Finite Group Databasebeta

← #27315226

Subgroups — #27315226

Order 13568.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810484C211yesmaximal, characteristic
2C2×#16810481C212yesmaximal
3#20272979C212yesmaximal, complemented
4#20272978C212yesmaximal, complemented
5#20272973C212yesmaximal, complemented
6#20272970C212yesmaximal, complemented
7#20272960C211yesmaximal, characteristic
8#20272958C212yesmaximal
9C53×#1439C211yesmaximal, characteristic
10C24×Dic53C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719919C2212yes—
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2211yescharacteristic
17#16810484C418yescomplemented
18#16810481—28no—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C22×C4⋊C4×C53C2211yescharacteristic
29C22×C22⋊C4×C53C2212yes—
30C22×C22⋊C4×C53C2211yescharacteristic
31C2×#95×C53C2212yes—
32C2×#95×C53C2211yescharacteristic
33C23×Dic53—41no—
34C23×Dic53—41no—
35C23×Dic53—41no—
36C23×Dic53—41no—
37C23×Dic53C2×C414yes—
38C23×Dic53D414yes—
39C23×Dic53Q814yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C2×C4⋊Dic53—42no—
45C2×C4⋊Dic53—41no—
46C2×#11719919—24no—
47C2×#11719919—24no—
48C2×#11719919—42no—
49C2×#11719919—42no—
50C2×#11719919—42no—
51C2×#11719919—42no—
52C2×#11719919—42no—
53C2×#11719919—42no—
54C2×#11719919—42no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—41no—
57C2×C22⋊Dic53—41no—
58C2×C22⋊Dic53—42no—
59C2×C22⋊Dic53—42no—
60C2×C22⋊Dic53C2×C414yes—
61#13133124—24no—
62#13133124—24no—
63#13133124—42no—
64#13133124—42no—
65#13133124—42no—
66#13133124—42no—
67#13133124—41no—
68#13133124C2×C418yes—
69#13133124C2×C418yes—
70C25×C53C2311yescharacteristic
71C23×C4×C53C2312yes—
72C23×C4×C53C2312yes—
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2311yescharacteristic
75C2×C4⋊C4×C53—41no—
76C2×C4⋊C4×C53—42no—
77C2×C22⋊C4×C53—24no—
78C2×C22⋊C4×C53—41no—
79C2×C22⋊C4×C53—41no—
80C2×C22⋊C4×C53—42no—
81C2×C22⋊C4×C53—42no—
82C2×C22⋊C4×C53C2×C414yes—
83#95×C53—41no—
84#95×C53—42no—
85C24×C53—22no—
86C24×C53—22no—
87C24×C53—21no—
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53C22×C412yes—
94C24×C53C2×D412yes—
95C24×C53C2×Q812yes—
96C24×C53C2411yescharacteristic
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
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—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×Dic53—22no—
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—44no—
146C22×Dic53—44no—
147C22×Dic53—44no—
148C22×Dic53—44no—
149C22×Dic53—44no—
150C22×Dic53—44no—
151C22×Dic53C4⋊C4116yescomplemented
152C22×Dic53C22×C414yes—
153C22×Dic53C22×C414yes—
154C22×Dic53C2×D414yes—
155C22×Dic53C2×Q814yes—
156C4⋊C4×C53—44no—
157C22⋊C4×C53—44no—
158C22⋊C4×C53—48no—
159C22⋊Dic53—44no—
160C22⋊Dic53—48no—
161C4⋊Dic53—44no—
162#11719919—48no—
163#11719919—48no—
164#11719919—48no—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—24no—
168C23×C53—24no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—24no—
172C23×C53—24no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—24no—
176C23×C53—24no—
177C23×C53—41no—
178C23×C53—42no—
179C23×C53—41no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—41no—
185C23×C53—42no—
186C23×C53—41no—
187C23×C53C2×C4⋊C414yes—
188C23×C53C23×C411yescharacteristic
189C23×C53C22×D411yescharacteristic
190C23×C53C22×Q811yescharacteristic
191C23×C53H(5,𝔽2)11yescharacteristic
192C23×C53H(5,𝔽2)11yescharacteristic
193C23×C53H(5,𝔽2)12yes—
194C23×C53H(5,𝔽2)12yes—
195C23×C53H(5,𝔽2)11yescharacteristic
196C23×C53H(5,𝔽2)11yescharacteristic
197C23×C532−1+412yes—
198C23×C532−1+412yes—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
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×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—416no—
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—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53C2×C4⋊C418yes—
252#14674—531nomaximal, Sylow
253C22×C53—24no—
254C22×C53—28no—
255C22×C53—44no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—44no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—44no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53C22×C4⋊C411yescharacteristic
265C22×C53#51811yescharacteristic
266C22×C53#51811yescharacteristic
267C22×C53#52012yes—
268C22×C53#52012yes—
269C22×C53#53411yescharacteristic
270C22×C53#53411yescharacteristic
271C22×C53#53512yes—
272C22×C53#53612yes—
273C22×C53#54311yescharacteristic
274C22×C53#54311yescharacteristic
275C22×C53#54412yes—
276C22×C53#54412yes—
277C22×C53#56011yescharacteristic
278C22×C53#56012yes—
279C22×C53#56011yescharacteristic
280C22×C53#56212yes—
281C22×C53#56212yes—
282C22×C53#56212yes—
283C22×C53#56212yes—
284C22×C53#56311yescharacteristic
285C22×C53#56312yes—
286C22×C53#56311yescharacteristic
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—82no—
290C4×C53—82no—
291Dic53—28no—
292Dic53—44no—
293Dic53—82no—
294Dic53—82no—
295C2×#381—531no—
296C2×#380—532no—
297#1444—532no—
298#1444—532no—
299#1442—532no—
300#1442—532no—
301#1439—532no—
302#1439—531no—
303#1439D5311yescharacteristic
304C2×C53—44no—
305C2×C53#231511yescharacteristic
306C2×C53#231712yes—
307C2×C53#237611yescharacteristic
308C2×C53#237612yes—
309C2×C53#237611yescharacteristic
310C2×C53#281511yescharacteristic
311C2×C53#281612yes—
312C2×C53#281711yescharacteristic
313C2×C53#284111yescharacteristic
314C2×C53#284112yes—
315C2×C53#284111yescharacteristic
316C24×C4—531no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4—532no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4C2×D5311yescharacteristic
322C22×C22⋊C4—531no—
323C22×C22⋊C4—532no—
324C22×C22⋊C4C2×D5312yes—
325C22×C22⋊C4C2×D5311yescharacteristic
326#381—538no—
327#380—1068no—
328C2×#95—532no—
329C2×#95—532no—
330C2×#95—532no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95—531no—
334C2×#95—532no—
335C2×#95—532no—
336C2×#95—532no—
337C2×#95C2×D5311yescharacteristic
338C2×#95C2×D5312yes—
339C53#1467411yescharacteristic, complemented, Sylow
340C25C22×D5311yescharacteristic
341C23×C4—532no—
342C23×C4—534no—
343C23×C4—534no—
344C23×C4—534no—
345C23×C4—532no—
346C23×C4—532no—
347C23×C4—532no—
348C23×C4—2121no—
349C23×C4—2121no—
350C23×C4—2121no—
351C23×C4—2121no—
352C23×C4C22×D5312yes—
353C23×C4C22×D5312yes—
354C23×C4C22×D5311yescharacteristic
355C23×C4C22×D5311yescharacteristic
356C2×C4⋊C4—42no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—1064no—
359C2×C4⋊C4—1064no—
360C2×C4⋊C4—2122no—
361C2×C4⋊C4—2122no—
362C2×C4⋊C4—2122no—
363C2×C4⋊C4—2122no—
364C2×C4⋊C4—2122no—
365C2×C4⋊C4—2122no—
366C2×C4⋊C4—2122no—
367C2×C4⋊C4—2122no—
368C2×C4⋊C4—2121no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—41no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—534no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—2121no—
377C2×C22⋊C4—2121no—
378C2×C22⋊C4—2122no—
379C2×C22⋊C4—2122no—
380C2×C22⋊C4C4×D5314yes—
381#95—41no—
382#95—42no—
383#95—538no—
384#95—538no—
385#95—1064no—
386#95—1064no—
387#95—2121no—
388#95—2122no—
389#95—2122no—
390#95—2122no—
391#95—2122no—
392C24—22no—
393C24—22no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24C2×C4×D5312yes—
401C24D4×D5312yes—
402C24Q8×D5312yes—
403C24C23×D5311yescharacteristic
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
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—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—534no—
427C22×C4—534no—
428C22×C4—534no—
429C22×C4—5316no—
430C22×C4—534no—
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—2124no—
456C22×C4—2124no—
457C22×C4—2124no—
458C22×C4—2124no—
459C22×C4—2124no—
460C22×C4—2124no—
461C22×C4C2×C4×D5312yes—
462C22×C4C2×C4×D5312yes—
463C4⋊C4—44no—
464C4⋊C4—2128no—
465C4⋊C4—2128no—
466C4⋊C4—2128no—
467C4⋊C4—2124no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—2124no—
471C22⋊C4—2128no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23C4⋊C4×D5314yes—
495C23C22×C4×D5311yescharacteristic
496C23C2×D4×D5311yescharacteristic
497C23C2×Q8×D5311yescharacteristic
498C23#1313326211yescharacteristic
499C23#1313326211yescharacteristic
500C23#1313326212yes—
501C23#1313326212yes—
502C23#1313326311yescharacteristic
503C23#1313326311yescharacteristic
504C23#1313326412yes—
505C23#1313326412yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—538no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1064no—
538C2×C4—1064no—
539C2×C4—1064no—
540C2×C4—21216no—
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—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C22—24no—
560C22—28no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22C2×C4⋊C4×D5311yescharacteristic
571C22#1681109611yescharacteristic
572C22#1681109711yescharacteristic
573C22#1681110212yes—
574C22#1681110212yes—
575C22#1681118111yescharacteristic
576C22#1681118111yescharacteristic
577C22#1681118812yes—
578C22#1681119512yes—
579C22#1681124811yescharacteristic
580C22#1681124811yescharacteristic
581C22#1681125212yes—
582C22#1681125212yes—
583C22#1681136011yescharacteristic
584C22#1681136112yes—
585C22#1681136211yescharacteristic
586C22#1681136712yes—
587C22#1681136812yes—
588C22#1681136812yes—
589C22#1681137012yes—
590C22#1681137311yescharacteristic
591C22#1681137512yes—
592C22#1681137611yescharacteristic
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—1068no—
598C4—4242no—
599C4—4242no—
600C4—2124no—
601C2—44no—
602C2#2027732811yescharacteristic
603C2#2027733412yes—
604C2#2027785811yescharacteristic
605C2#2027786112yes—
606C2#2027786511yescharacteristic
607C2#2028274011yescharacteristic
608C2#2028274712yes—
609C2#2028275311yescharacteristic
610C2#2028297511yescharacteristic
611C2#2028297612yes—
612C2#2028297711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#14674
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #20272979C2semidirect
4. #20272978C2semidirect
5. #20272973C2semidirect
6. #20272970C2semidirect
17. #16810484C4semidirect
151. C22×Dic53C4⋊C4semidirect
339. C53#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#168104842
C2×#168104812
#202729792
#202729782
#202729732
#202729702
#202729602
#202729582
C53×#14392
#1467453