Finite Group Databasebeta

← #27544417

Subgroups — #27544417

Order 13568.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133197C211yesmaximal, characteristic
2C2×#16811100C211yesmaximal, characteristic, complemented
3C2×#117×D53C211yesmaximal, characteristic
4#20277741C214yesmaximal, complemented
5#20277719C214yesmaximal, complemented
6#20277586C214yesmaximal, complemented
7#20277561C214yesmaximal
8#20277395C214yesmaximal, complemented
9#20277299C211yesmaximal, characteristic, complemented
10#20277245C214yesmaximal, complemented
11#20277094C211yesmaximal, characteristic
12#20277072C211yesmaximal, characteristic, complemented
13C53×#2316C211yesmaximal, characteristic, complemented
14C23×Dic106C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C2×#13133197C4116yescomplemented
17C2×#13133197C2212yes—
18C2×#13133197C2212yes—
19C2×#13133197C2212yes—
20C2×#13133197C2212yes—
21#16811100—28no—
22C2×C4×Dic106C2214yescomplemented
23C2×#13133182C2214yescomplemented
24C2×#13133176C2211yescharacteristic
25C2×#13133176C2211yescharacteristic
26C2×#13133175C2212yes—
27C2×#13133174C2212yes—
28C2×#13133173C2214yes—
29C2×#13133173C2212yes—
30#117×D53—28no—
31C22×C4⋊Dic53C2212yes—
32C2×C4⋊C4×D53C2212yes—
33C22×#11719921C2212yes—
34C2×C22⋊C4×D53C2212yes—
35C2×C42×D53C2212yes—
36#16810529C2214yescomplemented
37#16810528C2214yescomplemented
38#16810519C2218yescomplemented
39#16810513C2218yes—
40#16810512C2218yes—
41#16810506C2218yescomplemented
42#16810506C2218yescomplemented
43#16810494C2218yes—
44#16810484C2214yes—
45#16810484C2212yes—
46#16810480C2212yes—
47#16810480C2212yes—
48#16810478C2212yes—
49C4⋊C4×Dic53C2214yescomplemented
50C4⋊C4×Dic53C2214yescomplemented
51C4⋊C4×Dic53C2218yescomplemented
52C4×#11719921C2212yes—
53C4×#11719921C2212yes—
54C4×C22⋊Dic53C2214yes—
55#16810470C2218yes—
56C2×#117×C53C2211yescharacteristic
57C2×#117×C53C2211yescharacteristic
58C22×C4⋊C4×C53C2211yescharacteristic
59C53×#389C2214yescomplemented
60C53×#389C2214yescomplemented
61C53×#380C2214yes—
62C24×D53C2311yescharacteristic
63C22×Dic106—24no—
64C22×Dic106—24no—
65C22×Dic106C2×C418yescomplemented
66C22×Dic106C2312yes—
67C22×Dic106C2312yes—
68C23×Dic53C2312yes—
69C23×Dic53C2311yescharacteristic
70C22×C4×D53—24no—
71C22×C4×D53—24no—
72C22×C4×D53C2×C418yes—
73C22×C4×D53C2314yes—
74C22×C4×D53C2311yescharacteristic
75C22×C4×D53C2311yescharacteristic
76C22×C4×D53C2311yescharacteristic
77C22×C4×D53C2311yescharacteristic
78#13133197C2×C4116yes—
79#13133197C2×C4116yes—
80#13133197C2×C4116yescomplemented
81#13133197C2×C4116yescomplemented
82C4×Dic106—216no—
83#13133182—216no—
84#13133176—24no—
85#13133176D418yes—
86#13133175—28no—
87#13133174—28no—
88#13133173—28no—
89#13133173—216no—
90C2×C4⋊Dic53C2×C4116yescomplemented
91C2×C4⋊Dic53C2312yes—
92C2×C4⋊Dic53C2314yes—
93C2×C4⋊Dic53C2314yes—
94C2×C4⋊Dic53C2312yes—
95C2×#11719919C2314yes—
96C2×#11719919C2318yes—
97C4⋊C4×D53—28no—
98C2×C22⋊Dic53C2312yes—
99C2×C22⋊Dic53C2312yes—
100C2×#11719921C2×C4116yes—
101C2×#11719921C2314yes—
102C2×#11719921C2314yes—
103C2×#11719921C2314yes—
104C2×#11719921C2314yes—
105C2×#11719921C2314yes—
106C22⋊C4×D53—28no—
107C2×C4×Dic53C2314yes—
108C2×C4×Dic53C2318yes—
109C2×C4×Dic53C2312yes—
110C2×C4×Dic53C2318yes—
111C2×C4×Dic53C2312yes—
112C42×D53—28no—
113#13133124C2318yes—
114#13133124C2314yes—
115#13133124C2314yes—
116#13133124C2318yes—
117#13133124C2318yes—
118C23×C4×C53C2312yes—
119C23×C4×C53C2311yescharacteristic
120#117×C53—24no—
121#117×C53—24no—
122C2×C4⋊C4×C53C2×C418yescomplemented
123C2×C4⋊C4×C53C2312yes—
124C2×C4⋊C4×C53C2312yes—
125C2×C4⋊C4×C53C2318yes—
126C2×C4⋊C4×C53C2312yes—
127C2×C4⋊C4×C53C2312yes—
128C2×C22⋊C4×C53C2312yes—
129C2×C22⋊C4×C53C2312yes—
130C2×C42×C53C2312yes—
131C2×C42×C53C2312yes—
132#95×C53C2318yes—
133C24×C53C2411yescharacteristic
134C23×D53—22no—
135C23×D53—22no—
136C23×D53—24no—
137C23×D53—42no—
138C23×D53C22×C414yes—
139C23×D53C2411yescharacteristic
140C23×D53C2411yescharacteristic
141C22×C4×C53—44no—
142C22×C4×C53C22×C418yes—
143C22×C4×C53C22×C414yes—
144C22×C4×C53C2×D414yes—
145C22×C4×C53#4114yes—
146C22×C4×C53C2414yes—
147C22×C4×C53C2414yes—
148C22×C4×C53C2414yes—
149C22×C4×C53C2411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—22no—
152C22×Dic53—24no—
153C22×Dic53—22no—
154C22×Dic53—24no—
155C22×Dic53C22×C418yes—
156C22×Dic53C22×C414yes—
157C22×Dic53C2414yes—
158C22×Dic53C2411yescharacteristic
159C22×Dic53C2414yes—
160C22×Dic53C2411yescharacteristic
161C22×Dic53C2414yes—
162C2×C4×D53—232no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×C4×D53—22no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53—22no—
171C2×C4×D53—28no—
172C2×C4×D53—28no—
173C2×C4×D53—44no—
174C2×C4×D53C22×C414yes—
175C2×C4×D53C22×C414yes—
176C2×C4×D53C22×C414yes—
177C2×C4×D53C22×C414yes—
178C2×Dic106—232no—
179C2×Dic106—24no—
180C2×Dic106—24no—
181C2×Dic106—24no—
182C2×Dic106—24no—
183C2×Dic106C22×C418yes—
184C2×Dic106C22×C418yes—
185C4⋊C4×C53—24no—
186C4⋊C4×C53—24no—
187C4⋊C4×C53—48no—
188C4⋊C4×C53C22×C418yes—
189C4⋊C4×C53C22×C418yes—
190C22⋊C4×C53—24no—
191C22⋊C4×C53—24no—
192C42×C53—24no—
193C42×C53—24no—
194C22⋊Dic53—24no—
195C22⋊Dic53C2×D418yes—
196#11719921—28no—
197#11719921—28no—
198#11719921—44no—
199#11719921C22×C4116yes—
200#11719921C22×C4116yes—
201C4⋊Dic53—24no—
202C4⋊Dic53C22×C4116yes—
203C4⋊Dic53C22×C4116yes—
204C4⋊Dic53C2×D418yes—
205#11719919—28no—
206#11719919—216no—
207C4×Dic53—24no—
208C4×Dic53—28no—
209C4×Dic53—216no—
210C4×Dic53—48no—
211C4×Dic53C2×D418yes—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53C23×C412yes—
215C23×C53C22×D412yes—
216C23×C53C2×#4112yes—
217C23×C53C2511yescharacteristic
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—22no—
221C22×D53—28no—
222C22×D53—21no—
223C22×D53—21no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—28no—
227C22×D53—44no—
228C22×D53—48no—
229C22×D53—48no—
230C22×D53C23×C412yes—
231C22×D53C23×C412yes—
232C22×D532−1+414yes—
233C2×C4×C53—28no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—48no—
241C2×C4×C53C4×D4116yes—
242C2×C4×C53C23×C418yes—
243C2×C4×C53C23×C412yes—
244C2×C4×C53C23×C412yes—
245C2×C4×C53C22×D412yes—
246C2×C4×C53C22×D412yes—
247C2×C4×C53C2×#4112yes—
248C2×C4×C53C2×#4112yes—
249C2×Dic53—216no—
250C2×Dic53—28no—
251C2×Dic53—24no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—24no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53C23×C418yes—
263C2×Dic53C23×C412yes—
264C2×Dic53C23×C412yes—
265C2×Dic53C22×D418yes—
266C4×D53—28no—
267C4×D53—28no—
268C4×D53—28no—
269C4×D53—28no—
270C4×D53—416no—
271Dic106—216no—
272Dic106—216no—
273#34203—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—24no—
277C22×C53—24no—
278C22×C53C2×C4×D414yes—
279C22×C53C24×C411yescharacteristic
280C22×C53C23×D411yescharacteristic
281C22×C53C22×#4111yescharacteristic
282C22×C53C2×2−1+412yes—
283C22×C53#58512yes—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—24no—
287C2×D53—24no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53—416no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53#51914yescomplemented
295C4×C53—28no—
296C4×C53—44no—
297C4×C53—44no—
298C4×C53C2×C4×D414yes—
299C4×C53C2×C4×D414yes—
300Dic53—28no—
301Dic53—22no—
302Dic53—22no—
303Dic53—28no—
304Dic53—44no—
305C22×C22⋊Q8—531no—
306C2×#519—531no—
307C22×#117—531no—
308#2362—534no—
309#2360—534no—
310#2344—534no—
311#2342—534no—
312#2323—534no—
313#2316D5311yescharacteristic, complemented
314#2310—531no—
315C4×C22⋊Q8—534no—
316#2280—531no—
317#2274—531no—
318C2×C53—24no—
319C2×C53C22×C4×D411yescharacteristic
320C2×C53C2×#51911yescharacteristic
321C2×C53#342811yescharacteristic
322C2×C53#348912yes—
323C2×C53#349512yes—
324D53—24no—
325D53—42no—
326C23×Q8—531no—
327C24×C4—531no—
328C2×C22⋊Q8—5316no—
329C2×C22⋊Q8—532no—
330C2×C22⋊Q8—532no—
331C2×C22⋊Q8—532no—
332C2×C22⋊Q8—532no—
333#519—1068no—
334C2×C4×Q8—534no—
335C2×C4×Q8—534no—
336C2×#117—531no—
337C2×#117—531no—
338C2×#117—532no—
339C2×#117—534no—
340C2×#117—532no—
341C2×#117—532no—
342C2×#117—1068no—
343C2×#117C2×D5311yescharacteristic
344C2×#117C2×D5311yescharacteristic
345C22×C4⋊C4—532no—
346C22×C4⋊C4—532no—
347C22×C4⋊C4C2×D5311yescharacteristic
348C22×C22⋊C4—532no—
349C22×C22⋊C4—532no—
350C22×C42—532no—
351#391—534no—
352#391—534no—
353#389—538no—
354#389C2×D5314yescomplemented
355#389C2×D5314yescomplemented
356#388—538no—
357#388—538no—
358#387—538no—
359#387—538no—
360#385—538no—
361#381—534no—
362#381—532no—
363#380—532no—
364#380—532no—
365#380C2×D5314yes—
366#379—532no—
367C4×C4⋊C4—534no—
368C4×C4⋊C4—538no—
369C4×C4⋊C4—534no—
370C4×C22⋊C4—534no—
371C4×C22⋊C4—532no—
372C4×C22⋊C4—532no—
373C4×C22⋊C4—538no—
374C53#3420311yescharacteristic, complemented, Sylow
375C25—531no—
376C22×Q8—538no—
377C22×Q8—532no—
378C22×Q8—532no—
379C22×Q8—1064no—
380C22×Q8—1064no—
381C23×C4—532no—
382C23×C4—531no—
383C23×C4—538no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—534no—
389C23×C4—1064no—
390C23×C4—1064no—
391C23×C4C22×D5312yes—
392C23×C4C22×D5311yescharacteristic
393C22⋊Q8—5316no—
394C22⋊Q8—5316no—
395C22⋊Q8—5316no—
396C22⋊Q8—5316no—
397C4×Q8—10616no—
398C4×Q8—10616no—
399#117—24no—
400#117—24no—
401#117—538no—
402#117—1068no—
403#117—1068no—
404#117—10616no—
405#117—1068no—
406#117—1064no—
407C2×C4⋊C4—5316no—
408C2×C4⋊C4—532no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—534no—
411C2×C4⋊C4—534no—
412C2×C4⋊C4—538no—
413C2×C4⋊C4—532no—
414C2×C4⋊C4—1068no—
415C2×C4⋊C4C4×D5318yescomplemented
416C2×C4⋊C4C22×D5312yes—
417C2×C4⋊C4C22×D5312yes—
418C2×C4⋊C4C22×D5318yes—
419C2×C4⋊C4C22×D5312yes—
420C2×C4⋊C4C22×D5312yes—
421C2×C22⋊C4—532no—
422C2×C22⋊C4—5316no—
423C2×C22⋊C4—534no—
424C2×C22⋊C4—534no—
425C2×C22⋊C4—532no—
426C2×C22⋊C4—534no—
427C2×C22⋊C4—534no—
428C2×C22⋊C4—534no—
429C2×C22⋊C4—1068no—
430C2×C22⋊C4C22×D5312yes—
431C2×C22⋊C4C22×D5312yes—
432C2×C42—532no—
433C2×C42—534no—
434C2×C42—538no—
435C2×C42—532no—
436C2×C42—538no—
437C2×C42—1068no—
438C2×C42C22×D5312yes—
439C2×C42C22×D5312yes—
440#95—534no—
441#95—538no—
442#95—534no—
443#95—538no—
444#95—538no—
445#95C22×D5318yes—
446C24—531no—
447C24—531no—
448C24—534no—
449C24—1064no—
450C24—1062no—
451C24—1062no—
452C24—2122no—
453C24C23×D5311yescharacteristic
454C2×Q8—538no—
455C2×Q8—538no—
456C2×Q8—10632no—
457C2×Q8—1064no—
458C2×Q8—1064no—
459C2×Q8—1064no—
460C2×Q8—1064no—
461C22×C4—44no—
462C22×C4—538no—
463C22×C4—534no—
464C22×C4—531no—
465C22×C4—534no—
466C22×C4—531no—
467C22×C4—534no—
468C22×C4—534no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—1068no—
474C22×C4—1068no—
475C22×C4—1062no—
476C22×C4—1064no—
477C22×C4—1062no—
478C22×C4—1064no—
479C22×C4—10632no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—2124no—
489C22×C4C2×C4×D5318yes—
490C22×C4C2×C4×D5314yes—
491C22×C4D4×D5314yes—
492C22×C4#1171994914yes—
493C22×C4C23×D5314yes—
494C22×C4C23×D5314yes—
495C22×C4C23×D5314yes—
496C22×C4C23×D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—48no—
501C4⋊C4—5316no—
502C4⋊C4—5316no—
503C4⋊C4—538no—
504C4⋊C4—1068no—
505C4⋊C4—10616no—
506C4⋊C4—1064no—
507C4⋊C4C2×C4×D5318yes—
508C4⋊C4C2×C4×D5318yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—538no—
512C22⋊C4—5316no—
513C22⋊C4—5316no—
514C22⋊C4—1068no—
515C22⋊C4—1068no—
516C22⋊C4—1064no—
517C22⋊C4—2124no—
518C42—24no—
519C42—24no—
520C42—538no—
521C42—1068no—
522C42—10616no—
523C42—1064no—
524C42—2128no—
525C23—22no—
526C23—22no—
527C23—532no—
528C23—534no—
529C23—532no—
530C23—1068no—
531C23—1061no—
532C23—1061no—
533C23—1062no—
534C23—1068no—
535C23—1061no—
536C23—1061no—
537C23—1062no—
538C23—1062no—
539C23—2124no—
540C23—2128no—
541C23—2128no—
542C23C22×C4×D5312yes—
543C23C2×D4×D5312yes—
544C23C2×#1171994912yes—
545C23C24×D5311yescharacteristic
546Q8—10616no—
547Q8—10616no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—538no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—538no—
560C2×C4—1068no—
561C2×C4—10616no—
562C2×C4—1061no—
563C2×C4—1064no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1068no—
571C2×C4—1068no—
572C2×C4—1068no—
573C2×C4—1068no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—21216no—
578C2×C4#13133179116yes—
579C2×C4C22×C4×D5318yes—
580C2×C4C22×C4×D5312yes—
581C2×C4C22×C4×D5312yes—
582C2×C4C2×D4×D5312yes—
583C2×C4C2×D4×D5312yes—
584C2×C4C2×#1171994912yes—
585C2×C4C2×#1171994912yes—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—534no—
591C22—1064no—
592C22—1062no—
593C22—1062no—
594C22—1064no—
595C22—1062no—
596C22—1062no—
597C22—1062no—
598C22—21216no—
599C22—2122no—
600C22—2122no—
601C22C2×#1313317914yes—
602C22C23×C4×D5311yescharacteristic
603C22C22×D4×D5311yescharacteristic
604C22C22×#1171994911yescharacteristic
605C222−1+4×D5312yes—
606C22#1681145812yes—
607C4—28no—
608C4—44no—
609C4—44no—
610C4—1068no—
611C4—1062no—
612C4—1062no—
613C4—1068no—
614C4—2124no—
615C4C2×#1313317914yes—
616C4C2×#1313317914yes—
617C2—24no—
618C2—1064no—
619C2—2122no—
620C2C22×#1313317911yescharacteristic
621C2#519×D5311yescharacteristic
622C2#2028791311yescharacteristic
623C2#2028839012yes—
624C2#2028847012yes—
subgroupG/H
center, FrattiniC23C24×D53
commutatorC22×C53C24×C4
FittingC53×#2316C2
socle, cosocleC23×C53C25
prime radicalC24×D53C23
hypercenter#2316D53
hypocenterC53#34203
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#16811100C2semidirect
4. #20277741C2semidirect
5. #20277719C2semidirect
6. #20277586C2semidirect
8. #20277395C2semidirect
9. #20277299C2semidirect
10. #20277245C2semidirect
12. #20277072C2semidirect
13. C53×#2316C2semidirect
16. C2×#13133197C4semidirect
22. C2×C4×Dic106C22semidirect
23. C2×#13133182C22semidirect
36. #16810529C22semidirect
37. #16810528C22semidirect
38. #16810519C22semidirect
41. #16810506C22semidirect
42. #16810506C22semidirect
49. C4⋊C4×Dic53C22semidirect
50. C4⋊C4×Dic53C22semidirect
51. C4⋊C4×Dic53C22semidirect
59. C53×#389C22semidirect
60. C53×#389C22semidirect
65. C22×Dic106C2×C4semidirect
80. #13133197C2×C4semidirect
81. #13133197C2×C4semidirect
90. C2×C4⋊Dic53C2×C4semidirect
122. C2×C4⋊C4×C53C2×C4semidirect
294. C2×D53#519semidirect
313. #2316D53semidirect
354. #389C2×D53semidirect
355. #389C2×D53semidirect
374. C53#34203semidirect
415. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331972
C2×#168111002
C2×#117×D532
#202777412
#202777192
#202775862
#202775612
#202773952
#202772992
#202772452
#202770942
#202770722
C53×#23162
#3420353