Finite Group Databasebeta

← #27350398

Subgroups — #27350398

Order 13568.

996 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811328C211yesmaximal, characteristic
2#20283751C212yesmaximal
3#20273080C212yesmaximal, complemented
4#20273077C212yesmaximal, complemented
5#20273069C212yesmaximal, complemented
6#20273057C211yesmaximal, characteristic, complemented
7#20273043C211yesmaximal, characteristic, complemented
8#20273037C211yesmaximal, characteristic, complemented
9C2×#16810349C211yesmaximal, characteristic, complemented
10#324×D53C211yesmaximal, characteristic, complemented
11C53×#1463C211yesmaximal, characteristic, complemented
12#16811328—22no—
13#16811328—22no—
14#16811328—22no—
15#16811328—22no—
16C2×#13133216C411yescharacteristic
17C2×#13133207C411yescharacteristic
18C2×#13133207C2211yescharacteristic
19C2×#13133197C411yescharacteristic
20C2×#13133196C411yescharacteristic
21C2×#13133196C411yescharacteristic
22C2×#13133196C2211yescharacteristic
23C2×#13133195C411yescharacteristic
24C2×#13133195C2211yescharacteristic
25C2×#13133182C2211yescharacteristic
26C2×C4×C22⋊D53C411yescharacteristic
27C2×C4×C22⋊D53C411yescharacteristic
28C2×C4×C22⋊D53C2211yescharacteristic
29C2×C4⋊C4×D53C2211yescharacteristic
30#16810699—22no—
31#16810692—22no—
32#16810691—22no—
33#16810686—22no—
34#16810677C2212yes—
35#16810673C2212yes—
36#16810585C2212yes—
37C2×C22⋊#88322C2212yes—
38C22⋊C8×D53C2212yes—
39#16810349C2211yescharacteristic
40#16810349C2211yescharacteristic
41#16810349C2212yes—
42#16810348C2212yes—
43#16810348C2212yes—
44#16810348C2212yes—
45#16810348C2212yes—
46C22×C4⋊C4×C53C2211yescharacteristic
47C53×#407C2212yes—
48C53×#324C2211yescharacteristic
49C53×#324C2211yescharacteristic
50C53×#324C2212yes—
51C22×Dic106C2×C411yescharacteristic
52C22×Dic106C2×C411yescharacteristic
53C22×Dic106C2×C411yescharacteristic
54C22×C22⋊D53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2311yescharacteristic
58C2×#11719912—22no—
59C22×#88322—22no—
60C2×C8×D53—22no—
61#13133216—22no—
62#13133216—22no—
63#13133207—22no—
64#13133207—22no—
65#13133207—22no—
66#13133207—22no—
67#13133197—22no—
68#13133197—22no—
69#13133196—22no—
70#13133196—22no—
71#13133196—22no—
72#13133196—22no—
73#13133196—22no—
74#13133196—22no—
75#13133195—22no—
76#13133195—22no—
77#13133195—22no—
78#13133195—22no—
79#13133182—22no—
80#13133182—22no—
81C4×C22⋊D53—21no—
82C4×C22⋊D53—21no—
83C4×C22⋊D53—21no—
84C4×C22⋊D53—21no—
85C4×C22⋊D53—21no—
86C4×C22⋊D53—21no—
87C4×C22⋊D53—21no—
88C4×C22⋊D53—21no—
89C4×C22⋊D53D412yes—
90C4×C22⋊D53D412yes—
91C4×C22⋊D53D412yes—
92C4×C22⋊D53D412yes—
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×C4⋊Dic53C2×C411yescharacteristic
95C2×C4⋊Dic53C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719919C2×C411yescharacteristic
100C2×#11719919C2×C411yescharacteristic
101C2×#11719919C2×C411yescharacteristic
102C2×#11719919C2×C411yescharacteristic
103C2×#11719919C2311yescharacteristic
104C2×#11719919C2311yescharacteristic
105C4⋊C4×D53—22no—
106C4⋊C4×D53—22no—
107C2×C22⋊Dic53C2×C411yescharacteristic
108C2×C22⋊Dic53C2×C411yescharacteristic
109C2×C22⋊Dic53C2×C411yescharacteristic
110C2×#11719921C2×C411yescharacteristic
111C2×#11719921C2×C411yescharacteristic
112C2×#11719921C2×C411yescharacteristic
113C2×C4×Dic53C2×C411yescharacteristic
114C2×C4×Dic53C2×C411yescharacteristic
115C2×C4×Dic53C2311yescharacteristic
116#13133146—22no—
117#13133146—22no—
118C22⋊#88322—22no—
119C22⋊#88322C2312yes—
120C22⋊#88322C2312yes—
121#13133129—22no—
122#13133129—22no—
123#13133127—22no—
124C8×Dic53—22no—
125C23×C4×C53C2×C411yescharacteristic
126C23×C4×C53C2×C411yescharacteristic
127C23×C4×C53C2311yescharacteristic
128C2×M16×C53—22no—
129C2×C4⋊C4×C53—22no—
130C2×C4⋊C4×C53—22no—
131C2×C4⋊C4×C53C2×C411yescharacteristic
132C2×C4⋊C4×C53C2×C411yescharacteristic
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C4⋊C4×C53C2311yescharacteristic
135C22⋊C8×C53—22no—
136C22⋊C8×C53C2312yes—
137C22⋊C8×C53C2312yes—
138C24×C53C22×C411yescharacteristic
139C23×D53C22×C411yescharacteristic
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—41no—
146C22×C4×C53—81no—
147C22×C4×C53—81no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2411yescharacteristic
156C2×C22⋊D53—24no—
157C2×C22⋊D53C22⋊C411yescharacteristic
158C2×C22⋊D53C22⋊C411yescharacteristic
159C2×C22⋊D53C22⋊C411yescharacteristic
160C2×C22⋊D53C22⋊C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C22×Dic53C22×C411yescharacteristic
164C22×Dic53C22×C411yescharacteristic
165C22×Dic53C22×C411yescharacteristic
166C22×Dic53C22×C411yescharacteristic
167C22×Dic53C22×C411yescharacteristic
168C2×C4×D53—21no—
169C2×C4×D53—21no—
170C2×C4×D53—22no—
171C2×C4×D53—21no—
172C2×C4×D53—21no—
173C2×C4×D53—41no—
174C2×C4×D53—81no—
175C2×C4×D53—81no—
176C2×C4×D53C2×D412yes—
177C2×C4×D53C2×D412yes—
178C2×Dic106—22no—
179C2×Dic106—22no—
180C2×Dic106—22no—
181C2×Dic106—22no—
182C2×Dic106—41no—
183C2×Dic106—81no—
184C2×Dic106—81no—
185M16×C53—82no—
186C2×C8×C53—22no—
187C2×C8×C53—22no—
188C4⋊C4×C53—22no—
189C4⋊C4×C53—22no—
190C4⋊C4×C53—22no—
191C4⋊C4×C53—22no—
192C22⋊Dic53—21no—
193C22⋊Dic53—21no—
194C22⋊Dic53—21no—
195C22⋊Dic53—21no—
196C22⋊Dic53C22⋊C412yes—
197C22⋊Dic53C22⋊C412yes—
198#11719921—21no—
199#11719921—21no—
200#11719921—21no—
201#11719921—21no—
202#11719921C22⋊C412yes—
203#11719921C22⋊C412yes—
204C4⋊Dic53—22no—
205C4⋊Dic53—22no—
206C4⋊Dic53—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—21no—
211#11719919—21no—
212#11719919—21no—
213#11719919—21no—
214#11719919—22no—
215#11719919—22no—
216#11719919—22no—
217#11719919C22⋊C412yes—
218#11719919C22⋊C412yes—
219C4×Dic53—21no—
220C4×Dic53—21no—
221C4×Dic53—21no—
222C4×Dic53—21no—
223C4×Dic53C2×D412yes—
224C4×Dic53C2×D412yes—
225C2×#88322—22no—
226C2×#88322—22no—
227C2×#88322—82no—
228#11719912—82no—
229C8×D53—82no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—22no—
233C23×C53—41no—
234C23×C53C2×C22⋊C411yescharacteristic
235C23×C53C2×C22⋊C411yescharacteristic
236C23×C53C23×C411yescharacteristic
237C22×D53—21no—
238C22×D53—21no—
239C22×D53—22no—
240C22×D53—41no—
241C22×D53C2×C22⋊C411yescharacteristic
242C22×D53C2×C22⋊C411yescharacteristic
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—81no—
259C2×C4×C53—81no—
260C2×C4×C53C2×C22⋊C412yes—
261C2×C4×C53#13112yes—
262C2×C4×C53C22×D412yes—
263C22⋊D53—22no—
264C22⋊D53—22no—
265C22⋊D53—22no—
266C22⋊D53#10114yes—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—22no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53C2×C22⋊C412yes—
282C2×Dic53C2×C22⋊C412yes—
283C2×Dic53C2×C22⋊C411yescharacteristic
284C2×Dic53C2×C22⋊C411yescharacteristic
285C4×D53—42no—
286C4×D53—42no—
287C4×D53—42no—
288C4×D53—42no—
289C4×D53—81no—
290C4×D53—81no—
291Dic106—44no—
292Dic106—82no—
293Dic106—82no—
294C8×C53—42no—
295C8×C53—42no—
296#88322—42no—
297#88322—42no—
298#17426—531nomaximal, Sylow
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—21no—
302C22×C53—21no—
303C22×C53—21no—
304C22×C53—21no—
305C22×C53—21no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53C2×#10112yes—
309C22×C53C22×C22⋊C411yescharacteristic
310C22×C53C2×#13111yescharacteristic
311C22×C53#56811yescharacteristic
312C2×D53—21no—
313C2×D53—21no—
314C2×D53—21no—
315C2×D53—21no—
316C2×D53—21no—
317C2×D53—42no—
318C2×D53—42no—
319C2×D53C2×#10112yes—
320C4×C53—22no—
321C4×C53—22no—
322C4×C53—22no—
323C4×C53—41no—
324C4×C53—41no—
325C4×C53—41no—
326Dic53—22no—
327Dic53—22no—
328Dic53—21no—
329Dic53—41no—
330Dic53—41no—
331Dic53—41no—
332Dic53—41no—
333Dic53C2×#10112yes—
334C2×#554—531no—
335#2974—532no—
336#1472—532no—
337#1472—532no—
338#1470—532no—
339#1467—531no—
340#1463—531no—
341#1463D5311yescharacteristic, complemented
342#1461—531no—
343C2×#324—531no—
344C2×#324—531no—
345C2×C53—21no—
346C2×C53—21no—
347C2×C53—22no—
348C2×C53—41no—
349C2×C53#288011yescharacteristic
350C2×C53#288311yescharacteristic
351C2×C53C22×#10111yescharacteristic
352D53—22no—
353D53—41no—
354#554—1062no—
355#554—1062no—
356#554—1062no—
357#554—1062no—
358C2×C4⋊Q8—531no—
359C2×#125—531no—
360C2×#125—531no—
361C2×C22⋊Q8—531no—
362C2×C22⋊Q8—531no—
363C2×C22⋊Q8—531no—
364C2×C22⋊Q8—531no—
365C2×C22⋊Q8—531no—
366C2×C22⋊Q8—531no—
367C2×C4×Q8—531no—
368C2×C4×D4—531no—
369C2×C4×D4—531no—
370C2×C4×D4—531no—
371C22×C4⋊C4—531no—
372C22×C4⋊C4C2×D5311yescharacteristic
373#436—1062no—
374#435—1062no—
375#435—1062no—
376C8×D4—1062no—
377#432—532no—
378#431—532no—
379#407—532no—
380#407C2×D5312yes—
381C2×C22⋊C8—532no—
382C2×C22⋊C8—532no—
383#324—531no—
384#324—531no—
385#324—532no—
386#324—532no—
387#324—532no—
388#324—532no—
389#324—532no—
390#324C2×D5311yescharacteristic
391#324C2×D5311yescharacteristic
392#324C2×D5312yes—
393C53#1742611yescharacteristic, complemented, Sylow
394C22×Q8—531no—
395C22×Q8—531no—
396C22×Q8—531no—
397C22×D4—531no—
398C23×C4—531no—
399C23×C4—531no—
400C23×C4—531no—
401C23×C4C4×D5311yescharacteristic
402C23×C4C4×D5311yescharacteristic
403C23×C4C22×D5311yescharacteristic
404C2×M16—22no—
405C2×M16—1062no—
406C22×C8—1062no—
407C22×C8—1062no—
408C4⋊Q8—1062no—
409C4⋊Q8—1062no—
410#125—1062no—
411#125—1062no—
412#125—1062no—
413#125—1062no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—1062no—
419C22⋊Q8—1062no—
420C22⋊Q8—1062no—
421C22⋊Q8—1062no—
422C22⋊Q8—1062no—
423C22⋊Q8—1062no—
424C22⋊Q8—1062no—
425C22⋊Q8—1062no—
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×D4—532no—
429C4×D4—532no—
430C4×D4—532no—
431C4×D4—532no—
432C4×D4—1061no—
433C4×D4—1061no—
434C4×D4—1061no—
435C4×D4—1061no—
436C4×D4—1061no—
437C4×D4—1061no—
438C4×D4—1061no—
439C4×D4—1061no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—531no—
445C2×C4⋊C4—531no—
446C2×C4⋊C4—531no—
447C2×C4⋊C4—531no—
448C2×C4⋊C4—531no—
449C2×C4⋊C4—531no—
450C2×C4⋊C4—531no—
451C2×C4⋊C4—531no—
452C2×C4⋊C4—531no—
453C2×C4⋊C4—531no—
454C2×C4⋊C4—1062no—
455C2×C4⋊C4—1062no—
456C2×C4⋊C4C4×D5311yescharacteristic
457C2×C4⋊C4C4×D5311yescharacteristic
458C2×C4⋊C4C22×D5311yescharacteristic
459C2×C4⋊C4C22×D5311yescharacteristic
460C2×C22⋊C4—531no—
461C2×C22⋊C4—531no—
462C2×C22⋊C4—531no—
463C2×C22⋊C4—531no—
464C2×C22⋊C4—531no—
465C2×C22⋊C4—531no—
466C2×C42—531no—
467C2×C42—531no—
468C2×C42—531no—
469C4⋊C8—1062no—
470C4⋊C8—1062no—
471C22⋊C8—22no—
472C22⋊C8—532no—
473C22⋊C8—532no—
474C22⋊C8—1062no—
475C22⋊C8—1062no—
476C22⋊C8—1062no—
477C22⋊C8C22×D5312yes—
478C22⋊C8C22×D5312yes—
479#97—1062no—
480C4×C8—1062no—
481C24—531no—
482C24C2×C4×D5311yescharacteristic
483C2×Q8—1062no—
484C2×Q8—1062no—
485C2×Q8—1062no—
486C2×Q8—1062no—
487C2×Q8—4241no—
488C2×Q8—4241no—
489C2×Q8—2121no—
490C2×D4—531no—
491C2×D4—531no—
492C2×D4—531no—
493C2×D4—531no—
494C2×D4—1064no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—41no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—532no—
507C22×C4—532no—
508C22×C4—531no—
509C22×C4—531no—
510C22×C4—531no—
511C22×C4—531no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—1061no—
516C22×C4—1062no—
517C22×C4—4241no—
518C22×C4—4241no—
519C22×C4—2121no—
520C22×C4C2×C4×D5311yescharacteristic
521C22×C4C2×C4×D5311yescharacteristic
522C22×C4C2×C4×D5311yescharacteristic
523C22×C4C2×C4×D5311yescharacteristic
524C22×C4C2×C4×D5311yescharacteristic
525C22×C4D4×D5312yes—
526C22×C4D4×D5312yes—
527C22×C4C23×D5311yescharacteristic
528M16—82no—
529M16—4242no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—1062no—
533C2×C8—1062no—
534C2×C8—4242no—
535C2×C8—4242no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—532no—
541C4⋊C4—532no—
542C4⋊C4—1062no—
543C4⋊C4—1062no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1062no—
549C4⋊C4—1062no—
550C4⋊C4—1062no—
551C4⋊C4—1062no—
552C4⋊C4—1062no—
553C4⋊C4—1062no—
554C4⋊C4—1062no—
555C22⋊C4—532no—
556C22⋊C4—532no—
557C22⋊C4—532no—
558C22⋊C4—532no—
559C22⋊C4—1061no—
560C22⋊C4—1061no—
561C22⋊C4—1061no—
562C22⋊C4—1061no—
563C22⋊C4—1061no—
564C22⋊C4—1061no—
565C22⋊C4—1061no—
566C22⋊C4—1061no—
567C42—532no—
568C42—532no—
569C42—1061no—
570C42—1061no—
571C42—1061no—
572C42—1061no—
573C23—22no—
574C23—21no—
575C23—21no—
576C23—41no—
577C23—531no—
578C23—531no—
579C23—1062no—
580C23—1061no—
581C23—1061no—
582C23—2121no—
583C23C22⋊C4×D5311yescharacteristic
584C23C22⋊C4×D5311yescharacteristic
585C23C22×C4×D5311yescharacteristic
586Q8—4242no—
587Q8—4242no—
588Q8—2124no—
589D4—534no—
590D4—1062no—
591D4—1062no—
592D4—1062no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—531no—
611C2×C4—531no—
612C2×C4—532no—
613C2×C4—532no—
614C2×C4—1062no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—4241no—
626C2×C4—4241no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4C22⋊C4×D5312yes—
635C2×C4#1313322512yes—
636C2×C4C2×D4×D5312yes—
637C8—42no—
638C8—42no—
639C8—2122no—
640C8—2122no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—42no—
649C22—42no—
650C22—532no—
651C22—1061no—
652C22—1061no—
653C22—1061no—
654C22—1061no—
655C22—1061no—
656C22—2122no—
657C22—2122no—
658C22#101×D5312yes—
659C22C2×C22⋊C4×D5311yescharacteristic
660C22C2×#1313322511yescharacteristic
661C22#1681138911yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—532no—
669C4—1062no—
670C4—1062no—
671C4—1061no—
672C4—2121no—
673C4—2121no—
674C4—2121no—
675C4—2121no—
676C2—21no—
677C2—21no—
678C2—22no—
679C2—41no—
680C2—1062no—
681C2—2121no—
682C2#2028316511yescharacteristic
683C2#2028317311yescharacteristic
684C2C2×#101×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1463C2
socleC22×C53C22×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×C4×C53C24
hypercenter#1463D53
hypocenterC53#17426
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20273080C2semidirect
4. #20273077C2semidirect
5. #20273069C2semidirect
6. #20273057C2semidirect
7. #20273043C2semidirect
8. #20273037C2semidirect
9. C2×#16810349C2semidirect
10. #324×D53C2semidirect
11. C53×#1463C2semidirect
341. #1463D53semidirect
393. C53#17426semidirect

Maximal subgroups

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

subgroupindex
C2×#168113282
#202837512
#202730802
#202730772
#202730692
#202730572
#202730432
#202730372
C2×#168103492
#324×D532
C53×#14632
#1742653