Finite Group Databasebeta

← #27349995

Subgroups — #27349995

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811095C211yesmaximal, characteristic
2#20283744C211yesmaximal, characteristic, complemented
3#20273055C212yesmaximal, complemented
4#20273051C212yesmaximal, complemented
5#20273047C212yesmaximal, complemented
6#20273041C212yesmaximal, complemented
7#20273031C212yesmaximal, complemented
8C2×#16810346C212yesmaximal, complemented
9C53×#2973C211yesmaximal, characteristic, complemented
10C22×D4×D53C411yescharacteristic
11#16811095—24no—
12#16811095—24no—
13C2×C4×D212C2211yescharacteristic
14C2×D4×Dic53C411yescharacteristic
15C2×#13133179C411yescharacteristic
16C2×C4×C22⋊D53C2211yescharacteristic
17C2×C4×C22⋊D53C2211yescharacteristic
18C2×#13133177C411yescharacteristic
19C2×#13133177C411yescharacteristic
20C2×#13133175C411yescharacteristic
21C2×#13133174C2211yescharacteristic
22C22×#11719921C2211yescharacteristic
23C22×#11719921C2211yescharacteristic
24C2×C22⋊C4×D53C411yescharacteristic
25C2×C22⋊C4×D53C411yescharacteristic
26#16810695—24no—
27#16810680C2212yescomplemented
28#16810587C2212yescomplemented
29C2×C22⋊#88322C2212yescomplemented
30#16810348C2214yescomplemented
31#16810346C2214yes—
32#16810346C2214yescomplemented
33#16810346C2214yescomplemented
34C2×C4×D4×C53C2211yescharacteristic
35C53×#435—24no—
36C53×#432C2212yescomplemented
37C53×#407C2212yescomplemented
38C2×C22⋊C8×C53C2212yescomplemented
39C24×D53C2×C411yescharacteristic
40C24×D53C2×C411yescharacteristic
41C22×D212C2×C411yescharacteristic
42C22×C22⋊D53C2×C411yescharacteristic
43C22×C22⋊D53C2×C411yescharacteristic
44C2×D4×D53—24no—
45C2×D4×D53—24no—
46C2×D4×D53—24no—
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C2×#11719917—22no—
55C22×#88322—22no—
56C4×D212—22no—
57C4×D212—22no—
58D4×Dic53—24no—
59#13133179—24no—
60C4×C22⋊D53—22no—
61C4×C22⋊D53—22no—
62C4×C22⋊D53—22no—
63C4×C22⋊D53—22no—
64#13133177—24no—
65#13133177—24no—
66#13133175—24no—
67#13133174—22no—
68#13133174—22no—
69C2×C4⋊Dic53C2×C411yescharacteristic
70C2×#11719919C2×C411yescharacteristic
71C2×#11719919C2311yescharacteristic
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2×C411yescharacteristic
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C2×#11719921C2×C411yescharacteristic
79C2×#11719921C2×C411yescharacteristic
80C2×#11719921C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×#11719921C2311yescharacteristic
84C2×#11719921C2311yescharacteristic
85C2×#11719921C2311yescharacteristic
86C22⋊C4×D53—24no—
87C22⋊C4×D53—24no—
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2×C411yescharacteristic
91C4⋊#88322—22no—
92C22⋊#88322—22no—
93C22⋊#88322—22no—
94C22⋊#88322C2314yes—
95#13133128—22no—
96C22×D4×C53C2×C411yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C2×M16×C53—22no—
100C22×C8×C53—22no—
101C4×D4×C53D414yes—
102C4×D4×C53D414yes—
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C22⋊C4×C53C2×C411yescharacteristic
105C2×C22⋊C4×C53C2×C411yescharacteristic
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—22no—
108C22⋊C8×C53—22no—
109C22⋊C8×C53—22no—
110C22⋊C8×C53C2314yes—
111#97×C53—22no—
112C24×C53C22×C411yescharacteristic
113C24×C53C22×C411yescharacteristic
114C23×D53—22no—
115C23×D53—22no—
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—22no—
119C23×D53—81no—
120C23×D53—81no—
121C23×D53—81no—
122C23×D53—81no—
123C23×D53M1614yescomplemented
124C23×D53C22×C411yescharacteristic
125C23×D53C22×C411yescharacteristic
126C23×D53C22×C411yescharacteristic
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53C22⋊C414yes—
130C22×C4×C53—81no—
131C22×C4×C53—81no—
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2411yescharacteristic
141C2×C22⋊D53—22no—
142C2×C22⋊D53—22no—
143C2×C22⋊D53—22no—
144C2×C22⋊D53—22no—
145C2×C22⋊D53—22no—
146C2×C22⋊D53M1614yescomplemented
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—81no—
150C22×Dic53—81no—
151C22×Dic53C22×C411yescharacteristic
152C22×Dic53C22×C411yescharacteristic
153C22×Dic53C22×C411yescharacteristic
154C22×Dic53C22×C411yescharacteristic
155C22×Dic53C22×C411yescharacteristic
156D4×D53—232no—
157C2×D212—22no—
158C2×D212—22no—
159C2×D212—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×C4×D53—81no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C2×C4×D53—81no—
171M16×C53—82no—
172C2×C8×C53—24no—
173C2×C8×C53—82no—
174C4⋊C4×C53C22⋊C414yes—
175C22⋊C4×C53C22⋊C414yes—
176C22⋊C4×C53C22⋊C414yes—
177C42×C53C2×D412yes—
178C42×C53C2×D412yes—
179C22⋊Dic53—22no—
180C22⋊Dic53—22no—
181#11719921—22no—
182#11719921—22no—
183#11719921—22no—
184#11719921—22no—
185#11719921—22no—
186#11719921—21no—
187#11719921—21no—
188#11719921—21no—
189#11719921—21no—
190#11719921—21no—
191#11719921—21no—
192C4⋊Dic53—22no—
193#11719919—22no—
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—22no—
197C4×Dic53—22no—
198C4×Dic53—22no—
199#11719917—82no—
200C2×#88322—24no—
201C2×#88322—82no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53C2×C22⋊C412yes—
209C23×C53C2×C22⋊C412yes—
210C23×C53C2×M1612yes—
211C23×C53C2×M1612yes—
212C23×C53C23×C411yescharacteristic
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—28no—
220C22×D53—28no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—82no—
228C22×D53—82no—
229C22×D53—82no—
230C22×D53—82no—
231C22×D53—84no—
232C22×D53—84no—
233C22×D53—84no—
234C22×D53—84no—
235C22×D53C2×M1612yes—
236C22×D53C2×M1612yes—
237C22×D53C2×M1612yes—
238D4×C53—28no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53C2×C22⋊C412yes—
250C2×C4×C53C2×C22⋊C412yes—
251C2×C4×C53C2×C22⋊C412yes—
252C2×C4×C53C2×C22⋊C412yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C22⋊D53—28no—
256C22⋊D53—28no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—82no—
268C2×Dic53—82no—
269C2×Dic53C2×M1612yes—
270D212—28no—
271C4×D53—28no—
272C4×D53—82no—
273C4×D53—82no—
274C4×D53—82no—
275C4×D53—82no—
276C8×C53—44no—
277#88322—44no—
278#17367—531nomaximal, Sylow
279C22×C53—21no—
280C22×C53—22no—
281C22×C53—21no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53#40714yes—
291C22×C53C22×C22⋊C411yescharacteristic
292C22×C53C22×M1611yescharacteristic
293C22×C53C22×M1611yescharacteristic
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—41no—
301C2×D53—41no—
302C2×D53—41no—
303C2×D53—41no—
304C2×D53—41no—
305C2×D53—41no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—88no—
310C2×D53—88no—
311C4×C53—22no—
312C4×C53—21no—
313C4×C53—21no—
314C4×C53—22no—
315C4×C53—22no—
316C4×C53—22no—
317Dic53—22no—
318Dic53—41no—
319Dic53—41no—
320Dic53—41no—
321Dic53—41no—
322C2×#518—531no—
323#2973—531no—
324#2973D5311yescharacteristic, complemented
325#1466—532no—
326#1465—532no—
327#1464—532no—
328#1462—532no—
329#1459—532no—
330C2×#323—532no—
331C2×C53—21no—
332C2×C53—21no—
333C2×C53—24no—
334C2×C53—42no—
335C2×C53C2×#40711yescharacteristic
336C2×C53#288311yescharacteristic
337C2×C53#288811yescharacteristic
338D53—42no—
339D53—42no—
340D53—42no—
341C23×D4—531no—
342#518—1064no—
343#518—1064no—
344C2×C4×D4—531no—
345C2×C4×D4—531no—
346C2×C4×D4—531no—
347C2×C4×D4—531no—
348C2×C4×D4—531no—
349C2×C4×D4—531no—
350C2×C4×D4—531no—
351C2×C4×D4C2×D5311yescharacteristic
352C2×#117—531no—
353C2×#117—531no—
354C22×C22⋊C4—531no—
355C22×C22⋊C4—531no—
356C22×C22⋊C4—531no—
357C22×C22⋊C4—531no—
358#435—24no—
359#435—1064no—
360#432—532no—
361#432C2×D5312yescomplemented
362#407—532no—
363#407C2×D5312yescomplemented
364C2×C22⋊C8—532no—
365C2×C22⋊C8C2×D5312yescomplemented
366#324—534no—
367#323—534no—
368#323—534no—
369#323—534no—
370C53#1736711yescharacteristic, complemented, Sylow
371C25—531no—
372C25—531no—
373C22×D4—531no—
374C22×D4—531no—
375C22×D4—531no—
376C22×D4—1064no—
377C22×D4—1064no—
378C22×D4—1064no—
379C22×D4C4×D5311yescharacteristic
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C2×M16—22no—
390C2×M16—1062no—
391C22×C8—22no—
392C22×C8—1062no—
393C4×D4—1064no—
394C4×D4—1062no—
395C4×D4—1062no—
396C4×D4—1062no—
397C4×D4—1062no—
398C4×D4—1064no—
399C4×D4—1064no—
400C4×D4—1064no—
401C4×D4—1062no—
402C4×D4—1062no—
403C4×D4D21214yes—
404C4×D4C22⋊D5314yes—
405#117—1064no—
406#117—1062no—
407#117—1062no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4C4×D5311yescharacteristic
412C2×C22⋊C4—531no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—531no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—531no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—1062no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4C4×D5311yescharacteristic
429C2×C22⋊C4C4×D5311yescharacteristic
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—531no—
433C2×C42C22×D5311yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—1062no—
436C22⋊C8—22no—
437C22⋊C8—22no—
438C22⋊C8—534no—
439C22⋊C8—1062no—
440C22⋊C8—1062no—
441C22⋊C8C22×D5314yes—
442#97—22no—
443#97—1062no—
444C24—534no—
445C24—531no—
446C24—531no—
447C24—531no—
448C24—1062no—
449C24—1062no—
450C24—1062no—
451C24—1062no—
452C24—1062no—
453C24—4241no—
454C24—4241no—
455C24—4241no—
456C24—4241no—
457C24C2×C4×D5311yescharacteristic
458C24C2×C4×D5311yescharacteristic
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—534no—
462C2×D4—1062no—
463C2×D4—1062no—
464C2×D4—1062no—
465C2×D4—10632no—
466C2×D4—1062no—
467C2×D4—1062no—
468C2×D4—1062no—
469C2×D4—1062no—
470C2×D4—1062no—
471C2×D4#1171992114yes—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—1062no—
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—4241no—
489C22×C4—4241no—
490C22×C4—4241no—
491C22×C4—4241no—
492C22×C4—4241no—
493C22×C4—4241no—
494C22×C4C2×C4×D5311yescharacteristic
495C22×C4C2×C4×D5311yescharacteristic
496C22×C4C2×C4×D5311yescharacteristic
497C22×C4C2×C4×D5311yescharacteristic
498C22×C4C2×D21212yes—
499C22×C4C2×D21212yes—
500C22×C4C2×C22⋊D5312yes—
501C22×C4C2×C22⋊D5312yes—
502C22×C4C23×D5311yescharacteristic
503M16—82no—
504M16—4242no—
505C2×C8—24no—
506C2×C8—82no—
507C2×C8—1064no—
508C2×C8—4242no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4#1171992114yes—
514C22⋊C4—1062no—
515C22⋊C4—1062no—
516C22⋊C4—1062no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1061no—
522C22⋊C4—1061no—
523C22⋊C4—1061no—
524C22⋊C4—1061no—
525C22⋊C4—1061no—
526C22⋊C4—1061no—
527C22⋊C4#1171992114yes—
528C22⋊C4#1171992114yes—
529C42—1062no—
530C42—1062no—
531C42—1062no—
532C42C2×D21212yes—
533C42C2×C22⋊D5312yes—
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—532no—
541C23—532no—
542C23—532no—
543C23—1068no—
544C23—1068no—
545C23—1061no—
546C23—1061no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23—4242no—
552C23—4242no—
553C23—4242no—
554C23—4242no—
555C23—4244no—
556C23—4244no—
557C23—4244no—
558C23—4244no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23—2121no—
564C23—2121no—
565C23C2×#1171992112yes—
566C23C2×#1171992112yes—
567C23M16×D5312yes—
568C23M16×D5312yes—
569C23C22×C4×D5311yescharacteristic
570D4—28no—
571D4—1068no—
572D4—1068no—
573D4—1068no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—532no—
585C2×C4—1068no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—4242no—
593C2×C4—4242no—
594C2×C4—4242no—
595C2×C4—4242no—
596C2×C4—4242no—
597C2×C4—4242no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4C2×#1171992112yes—
603C2×C4C2×#1171992112yes—
604C2×C4C2×#1171992112yes—
605C2×C4C2×#1171992112yes—
606C2×C4C22×C22⋊D5311yescharacteristic
607C2×C4C22×D21211yescharacteristic
608C8—44no—
609C8—2124no—
610C22—21no—
611C22—22no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—1062no—
622C22—1062no—
623C22—1062no—
624C22—4248no—
625C22—4248no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22—2122no—
632C22—2121no—
633C22—2121no—
634C22—2121no—
635C22—2121no—
636C22—2121no—
637C22—2121no—
638C22#1681058614yes—
639C22C22×#1171992111yescharacteristic
640C22C2×M16×D5311yescharacteristic
641C22C2×M16×D5311yescharacteristic
642C4—22no—
643C4—21no—
644C4—21no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—1062no—
649C4—2121no—
650C4—2121no—
651C4—2121no—
652C4—2121no—
653C2—21no—
654C2—21no—
655C2—24no—
656C2—42no—
657C2—2122no—
658C2—2122no—
659C2—2122no—
660C2C2×#1681058611yescharacteristic
661C2#2028317511yescharacteristic
662C2#2028319311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2973C2
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C4
cosocleC22×C4×C53C24
hypercenter#2973D53
hypocenterC53#17367
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20283744C2semidirect
3. #20273055C2semidirect
4. #20273051C2semidirect
5. #20273047C2semidirect
6. #20273041C2semidirect
7. #20273031C2semidirect
8. C2×#16810346C2semidirect
9. C53×#2973C2semidirect
27. #16810680C22semidirect
28. #16810587C22semidirect
29. C2×C22⋊#88322C22semidirect
30. #16810348C22semidirect
32. #16810346C22semidirect
33. #16810346C22semidirect
36. C53×#432C22semidirect
37. C53×#407C22semidirect
38. C2×C22⋊C8×C53C22semidirect
123. C23×D53M16semidirect
146. C2×C22⋊D53M16semidirect
324. #2973D53semidirect
361. #432C2×D53semidirect
363. #407C2×D53semidirect
365. C2×C22⋊C8C2×D53semidirect
370. C53#17367semidirect

Maximal subgroups

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

subgroupindex
C2×#168110952
#202837442
#202730552
#202730512
#202730472
#202730412
#202730312
C2×#168103462
C53×#29732
#1736753