4—534no—
391C23×C4C22×D5311yescharacteristic
392#127—538no—
393#125—538no—
394#123—5332no—
395#121—5316no—
396#121—5332no—
397C4×D4—5316no—
398C4×D4—1068no—
399C4×D4—1068no—
400C4×D4—2128no—
401C4×D4—2128no—
402#117—1064no—
403#117C4×D5318yescomplemented
404C2×C4⋊C4—532no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—532no—
407C2×C4⋊C4—534no—
408C2×C4⋊C4—534no—
409C2×C4⋊C4C22×D5312yes—
410C2×C4⋊C4C22×D5312yes—
411C2×C4⋊C4C22×D5314yes—
412C2×C4⋊C4C22×D5312yes—
413C2×C22⋊C4—534no—
414C2×C22⋊C4—534no—
415C2×C22⋊C4—532no—
416C2×C22⋊C4—534no—
417C2×C22⋊C4—534no—
418C2×C22⋊C4—534no—
419C2×C22⋊C4—534no—
420C2×C22⋊C4—534no—
421C2×C22⋊C4—534no—
422C2×C22⋊C4C22×D5314yes—
423C2×C22⋊C4C22×D5312yes—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—532no—
428C2×C42—534no—
429C2×C42—532no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—531no—
433C2×C42—532no—
434C2×C42—532no—
435C2×C42—534no—
436C2×C42—1064no—
437C2×C42—2128no—
438C2×C42C22×D5314yes—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#95—532no—
442#95—532no—
443#95—532no—
444#95—532no—
445#95—534no—
446#95—534no—
447#95—534no—
448#95—534no—
449#95C22×D5314yes—
450#95C22×D5314yes—
451#95C22×D5314yes—
452C24—534no—
453C24C23×D5311yescharacteristic
454C2×D4—5316no—
455C2×D4—538no—
456C2×D4—5316no—
457C2×D4—1064no—
458C2×D4—1064no—
459C2×D4—2124no—
460C2×D4—2124no—
461C2×D4—2124no—
462C2×D4—2124no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—532no—
468C22×C4—534no—
469C22×C4—5316no—
470C22×C4—531no—
471C22×C4—532no—
472C22×C4—531no—
473C22×C4—1068no—
474C22×C4—1068no—
475C22×C4—1068no—
476C22×C4—1068no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1064no—
480C22×C4—2124no—
481C22×C4—2124no—
482C22×C4—2124no—
483C22×C4—2124no—
484C22×C4—2122no—
485C22×C4C2×C4×D5314yes—
486C22×C4C23×D5314yes—
487C22×C4C23×D5314yes—
488C22×C4C23×D5312yes—
489C22×C4C23×D5312yes—
490C22×C4C23×D5311yescharacteristic
491C22×C4C23×D5311yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—538no—
496C4⋊C4—5316no—
497C4⋊C4—1064no—
498C4⋊C4—2124no—
499C4⋊C4—2124no—
500C4⋊C4C2×C4×D5318yes—
501C22⋊C4—44no—
502C22⋊C4—5316no—
503C22⋊C4—5316no—
504C22⋊C4—1068no—
505C22⋊C4—1068no—
506C22⋊C4—1064no—
507C22⋊C4—2124no—
508C22⋊C4—2124no—
509C22⋊C4—2124no—
510C22⋊C4—2124no—
511C22⋊C4C2×C4×D5318yes—
512C42—44no—
513C42—534no—
514C42—534no—
515C42—1062no—
516C42—1062no—
517C42—1064no—
518C42—1068no—
519C42—1064no—
520C42—1062no—
521C42—1062no—
522C42—2124no—
523C42—2122no—
524C42—2122no—
525C42C2×C4×D5314yes—
526C42C2×C4×D5314yes—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—538no—
532C23—1064no—
533C23—1064no—
534C23—1064no—
535C23—1064no—
536C23—1064no—
537C23—1064no—
538C23C22×C4×D5312yes—
539C23C24×D5311yescharacteristic
540D4—10616no—
541D4—21216no—
542D4—21216no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—532no—
557C2×C4—534no—
558C2×C4—538no—
559C2×C4—532no—
560C2×C4—534no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1064no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—21216no—
574C2×C4—21216no—
575C2×C4—21216no—
576C2×C4—2124no—
577C2×C4—2124no—
578C2×C4C22×C4×D5314yes—
579C2×C4C22×C4×D5314yes—
580C2×C4C22×C4×D5312yes—
581C2×C4C22×C4×D5312yes—
582C2×C4#41×D5312yes—
583C2×C4#41×D5312yes—
584C22—22no—
585C22—24no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—1068no—
590C22—1068no—
591C22—1068no—
592C22—2124no—
593C22—2124no—
594C22—2124no—
595C22—2124no—
596C22C23×C4×D5311yescharacteristic
597C22C2×#41×D5311yescharacteristic
598C22C2×#41×D5311yescharacteristic
599C22H(5,𝔽2)×D5312yes—
600C22#1681146012yes—
601C4—22no—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—534no—
607C4—1062no—
608C4—1064no—
609C4—1068no—
610C4—2121no—
611C4—2121no—
612C4—2122no—
613C4#1681109114yes—
614C2—42no—
615C2—2128no—
616C2C2×#1681109111yescharacteristic
617C2#2028792112yes—
618C2#540×D5311yescharacteristic
619C2#2028810611yescharacteristic
620C2#2028813512yes—
subgroupG/H
center, FrattiniC23C24×D53
commutatorC22×C53C24×C4
FittingC53×#2314C2
socle, cosocleC23×C53C25
prime radicalC2×#16811228C2
hypercenter#2314D53
hypocenterC53C4×#540
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #20277511C2semidirect
3. #20277478C2semidirect
4. #20277454C2semidirect
5. #20277438C2semidirect
6. #20277404C2semidirect
7. #20277353C2semidirect
8. #20277277C2semidirect
9. #20277270C2semidirect
10. #20277236C2semidirect
11. #20277227C2semidirect
12. #20277221C2semidirect
13. C4×#13133179C2semidirect
14. #117×Dic53C2semidirect
15. C53×#2314C2semidirect
16. #16811228C4semidirect
80. #13133214C2×C4semidirect
81. #13133209C2×C4semidirect
82. #13133202C2×C4semidirect
83. #13133191C2×C4semidirect
84. #13133190C2×C4semidirect
85. C4×D212C2×C4semidirect
128. #117×C53C2×C4semidirect
303. Dic53#540semidirect
311. #2314D53semidirect
382. C53C4×#540semidirect
403. #117C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112282
#202775112
#202774782
#202774542
#202774382
#202774042
#202773532
#202772772
#202772702
#202772362
#202772272
#202772212
C4×#131331792
#117×Dic532
C53×#23142
C4×#54053