Finite Group Databasebeta

← #27311311

Subgroups — #27311311

Order 13568.

1542 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
1C22×C4×Dic106C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic
3C2×#16810487C211yesmaximal, characteristic
4#20272985C212yesmaximal
5#20272981C212yesmaximal
6#20272967C212yesmaximal
7#20272966C212yesmaximal
8#20272942C211yesmaximal, characteristic
9#20272939C211yesmaximal, characteristic
10#20272939C211yesmaximal, characteristic
11C53×#1434C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×C4×Dic106—24no—
14C2×C4×Dic106—24no—
15C2×C4×Dic106C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810490—24no—
23#16810487—24no—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C42×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C22×Dic106—24no—
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—24no—
44C22×Dic106C2×C414yes—
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C4×Dic106—232no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2×C414yes—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—24no—
60C2×#11719919—42no—
61C2×#11719919—42no—
62C2×#11719919C2×C414yes—
63C2×#11719919C2×C414yes—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53C2×C414yes—
67C2×C4×Dic53C2×C414yes—
68C2×C4×Dic53D414yes—
69C2×C4×Dic53D414yes—
70#13133124—22no—
71#13133124—22no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—42no—
79#13133124—42no—
80#13133124—42no—
81C23×C4×C53C2312yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C2×C42×C53—22no—
87C2×C42×C53—22no—
88C2×C42×C53C2×C414yes—
89#95×C53—22no—
90#95×C53—22no—
91#95×C53—24no—
92#95×C53—42no—
93C24×C53C2411yescharacteristic
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
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×C4×C53C22×C412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53#4114yes—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
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×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53#4114yes—
163C2×Dic106—216no—
164C2×Dic106—216no—
165C2×Dic106—28no—
166C2×Dic106—28no—
167C2×Dic106—28no—
168C42×C53—28no—
169C4⋊Dic53—28no—
170#11719919—28no—
171#11719919—28no—
172#11719919—48no—
173C4×Dic53—28no—
174C4×Dic53C22⋊C4116yes—
175C23×C53C23×C411yescharacteristic
176C23×C53C22×D411yescharacteristic
177C23×C53C22×D411yescharacteristic
178C23×C53C2×#4112yes—
179C23×C53C2×#4112yes—
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C53H(5,𝔽2)11yescharacteristic
182C23×C532−1+412yes—
183C23×C532−1+412yes—
184C23×C532−1+411yescharacteristic
185C23×C532−1+411yescharacteristic
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—24no—
193C2×C4×C53—24no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
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×C53C2×C22⋊C414yes—
216C2×C4×C53#11714yes—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
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×Dic53—42no—
252C2×Dic53C2×C22⋊C414yes—
253C2×Dic53C2×C22⋊C414yes—
254C2×Dic53#11714yes—
255Dic106—232no—
256#14346—531nomaximal, Sylow
257C22×C53C22×C22⋊C411yescharacteristic
258C22×C53C2×#11711yescharacteristic
259C22×C53C2×#11711yescharacteristic
260C22×C53#51911yescharacteristic
261C22×C53#51911yescharacteristic
262C22×C53#52011yescharacteristic
263C22×C53#52011yescharacteristic
264C22×C53#52512yes—
265C22×C53#52512yes—
266C22×C53#53511yescharacteristic
267C22×C53#53612yes—
268C22×C53#53612yes—
269C22×C53#53611yescharacteristic
270C22×C53#53711yescharacteristic
271C22×C53#53711yescharacteristic
272C22×C53#54112yes—
273C22×C53#54112yes—
274C22×C53#54112yes—
275C22×C53#54112yes—
276C22×C53#54212yes—
277C22×C53#54212yes—
278C22×C53#54212yes—
279C22×C53#54212yes—
280C4×C53—24no—
281C4×C53—24no—
282C4×C53—42no—
283C4×C53—82no—
284C4×C53—82no—
285Dic53—24no—
286Dic53—24no—
287Dic53—24no—
288Dic53—42no—
289Dic53—82no—
290Dic53—82no—
291C22×C4×Q8—531no—
292C2×#383—531no—
293C2×#382—531no—
294#1446—532no—
295#1445—532no—
296#1441—532no—
297#1441—532no—
298#1434—531no—
299#1434—531no—
300#1434—531no—
301#1434D5311yescharacteristic
302C2×C53#228011yescharacteristic
303C2×C53#231011yescharacteristic
304C2×C53#231111yescharacteristic
305C2×C53#232011yescharacteristic
306C2×C53#232011yescharacteristic
307C2×C53#232011yescharacteristic
308C2×C53#232011yescharacteristic
309C2×C53#263212yes—
310C2×C53#263412yes—
311C2×C53#263712yes—
312C2×C53#263712yes—
313C23×Q8—531no—
314C2×C4×Q8—538no—
315C2×C4×Q8—1064no—
316C2×C4×Q8—1064no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—532no—
321C22×C42—531no—
322C22×C42—531no—
323C22×C42C2×D5311yescharacteristic
324#383—1064no—
325#382—1064no—
326C2×#95—531no—
327C2×#95—531no—
328C2×#95—532no—
329C2×#95—532no—
330C2×#95—532no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95—532no—
334C2×#95—532no—
335C2×#95—532no—
336C2×#95—532no—
337C2×#95C2×D5311yescharacteristic
338C2×#95C2×D5311yescharacteristic
339C2×#95C2×D5312yes—
340C2×#95C2×D5312yes—
341C53#1434611yescharacteristic, complemented, Sylow
342C22×Q8—534no—
343C22×Q8—1062no—
344C22×Q8—1062no—
345C22×Q8—1064no—
346C22×Q8—1064no—
347C23×C4—531no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—531no—
351C23×C4—532no—
352C23×C4—532no—
353C23×C4C22×D5312yes—
354C23×C4C22×D5312yes—
355C23×C4C22×D5311yescharacteristic
356C23×C4C22×D5311yescharacteristic
357C23×C4C22×D5311yescharacteristic
358C4×Q8—10632no—
359C2×C4⋊C4—534no—
360C2×C4⋊C4—534no—
361C2×C4⋊C4—534no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—2122no—
370C2×C4⋊C4—2122no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—534no—
374C2×C42—534no—
375C2×C42—534no—
376C2×C42—534no—
377C2×C42—1062no—
378C2×C42—1062no—
379C2×C42C4×D5314yes—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—42no—
384#95—1064no—
385#95—1064no—
386#95—1064no—
387#95—1064no—
388#95—1064no—
389#95—1064no—
390#95—1062no—
391#95—1062no—
392#95—2122no—
393#95—2122no—
394#95—2122no—
395C24C23×D5311yescharacteristic
396C2×Q8—1068no—
397C2×Q8—1068no—
398C2×Q8—10616no—
399C2×Q8—10616no—
400C2×Q8—1068no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
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—532no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—532no—
431C22×C4—532no—
432C22×C4—532no—
433C22×C4—534no—
434C22×C4—532no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4C2×C4×D5312yes—
465C22×C4C2×C4×D5312yes—
466C22×C4C2×C4×D5312yes—
467C22×C4#1171994514yes—
468C22×C4D4×D5312yes—
469C22×C4D4×D5312yes—
470C4⋊C4—1068no—
471C4⋊C4—1068no—
472C4⋊C4—1068no—
473C4⋊C4—2128no—
474C42—28no—
475C42—5316no—
476C42—1068no—
477C23C22×C4×D5311yescharacteristic
478C23C2×D4×D5311yescharacteristic
479C23C2×D4×D5311yescharacteristic
480C23#41×D5312yes—
481C23C2×#1171994512yes—
482C23#1313326211yescharacteristic
483C23#1313326311yescharacteristic
484C23#1313326412yes—
485C23#1313326412yes—
486C23#1313326411yescharacteristic
487C23#1313326511yescharacteristic
488Q8—10632no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—534no—
519C2×C4—534no—
520C2×C4—534no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—1062no—
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—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4C22⋊C4×D5314yes—
557C2×C4#1313317414yes—
558C22C2×C22⋊C4×D5311yescharacteristic
559C22#117×D5311yescharacteristic
560C22C2×#1313317411yescharacteristic
561C22#1681110011yescharacteristic
562C22#1681110011yescharacteristic
563C22#1681110311yescharacteristic
564C22#1681110411yescharacteristic
565C22#1681113212yes—
566C22#1681113212yes—
567C22#1681118911yescharacteristic
568C22#1681119512yes—
569C22#1681119512yes—
570C22#1681119611yescharacteristic
571C22#1681120211yescharacteristic
572C22#1681120311yescharacteristic
573C22#1681123012yes—
574C22#1681123112yes—
575C22#1681123212yes—
576C22#1681123512yes—
577C22#1681123712yes—
578C22#1681123812yes—
579C22#1681123912yes—
580C22#1681124412yes—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—1064no—
587C4—1064no—
588C4—1064no—
589C4—4242no—
590C4—4242no—
591C4—2122no—
592C2#2027709611yescharacteristic
593C2#2027730011yescharacteristic
594C2#2027730411yescharacteristic
595C2#2027736111yescharacteristic
596C2#2027736311yescharacteristic
597C2#2027736511yescharacteristic
598C2#2027736711yescharacteristic
599C2#2028057912yes—
600C2#2028059512yes—
601C2#2028062712yes—
602C2#2028062912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1434D53
hypocenterC53#14346
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic106C4semidirect
341. C53#14346semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1062
C2×#168104902
C2×#168104872
#202729852
#202729812
#202729672
#202729662
#202729422
#202729392
C53×#14342
#1434653