Finite Group Databasebeta

← #27315005

Subgroups — #27315005

Order 13568.

1226 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810559C211yesmaximal, characteristic
2#20272976C212yesmaximal, complemented
3#20272967C212yesmaximal, complemented
4#20272966C212yesmaximal, complemented
5#20272962C212yesmaximal
6#20272958C212yesmaximal
7#20272952C211yesmaximal, characteristic
8#20272950C212yesmaximal, complemented
9C53×#1440C211yesmaximal, characteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×#11719919C2212yes—
12C22×#11719919C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C4×Dic53C2211yescharacteristic
16#16810559C418yescomplemented
17C2×#13133124C2212yes—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C22×C22⋊C4×C53C2211yescharacteristic
28C22×C22⋊C4×C53C2211yescharacteristic
29C22×C22⋊C4×C53C2211yescharacteristic
30C2×#95×C53C2212yes—
31C2×#95×C53C2212yes—
32C23×Dic53C2312yes—
33C23×Dic53C2312yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C2×C4⋊Dic53—41no—
37C2×C4⋊Dic53—41no—
38C2×C4⋊Dic53C2×C414yes—
39C2×#11719919—42no—
40C2×#11719919—42no—
41C2×#11719919—42no—
42C2×#11719919—42no—
43C2×#11719919—42no—
44C2×#11719919C2×C418yes—
45C2×C22⋊Dic53—24no—
46C2×C22⋊Dic53—42no—
47C2×C22⋊Dic53—42no—
48C2×C22⋊Dic53—42no—
49C2×C22⋊Dic53—42no—
50C2×C22⋊Dic53C2×C418yes—
51C2×C4×Dic53—41no—
52C2×C4×Dic53—42no—
53#13133124—24no—
54#13133124—24no—
55#13133124—42no—
56#13133124—42no—
57#13133124—42no—
58#13133124—42no—
59#13133124—42no—
60#13133124—42no—
61#13133124—41no—
62#13133124C2×C414yes—
63C25×C53C2311yescharacteristic
64C23×C4×C53C2312yes—
65C23×C4×C53C2312yes—
66C23×C4×C53C2311yescharacteristic
67C23×C4×C53C2311yescharacteristic
68C2×C22⋊C4×C53—41no—
69C2×C22⋊C4×C53—41no—
70C2×C22⋊C4×C53—42no—
71C2×C22⋊C4×C53—42no—
72C2×C22⋊C4×C53—41no—
73C2×C22⋊C4×C53—41no—
74C2×C22⋊C4×C53C2×C414yes—
75#95×C53—42no—
76#95×C53—42no—
77C24×C53—22no—
78C24×C53—21no—
79C24×C53—21no—
80C24×C53—22no—
81C24×C53—22no—
82C24×C53—22no—
83C24×C53—22no—
84C24×C53—22no—
85C24×C53C22×C412yes—
86C24×C53C2411yescharacteristic
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
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×C53C22×C412yes—
110C22×C4×C53C22×C412yes—
111C22×Dic53—22no—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—22no—
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×Dic53C22×C414yes—
137C22×Dic53C22×C414yes—
138C22×Dic53#4114yes—
139C22⋊C4×C53—44no—
140C22⋊C4×C53—44no—
141C22⋊C4×C53—44no—
142C22⋊Dic53—48no—
143C22⋊Dic53—48no—
144C4⋊Dic53—44no—
145#11719919—48no—
146#11719919—48no—
147C4×Dic53—44no—
148C23×C53—22no—
149C23×C53—24no—
150C23×C53—22no—
151C23×C53—22no—
152C23×C53—24no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—42no—
157C23×C53—42no—
158C23×C53—42no—
159C23×C53—42no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—42no—
164C23×C53—42no—
165C23×C53—42no—
166C23×C53—42no—
167C23×C53—42no—
168C23×C53—41no—
169C23×C53C23×C411yescharacteristic
170C23×C53C2×#4112yes—
171C23×C53H(5,𝔽2)11yescharacteristic
172C23×C53H(5,𝔽2)12yes—
173C23×C53H(5,𝔽2)11yescharacteristic
174C23×C53H(5,𝔽2)12yes—
175C23×C53H(5,𝔽2)12yes—
176C23×C532−1+412yes—
177C23×C532−1+412yes—
178C2×C4×C53—24no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—24no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—42no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×Dic53—24no—
203C2×Dic53—24no—
204C2×Dic53—24no—
205C2×Dic53—24no—
206C2×Dic53—24no—
207C2×Dic53—24no—
208C2×Dic53—24no—
209C2×Dic53—24no—
210C2×Dic53—24no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230#14646—531nomaximal, Sylow
231C22×C53—42no—
232C22×C53—44no—
233C22×C53—42no—
234C22×C53—44no—
235C22×C53—42no—
236C22×C53—82no—
237C22×C53—82no—
238C22×C53—82no—
239C22×C53—82no—
240C22×C53C4×#4111yescharacteristic
241C22×C53#51812yes—
242C22×C53#51811yescharacteristic
243C22×C53#51911yescharacteristic
244C22×C53#52012yes—
245C22×C53#53812yes—
246C22×C53#53912yes—
247C22×C53#53912yes—
248C22×C53#54012yes—
249C22×C53#54212yes—
250C22×C53#54212yes—
251C22×C53#56012yes—
252C22×C53#56011yescharacteristic
253C22×C53#56011yescharacteristic
254C22×C53#56011yescharacteristic
255C22×C53#56212yes—
256C22×C53#56212yes—
257C22×C53#56212yes—
258C22×C53#56312yes—
259C22×C53#56312yes—
260C22×C53#56311yescharacteristic
261C4×C53—42no—
262C4×C53—42no—
263C4×C53—82no—
264C4×C53—82no—
265Dic53—44no—
266Dic53—44no—
267Dic53—44no—
268Dic53—82no—
269C2×#399—531no—
270#1443—532no—
271#1441—532no—
272#1441—532no—
273#1440—532no—
274#1440D5311yescharacteristic
275#1439—532no—
276#1437—531no—
277#1436—532no—
278C2×C53—82no—
279C2×C53#233211yescharacteristic
280C2×C53#233512yes—
281C2×C53#237611yescharacteristic
282C2×C53#237812yes—
283C2×C53#238011yescharacteristic
284C2×C53#282712yes—
285C2×C53#282912yes—
286C2×C53#283712yes—
287C2×C53#283812yes—
288C22×C4⋊C4—532no—
289C22×C4⋊C4—531no—
290C22×C4⋊C4—532no—
291C22×C22⋊C4—532no—
292C22×C22⋊C4—532no—
293C22×C22⋊C4C2×D5311yescharacteristic
294C22×C22⋊C4C2×D5311yescharacteristic
295C22×C22⋊C4C2×D5311yescharacteristic
296C22×C42—531no—
297#399—538no—
298C2×#95—531no—
299C2×#95—532no—
300C2×#95—532no—
301C2×#95—532no—
302C2×#95—532no—
303C2×#95—531no—
304C2×#95—532no—
305C2×#95—532no—
306C2×#95—532no—
307C2×#95—532no—
308C2×#95C2×D5312yes—
309C2×#95C2×D5312yes—
310C53#1464611yescharacteristic, complemented, Sylow
311C25C22×D5311yescharacteristic
312C23×C4—532no—
313C23×C4—532no—
314C23×C4—532no—
315C23×C4—532no—
316C23×C4C22×D5312yes—
317C23×C4C22×D5312yes—
318C23×C4C22×D5311yescharacteristic
319C23×C4C22×D5311yescharacteristic
320C2×C4⋊C4—538no—
321C2×C4⋊C4—534no—
322C2×C4⋊C4—2122no—
323C2×C4⋊C4—2122no—
324C2×C4⋊C4—2122no—
325C2×C4⋊C4—2122no—
326C2×C4⋊C4—2122no—
327C2×C4⋊C4—2121no—
328C2×C4⋊C4—2121no—
329C2×C22⋊C4—41no—
330C2×C22⋊C4—41no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—41no—
334C2×C22⋊C4—41no—
335C2×C22⋊C4—538no—
336C2×C22⋊C4—1064no—
337C2×C22⋊C4—2122no—
338C2×C22⋊C4—2122no—
339C2×C22⋊C4—2122no—
340C2×C22⋊C4—2122no—
341C2×C22⋊C4C4×D5314yes—
342C2×C42—2121no—
343C2×C42—2122no—
344#95—42no—
345#95—42no—
346#95—534no—
347#95—1064no—
348#95—1064no—
349#95—2122no—
350#95—2121no—
351#95—2122no—
352#95—2122no—
353#95—2122no—
354#95—2122no—
355#95—2122no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24C2×C4×D5312yes—
365C24C23×D5311yescharacteristic
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—534no—
389C22×C4—534no—
390C22×C4—534no—
391C22×C4—1062no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—1062no—
396C22×C4—1062no—
397C22×C4—1062no—
398C22×C4—1062no—
399C22×C4—1062no—
400C22×C4—1062no—
401C22×C4—1062no—
402C22×C4—1062no—
403C22×C4—1062no—
404C22×C4—1062no—
405C22×C4—1062no—
406C22×C4—1062no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—1062no—
410C22×C4—1062no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4—1062no—
416C22×C4C2×C4×D5312yes—
417C22×C4C2×C4×D5312yes—
418C4⋊C4—2128no—
419C4⋊C4—2124no—
420C4⋊C4—2128no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—2128no—
425C22⋊C4—2128no—
426C42—2124no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23C22×C4×D5311yescharacteristic
449C23#41×D5312yes—
450C23#1313326212yes—
451C23#1313326211yescharacteristic
452C23#1313326211yescharacteristic
453C23#1313326212yes—
454C23#1313326312yes—
455C23#1313326412yes—
456C23#1313326412yes—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—1064no—
482C2×C4—1064no—
483C2×C4—1064no—
484C2×C4—1064no—
485C2×C4—1064no—
486C2×C4—1064no—
487C2×C4—1064no—
488C2×C4—1064no—
489C2×C4—1064no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C22—42no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22#1681109111yescharacteristic
519C22#1681109512yes—
520C22#1681109611yescharacteristic
521C22#1681110011yescharacteristic
522C22#1681110212yes—
523C22#1681120712yes—
524C22#1681121412yes—
525C22#1681121612yes—
526C22#1681122312yes—
527C22#1681124112yes—
528C22#1681124112yes—
529C22#1681136112yes—
530C22#1681136211yescharacteristic
531C22#1681136211yescharacteristic
532C22#1681136311yescharacteristic
533C22#1681136812yes—
534C22#1681136812yes—
535C22#1681137012yes—
536C22#1681137312yes—
537C22#1681137512yes—
538C22#1681137611yescharacteristic
539C4—42no—
540C4—42no—
541C4—82no—
542C4—82no—
543C4—4242no—
544C4—2124no—
545C4—2124no—
546C4—2124no—
547C2—82no—
548C2#2027746811yescharacteristic
549C2#2027750312yes—
550C2#2027785811yescharacteristic
551C2#2027787512yes—
552C2#2027789011yescharacteristic
553C2#2028282012yes—
554C2#2028285012yes—
555C2#2028292212yes—
556C2#2028294312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1440C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1440D53
hypocenterC53#14646
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
2. #20272976C2semidirect
3. #20272967C2semidirect
4. #20272966C2semidirect
8. #20272950C2semidirect
16. #16810559C4semidirect
310. C53#14646semidirect

Maximal subgroups

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

subgroupindex
C2×#168105592
#202729762
#202729672
#202729662
#202729622
#202729582
#202729522
#202729502
C53×#14402
#1464653