Finite Group Databasebeta

← #27350244

Subgroups — #27350244

Order 13568.

952 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811215C211yesmaximal, characteristic
2#20283754C211yesmaximal, characteristic, complemented
3#20283742C211yesmaximal, characteristic, complemented
4#20273081C212yesmaximal, complemented
5#20273077C212yesmaximal, complemented
6#20273069C212yesmaximal, complemented
7#20273060C212yesmaximal, complemented
8#20273053C211yesmaximal, characteristic, complemented
9#20273044C211yesmaximal, characteristic, complemented
10#20273044C211yesmaximal, characteristic, complemented
11C53×#1465C211yesmaximal, characteristic, complemented
12#16811215—22no—
13#16811215—22no—
14#16811215—22no—
15#16811215—22no—
16C2×#13133211C411yescharacteristic
17C2×#13133210C2211yescharacteristic
18C2×#13133209C411yescharacteristic
19C2×#13133207C2211yescharacteristic
20C2×#13133201C411yescharacteristic
21C2×#13133200C411yescharacteristic
22C2×#13133199C2211yescharacteristic
23C2×#13133198C411yescharacteristic
24C2×#13133196C411yescharacteristic
25C2×#13133195C2211yescharacteristic
26C2×#13133194C411yescharacteristic
27C2×#13133189C411yescharacteristic
28C2×C4×C22⋊D53C2211yescharacteristic
29C22×#11719919C2211yescharacteristic
30#16810699—22no—
31#16810698—22no—
32#16810697—22no—
33#16810692—22no—
34#16810677C2212yes—
35#16810673C2212yes—
36#16810587C2211yescharacteristic
37#16810587C2211yescharacteristic
38#16810585C2212yes—
39C22⋊C8×D53C2212yes—
40#16810349C2212yes—
41#16810348C2212yes—
42#16810348C2212yes—
43#16810348C2212yes—
44#16810348C2212yes—
45#16810347C2212yes—
46C2×C4×D4×C53C2211yescharacteristic
47C53×#407C2211yescharacteristic
48C53×#407C2211yescharacteristic
49C53×#324C2212yes—
50C53×#323C2212yes—
51C22×Dic106C2×C411yescharacteristic
52C22×C22⋊D53C2×C411yescharacteristic
53C22×C22⋊D53C2×C411yescharacteristic
54C23×Dic53C2×C411yescharacteristic
55C23×Dic53C2×C411yescharacteristic
56C22×C4×D53C2311yescharacteristic
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719912—22no—
60C2×C8×D53—22no—
61#13133211—22no—
62#13133211—22no—
63#13133210—22no—
64#13133210—22no—
65#13133209—22no—
66#13133209—22no—
67#13133207—22no—
68#13133207—22no—
69#13133201—22no—
70#13133201—22no—
71#13133200—22no—
72#13133200—22no—
73#13133199—22no—
74#13133199—22no—
75#13133198—22no—
76#13133198—22no—
77#13133196—22no—
78#13133196—22no—
79#13133195—22no—
80#13133195—22no—
81#13133194—22no—
82#13133194—22no—
83#13133189—22no—
84#13133189—22no—
85C4×C22⋊D53D412yes—
86C4×C22⋊D53D412yes—
87C4×C22⋊D53D412yes—
88C4×C22⋊D53D412yes—
89C2×C4⋊Dic53C2×C411yescharacteristic
90C2×C4⋊Dic53C2×C411yescharacteristic
91C2×C4⋊Dic53C2×C411yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719919C2×C411yescharacteristic
100C2×#11719919C2×C411yescharacteristic
101C2×#11719919C2×C411yescharacteristic
102C2×#11719919C2×C411yescharacteristic
103C2×#11719919C2×C411yescharacteristic
104C2×#11719919C2311yescharacteristic
105C2×#11719919C2311yescharacteristic
106C2×C22⋊Dic53C2×C411yescharacteristic
107C2×C22⋊Dic53C2×C411yescharacteristic
108C2×C22⋊Dic53C2×C411yescharacteristic
109C2×C22⋊Dic53C2×C411yescharacteristic
110C2×#11719921C2×C411yescharacteristic
111C2×#11719921C2×C411yescharacteristic
112C2×#11719921C2×C411yescharacteristic
113C2×#11719921C2×C411yescharacteristic
114C2×C4×Dic53C2311yescharacteristic
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322—21no—
119C22⋊#88322C2312yes—
120C22⋊#88322C2312yes—
121#13133129—22no—
122#13133129—22no—
123#13133127—22no—
124C8×Dic53—22no—
125C22×D4×C53C2×C411yescharacteristic
126C23×C4×C53C2×C411yescharacteristic
127C23×C4×C53C2311yescharacteristic
128C2×M16×C53—21no—
129C2×M16×C53—21no—
130C4×D4×C53—21no—
131C4×D4×C53—21no—
132C4×D4×C53—21no—
133C4×D4×C53—21no—
134C2×C4⋊C4×C53C2311yescharacteristic
135C2×C22⋊C4×C53C2×C411yescharacteristic
136C2×C22⋊C4×C53C2311yescharacteristic
137C2×C42×C53C2×C411yescharacteristic
138C22⋊C8×C53—21no—
139C22⋊C8×C53—21no—
140C22⋊C8×C53C2312yes—
141C22⋊C8×C53C2312yes—
142C24×C53C22×C411yescharacteristic
143C24×C53C22×C411yescharacteristic
144C23×D53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—42no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—81no—
151C22×C4×C53—81no—
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C22×C411yescharacteristic
154C22×C4×C53C22×C411yescharacteristic
155C22×C4×C53C22×C411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2411yescharacteristic
161C2×C22⋊D53—22no—
162C2×C22⋊D53—42no—
163C2×C22⋊D53—81no—
164C2×C22⋊D53C22⋊C412yes—
165C2×C22⋊D53C22⋊C412yes—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—42no—
171C22×Dic53—81no—
172C22×Dic53C22×C411yescharacteristic
173C22×Dic53C22×C411yescharacteristic
174C22×Dic53C22×C411yescharacteristic
175C22×Dic53C22×C411yescharacteristic
176C22×Dic53C22×C411yescharacteristic
177C22×Dic53C22×C411yescharacteristic
178C22×Dic53C22×C411yescharacteristic
179C2×C4×D53—81no—
180C2×C4×D53C2×D412yes—
181C2×C4×D53C2×D412yes—
182C2×Dic106—22no—
183C2×Dic106—81no—
184M16×C53—81no—
185M16×C53—81no—
186C2×C8×C53—22no—
187C2×C8×C53—22no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C22⋊C4×C53—21no—
191C22⋊C4×C53—21no—
192C22⋊C4×C53—21no—
193C22⋊C4×C53—21no—
194C42×C53—21no—
195C42×C53—21no—
196C22⋊Dic53—22no—
197C22⋊Dic53—22no—
198C22⋊Dic53—22no—
199C22⋊Dic53C22⋊C412yes—
200C22⋊Dic53C22⋊C412yes—
201#11719921—22no—
202#11719921—22no—
203#11719921—22no—
204#11719921C22⋊C412yes—
205#11719921C22⋊C412yes—
206C4⋊Dic53—22no—
207C4⋊Dic53—22no—
208C4⋊Dic53—22no—
209#11719919—21no—
210#11719919—21no—
211#11719919—21no—
212#11719919—21no—
213#11719919—21no—
214#11719919—21no—
215#11719919—21no—
216#11719919—21no—
217#11719919—22no—
218#11719919—22no—
219#11719919—22no—
220#11719919—22no—
221#11719919—22no—
222#11719919C22⋊C412yes—
223#11719919C22⋊C412yes—
224C4×Dic53C2×D412yes—
225C4×Dic53C2×D412yes—
226#11719917—81no—
227#11719917—81no—
228C2×#88322—22no—
229C2×#88322—22no—
230#11719912—82no—
231C8×D53—82no—
232C23×C53—22no—
233C23×C53—21no—
234C23×C53—22no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53—41no—
238C23×C53—41no—
239C23×C53C2×C22⋊C411yescharacteristic
240C23×C53C2×C22⋊C411yescharacteristic
241C23×C53C23×C411yescharacteristic
242C22×D53—22no—
243C22×D53—41no—
244C22×D53—41no—
245C22×D53C2×C22⋊C412yes—
246D4×C53—42no—
247D4×C53—42no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—22no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—81no—
261C2×C4×C53—81no—
262C2×C4×C53C2×C22⋊C411yescharacteristic
263C2×C4×C53C2×C22⋊C411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×D411yescharacteristic
266C22⋊D53—42no—
267C22⋊D53—42no—
268C22⋊D53—82no—
269C2×Dic53—22no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—81no—
283C2×Dic53—81no—
284C2×Dic53C2×C22⋊C412yes—
285C2×Dic53C2×C22⋊C412yes—
286C2×Dic53C2×C22⋊C412yes—
287C4×D53—42no—
288C4×D53—42no—
289Dic106—82no—
290C8×C53—42no—
291C8×C53—42no—
292#88322—42no—
293#88322—42no—
294#17406—531nomaximal, Sylow
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—21no—
302C22×C53—21no—
303C22×C53—44no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—82no—
308C22×C53C22×C22⋊C411yescharacteristic
309C22×C53#56812yes—
310C2×D53—22no—
311C2×D53—22no—
312C2×D53—22no—
313C2×D53—82no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—21no—
318C4×C53—41no—
319C4×C53—41no—
320C4×C53—41no—
321Dic53—22no—
322Dic53—22no—
323Dic53—22no—
324Dic53—41no—
325Dic53—41no—
326Dic53—41no—
327Dic53—41no—
328C2×#539—531no—
329#2975—531no—
330#2973—531no—
331#1473—532no—
332#1472—532no—
333#1470—532no—
334#1468—532no—
335#1465—531no—
336#1465D5311yescharacteristic, complemented
337#1463—531no—
338#1463—531no—
339C2×C53—22no—
340C2×C53—41no—
341C2×C53—41no—
342C2×C53—81no—
343C2×C53#287911yescharacteristic
344C2×C53#288311yescharacteristic
345C2×C53#288911yescharacteristic
346D53—42no—
347#539—1062no—
348#539—1062no—
349#539—1062no—
350#539—1062no—
351C2×#126—531no—
352C2×#126—531no—
353C2×#125—531no—
354C2×#125—531no—
355C2×#123—531no—
356C2×#123—531no—
357C2×#123—531no—
358C2×#123—531no—
359C2×C22⋊Q8—531no—
360C2×C22⋊Q8—531no—
361C2×C22⋊Q8—531no—
362C2×#121—531no—
363C2×C4×D4—531no—
364C2×C4×D4C2×D5311yescharacteristic
365C22×C4⋊C4—531no—
366#436—1062no—
367#436—1062no—
368#435—1062no—
369#435—1062no—
370#432—532no—
371#431—532no—
372#407—531no—
373#407—531no—
374#407—532no—
375#407C2×D5311yescharacteristic
376#407C2×D5311yescharacteristic
377C2×C22⋊C8—532no—
378#324—532no—
379#324—532no—
380#324—532no—
381#324—532no—
382#324—532no—
383#324C2×D5312yes—
384#323—532no—
385#323C2×D5312yes—
386C53#1740611yescharacteristic, complemented, Sylow
387C22×Q8—531no—
388C22×D4—531no—
389C22×D4—531no—
390C22×D4C4×D5311yescharacteristic
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4C4×D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—1062no—
399C2×M16—1061no—
400C2×M16—1061no—
401C22×C8—1062no—
402#126—1062no—
403#126—1062no—
404#126—1062no—
405#126—1062no—
406#125—1062no—
407#125—1062no—
408#125—1062no—
409#125—1062no—
410#123—1062no—
411#123—1062no—
412#123—1062no—
413#123—1062no—
414#123—1062no—
415#123—1062no—
416#123—1062no—
417#123—1062no—
418C22⋊Q8—1062no—
419C22⋊Q8—1062no—
420C22⋊Q8—1062no—
421C22⋊Q8—1062no—
422C22⋊Q8—1062no—
423C22⋊Q8—1062no—
424#121—1062no—
425#121—1062no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—532no—
431C4×D4—532no—
432C4×D4—532no—
433C4×D4—532no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—531no—
445C2×C4⋊C4—531no—
446C2×C4⋊C4—531no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4—1061no—
450C2×C4⋊C4—1061no—
451C2×C4⋊C4C22×D5311yescharacteristic
452C2×C22⋊C4—531no—
453C2×C22⋊C4—531no—
454C2×C22⋊C4—531no—
455C2×C22⋊C4—531no—
456C2×C22⋊C4—531no—
457C2×C22⋊C4—531no—
458C2×C22⋊C4—531no—
459C2×C22⋊C4—531no—
460C2×C22⋊C4C4×D5311yescharacteristic
461C2×C22⋊C4C22×D5311yescharacteristic
462C2×C42—531no—
463C2×C42C4×D5311yescharacteristic
464C4⋊C8—1062no—
465C4⋊C8—1062no—
466C22⋊C8—21no—
467C22⋊C8—21no—
468C22⋊C8—532no—
469C22⋊C8—532no—
470C22⋊C8—1062no—
471C22⋊C8—1062no—
472C22⋊C8—1061no—
473C22⋊C8—1061no—
474C22⋊C8C22×D5312yes—
475C22⋊C8C22×D5312yes—
476#97—1062no—
477C4×C8—1062no—
478C24—531no—
479C24C2×C4×D5311yescharacteristic
480C24C2×C4×D5311yescharacteristic
481C2×Q8—1062no—
482C2×Q8—4241no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—532no—
487C2×D4—532no—
488C2×D4—1062no—
489C2×D4—4241no—
490C2×D4—2122no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—532no—
499C22×C4—532no—
500C22×C4—531no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—4241no—
509C22×C4—4241no—
510C22×C4—2122no—
511C22×C4C2×C4×D5311yescharacteristic
512C22×C4C2×C4×D5311yescharacteristic
513C22×C4C2×C4×D5311yescharacteristic
514C22×C4C2×C4×D5311yescharacteristic
515C22×C4D4×D5311yescharacteristic
516C22×C4D4×D5311yescharacteristic
517C22×C4D4×D5311yescharacteristic
518C22×C4D4×D5311yescharacteristic
519C22×C4C23×D5311yescharacteristic
520M16—81no—
521M16—81no—
522M16—4242no—
523M16—4241no—
524M16—4241no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—1062no—
528C2×C8—1062no—
529C2×C8—4242no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—532no—
533C4⋊C4—532no—
534C4⋊C4—1062no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—1062no—
538C4⋊C4—1062no—
539C4⋊C4—1062no—
540C4⋊C4—1062no—
541C4⋊C4—1062no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—532no—
555C22⋊C4—532no—
556C22⋊C4—532no—
557C22⋊C4—532no—
558C22⋊C4—1062no—
559C22⋊C4—1062no—
560C22⋊C4—1062no—
561C22⋊C4—1062no—
562C22⋊C4—1062no—
563C22⋊C4—1062no—
564C42—21no—
565C42—21no—
566C42—532no—
567C42—532no—
568C23—22no—
569C23—21no—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—532no—
576C23—1062no—
577C23—2121no—
578C23—2121no—
579C23C22⋊C4×D5311yescharacteristic
580C23C22⋊C4×D5311yescharacteristic
581C23C22×C4×D5311yescharacteristic
582Q8—4242no—
583D4—42no—
584D4—42no—
585D4—4242no—
586D4—2122no—
587D4—2122no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—532no—
603C2×C4—532no—
604C2×C4—532no—
605C2×C4—1062no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—4241no—
613C2×C4—4241no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4C22⋊C4×D5311yescharacteristic
623C2×C4C22⋊C4×D5311yescharacteristic
624C2×C4C2×D4×D5311yescharacteristic
625C2×C4C2×D4×D5311yescharacteristic
626C8—42no—
627C8—42no—
628C8—2122no—
629C8—2122no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—44no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—82no—
643C22—1062no—
644C22—1062no—
645C22—1062no—
646C22—4242no—
647C22C2×C22⋊C4×D5311yescharacteristic
648C22#1681138912yes—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—1062no—
657C4—1062no—
658C4—1062no—
659C4—2121no—
660C4—2121no—
661C4—2121no—
662C4—2121no—
663C2—22no—
664C2—41no—
665C2—41no—
666C2—81no—
667C2—2122no—
668C2#2028315811yescharacteristic
669C2#2028317311yescharacteristic
670C2#2028319611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1465C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133189C4
cosocleC22×C4×C53C24
hypercenter#1465D53
hypocenterC53#17406
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #20283754C2semidirect
3. #20283742C2semidirect
4. #20273081C2semidirect
5. #20273077C2semidirect
6. #20273069C2semidirect
7. #20273060C2semidirect
8. #20273053C2semidirect
9. #20273044C2semidirect
10. #20273044C2semidirect
11. C53×#1465C2semidirect
336. #1465D53semidirect
386. C53#17406semidirect

Maximal subgroups

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

subgroupindex
C2×#168112152
#202837542
#202837422
#202730812
#202730772
#202730692
#202730602
#202730532
#202730442
C53×#14652
#1740653