Finite Group Databasebeta

← #27305980

Subgroups — #27305980

Order 13568.

2438 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133173C211yesmaximal, characteristic
2C23×#11719919C211yesmaximal, characteristic
3C2×#16810486C211yesmaximal, characteristic
4C2×#16810484C211yesmaximal, characteristic
5#20272966C214yesmaximal, complemented
6#20272946C214yesmaximal, complemented
7#20272937C211yesmaximal, characteristic
8#20272937C211yesmaximal, characteristic
9C2×C53×#381C211yesmaximal, characteristic
10C24×Dic53C2211yescharacteristic
11C24×Dic53C2211yescharacteristic
12C2×#13133173—24no—
13C2×#13133173—24no—
14C2×#13133173—24no—
15C22×#11719919—24no—
16C22×#11719919—24no—
17C22×#11719919C418yescomplemented
18C22×#11719919C2212yes—
19C22×#11719919C2212yes—
20C22×#11719919C2212yes—
21C22×C22⋊Dic53C2211yescharacteristic
22C22×C22⋊Dic53C2211yescharacteristic
23C22×C4×Dic53C2212yes—
24#16810486—24no—
25#16810484—24no—
26C2×#13133124C2214yes—
27C2×#13133124C2214yes—
28C2×#13133124C2212yes—
29C2×#13133124C2214yes—
30C2×#13133124C2212yes—
31C24×C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2211yescharacteristic
34C53×#381—24no—
35C2×#95×C53C2214yes—
36C23×Dic53—22no—
37C23×Dic53—22no—
38C23×Dic53—22no—
39C23×Dic53—22no—
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—28no—
43C23×Dic53C2×C414yes—
44C23×Dic53C2×C414yes—
45C23×Dic53D414yes—
46C23×Dic53D414yes—
47C23×Dic53C2312yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2312yes—
52#13133173—232no—
53C2×#11719919—24no—
54C2×#11719919—232no—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919C2×C418yes—
62C2×#11719919C2×C418yes—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—22no—
68C2×C22⋊Dic53D414yes—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—28no—
75#13133124—28no—
76#13133124—28no—
77C25×C53C2311yescharacteristic
78C23×C4×C53—22no—
79C23×C4×C53—22no—
80C23×C4×C53—22no—
81C23×C4×C53—22no—
82C23×C4×C53—22no—
83C23×C4×C53—22no—
84C23×C4×C53C2×C414yes—
85C23×C4×C53C2312yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53D414yes—
94#95×C53—28no—
95C24×C53—24no—
96C24×C53—24no—
97C24×C53C22×C412yes—
98C24×C53C2×D412yes—
99C24×C53C2×D412yes—
100C24×C53C2×D412yes—
101C24×C53C2×D412yes—
102C24×C53#4112yes—
103C24×C53#4112yes—
104C24×C53C2411yescharacteristic
105C22×C4×C53—28no—
106C22×C4×C53—28no—
107C22×C4×C53—28no—
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—24no—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—24no—
122C22×C4×C53—24no—
123C22×C4×C53—44no—
124C22×C4×C53—44no—
125C22×C4×C53—44no—
126C22×C4×C53—44no—
127C22×C4×C53C22×C414yes—
128C22×C4×C53C2×D414yes—
129C22×Dic53—216no—
130C22×Dic53—216no—
131C22×Dic53—28no—
132C22×Dic53—28no—
133C22×Dic53—28no—
134C22×Dic53—28no—
135C22×Dic53—28no—
136C22×Dic53—28no—
137C22×Dic53—28no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—24no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—216no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—24no—
153C22×Dic53—24no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—24no—
157C22×Dic53—22no—
158C22×Dic53—24no—
159C22×Dic53C22⋊C4116yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C414yes—
163C22×Dic53C2×D414yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D412yes—
168C22×Dic53#4118yes—
169C22⋊C4×C53—28no—
170C22⋊C4×C53—44no—
171C22⋊Dic53—28no—
172C22⋊Dic53—44no—
173#11719919—216no—
174#11719919—216no—
175#11719919—216no—
176C4×Dic53—216no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—28no—
184C23×C53—28no—
185C23×C53—28no—
186C23×C53—28no—
187C23×C53C2×C22⋊C414yes—
188C23×C53C4×D414yes—
189C23×C53C4×D414yes—
190C23×C53#12014yes—
191C23×C53#12114yes—
192C23×C53#12114yes—
193C23×C53#12314yes—
194C23×C53C23×C411yescharacteristic
195C23×C53C22×D411yescharacteristic
196C23×C53C22×D411yescharacteristic
197C23×C53C22×D411yescharacteristic
198C23×C53C22×D411yescharacteristic
199C23×C53C2×#4114yes—
200C23×C53C2×#4111yescharacteristic
201C23×C53C2×#4111yescharacteristic
202C23×C532−1+414yes—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—28no—
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—44no—
215C2×C4×C53—416no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—44no—
220C2×C4×C53—44no—
221C2×Dic53—232no—
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—28no—
237C2×Dic53—28no—
238C2×Dic53—232no—
239C2×Dic53—24no—
240C2×Dic53—28no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—44no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—44no—
248C2×Dic53—44no—
249C2×Dic53C2×C22⋊C414yes—
250C2×Dic53C2×C22⋊C414yes—
251C2×Dic53#11718yescomplemented
252#13870—531nomaximal, Sylow
253C22×C53—24no—
254C22×C53—28no—
255C22×C53—28no—
256C22×C53—28no—
257C22×C53—28no—
258C22×C53—28no—
259C22×C53—28no—
260C22×C53—24no—
261C22×C53#38618yes—
262C22×C53C22×C22⋊C411yescharacteristic
263C22×C53C2×#11712yes—
264C22×C53C2×C4×D411yescharacteristic
265C22×C53C2×C4×D411yescharacteristic
266C22×C53#51912yes—
267C22×C53C2×#12011yescharacteristic
268C22×C53C2×#12111yescharacteristic
269C22×C53C2×#12111yescharacteristic
270C22×C53C2×#12311yescharacteristic
271C22×C53#52512yes—
272C22×C53#52512yes—
273C22×C53#53612yes—
274C22×C53#53612yes—
275C22×C53#54714yes—
276C22×C53#54714yes—
277C22×C53#55214yes—
278C22×C53#55214yes—
279C4×C53—44no—
280C4×C53—44no—
281C4×C53—82no—
282Dic53—28no—
283Dic53—24no—
284Dic53—24no—
285Dic53—44no—
286Dic53—82no—
287C22×#117—531no—
288C23×C4⋊C4—531no—
289C2×#381—531no—
290C2×#381—531no—
291C2×#381D5311yescharacteristic
292#1441—534no—
293#1436—534no—
294C22⋊#95—531no—
295C22⋊#95—531no—
296C2×C53—28no—
297C2×C53#228011yescharacteristic
298C2×C53C2×#38611yescharacteristic
299C2×C53#231011yescharacteristic
300C2×C53#234412yes—
301C2×C53#234412yes—
302C2×C53#240612yes—
303C2×C53#241312yes—
304C2×C53#241312yes—
305C2×C53#248412yes—
306C24×C4—531no—
307C24×C4—531no—
308C24×C4C2×D5311yescharacteristic
309C2×#117—1064no—
310C2×#117—1064no—
311C2×#117—1064no—
312C22×C4⋊C4—538no—
313C22×C4⋊C4—532no—
314C22×C4⋊C4—532no—
315C22×C4⋊C4—532no—
316C22×C4⋊C4—1064no—
317C22×C4⋊C4—1064no—
318C22×C22⋊C4—531no—
319C22×C22⋊C4—531no—
320C22×C22⋊C4C2×D5311yescharacteristic
321C22×C22⋊C4C2×D5311yescharacteristic
322C22×C42—532no—
323#381—24no—
324#381—1064no—
325#381—1064no—
326C2×#95—534no—
327C2×#95—534no—
328C2×#95—532no—
329C2×#95—534no—
330C2×#95—532no—
331C2×#95C2×D5314yes—
332C53#1387011yescharacteristic, complemented, Sylow
333C25C22×D5311yescharacteristic
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—22no—
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—534no—
341C23×C4—534no—
342C23×C4—534no—
343C23×C4—531no—
344C23×C4—532no—
345C23×C4—531no—
346C23×C4—534no—
347C23×C4—532no—
348C23×C4—532no—
349C23×C4—1068no—
350C23×C4—1062no—
351C23×C4—1062no—
352C23×C4—1062no—
353C23×C4—1062no—
354C23×C4—1062no—
355C23×C4—1062no—
356C23×C4C4×D5314yes—
357C23×C4C22×D5312yes—
358C23×C4C22×D5312yes—
359C23×C4C22×D5312yes—
360#117—10632no—
361C2×C4⋊C4—538no—
362C2×C4⋊C4—538no—
363C2×C4⋊C4—10632no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—534no—
377C2×C22⋊C4—1062no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1062no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4C22⋊D5314yes—
383C2×C42—1064no—
384C2×C42—1064no—
385C2×C42—1064no—
386#95—28no—
387#95—1068no—
388#95—1068no—
389#95—1068no—
390#95—1064no—
391#95—1064no—
392C24—24no—
393C24—24no—
394C24C2×C4×D5312yes—
395C24#1171994512yes—
396C24D4×D5312yes—
397C24D4×D5312yes—
398C24D4×D5312yes—
399C24#1171994712yes—
400C24C2×C22⋊D5312yes—
401C24C23×D5311yescharacteristic
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—5316no—
425C22×C4—532no—
426C22×C4—532no—
427C22×C4—534no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—532no—
431C22×C4—532no—
432C22×C4—538no—
433C22×C4—534no—
434C22×C4—10616no—
435C22×C4—10616no—
436C22×C4—10616no—
437C22×C4—1068no—
438C22×C4—1068no—
439C22×C4—1068no—
440C22×C4—1068no—
441C22×C4—1068no—
442C22×C4—1068no—
443C22×C4—1068no—
444C22×C4—1061no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1064no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1064no—
458C22×C4—1064no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—1064no—
464C22×C4C2×C4×D5314yes—
465C22×C4C2×C22⋊D5314yes—
466C4⋊C4—10616no—
467C4⋊C4—10616no—
468C4⋊C4—10616no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—1068no—
472C22⋊C4—2124no—
473C42—10616no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23C22⋊C4×D5314yes—
485C23#1313317714yes—
486C23C4×C22⋊D5314yes—
487C23#1313318514yes—
488C23#1313318814yes—
489C23#1313319014yes—
490C23#1313319914yes—
491C23C22×C4×D5311yescharacteristic
492C23C2×D4×D5311yescharacteristic
493C23C2×D4×D5311yescharacteristic
494C23C2×D4×D5311yescharacteristic
495C23C22×C22⋊D5311yescharacteristic
496C23#41×D5314yes—
497C23C2×#1171994711yescharacteristic
498C23C2×#1171994511yescharacteristic
499C23#1313326414yes—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—416no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—534no—
519C2×C4—534no—
520C2×C4—538no—
521C2×C4—10632no—
522C2×C4—10632no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—1068no—
538C2×C4—1068no—
539C2×C4—1068no—
540C2×C4—1064no—
541C2×C4—2124no—
542C2×C4—2124no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2124no—
547C2×C4—2122no—
548C2×C4—2122no—
549C22—28no—
550C22—24no—
551C22—28no—
552C22—28no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—28no—
557C22#1681049918yes—
558C22C2×C22⋊C4×D5311yescharacteristic
559C22#117×D5312yes—
560C22C2×#1313317711yescharacteristic
561C22C2×C4×C22⋊D5311yescharacteristic
562C22#1681110012yes—
563C22C2×#1313318511yescharacteristic
564C22C2×#1313318811yescharacteristic
565C22C2×#1313319011yescharacteristic
566C22C2×#1313319911yescharacteristic
567C22#1681113112yes—
568C22#1681113112yes—
569C22#1681119512yes—
570C22#1681119512yes—
571C22#1681127114yes—
572C22#1681127214yes—
573C22#1681130614yes—
574C22#1681131014yes—
575C4—44no—
576C4—44no—
577C4—82no—
578C4—1068no—
579C4—1064no—
580C4—1064no—
581C4—4242no—
582C4—2124no—
583C2—28no—
584C2#2027709511yescharacteristic
585C2C2×#1681049911yescharacteristic
586C2#2027730011yescharacteristic
587C2#2027757912yes—
588C2#2027758512yes—
589C2#2027804712yes—
590C2#2027811812yes—
591C2#2027812412yes—
592C2#2027899912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#381C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#381D53
hypocenterC53#13870
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
5. #20272966C2semidirect
6. #20272946C2semidirect
17. C22×#11719919C4semidirect
251. C2×Dic53#117semidirect
332. C53#13870semidirect

Maximal subgroups

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

subgroupindex
C22×#131331732
C23×#117199192
C2×#168104862
C2×#168104842
#202729662
#202729462
#202729372
C2×C53×#3812
#1387053