Finite Group Databasebeta

← #27388979

Subgroups — #27388979

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810678C211yesmaximal, characteristic
2#20277119C211yesmaximal, characteristic
3C4×C4⋊C4×D53C211yesmaximal, characteristic
4#20275477C211yesmaximal, characteristic, complemented
5#20275475C211yesmaximal, characteristic, complemented
6#20275472C211yesmaximal, characteristic, complemented
7#20275470C211yesmaximal, characteristic, complemented
8#20274909C211yesmaximal, characteristic
9#20274906C211yesmaximal, characteristic
10#20274904C211yesmaximal, characteristic
11#20274901C211yesmaximal, characteristic
12#20274613C211yesmaximal, characteristic, complemented
13#20274613C211yesmaximal, characteristic, complemented
14#20274612C211yesmaximal, characteristic, complemented
15C53×#1776C211yesmaximal, characteristic, complemented
16C2×#13133174—21no—
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C42×D53—21no—
20C2×C42×D53C2211yescharacteristic
21#16810678C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133129C2211yescharacteristic
31C2×#13133129C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810492C2211yescharacteristic
35#16810487—21no—
36#16810482—21no—
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×#11719919—21no—
42C4×#11719919—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810474—21no—
46#95×D53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133174—44no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53—22no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91C42×D53—44no—
92C42×D53C2×C414yes—
93C42×D53Q814yes—
94#13133151—22no—
95#13133150—22no—
96#13133150—22no—
97#13133149—22no—
98#13133148—22no—
99#13133148—22no—
100#13133146C2×C414yescomplemented
101#13133146C2×C414yescomplemented
102#13133129C2×C414yes—
103#13133129C2×C414yes—
104#13133128C2×C414yescomplemented
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#107×C53—22no—
121#106×C53—22no—
122#97×C53C2×C414yescomplemented
123#95×C53—21no—
124#95×C53—21no—
125C23×D53C2×D411yescharacteristic
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—21no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—24no—
149C2×C4×D53—24no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53C4⋊C414yes—
155C2×C4×D53C4⋊C414yes—
156C2×C4×D53C22×C412yes—
157C2×C4×D53C2×Q812yes—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53#4112yes—
171C4⋊C4×C53#4112yes—
172C42×C53—42no—
173C42×C53—42no—
174C42×C53C22×C412yes—
175C42×C53C2×Q812yes—
176#11719921—42no—
177#11719921—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53#4112yes—
185C4⋊Dic53#4112yes—
186#11719919—24no—
187#11719919—24no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53C4⋊C414yes—
197C4×Dic53C4⋊C414yes—
198C4×Dic53C22×C412yes—
199C4×Dic53C2×Q812yes—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C412yes—
205C2×#88322C22×C412yes—
206C23×C53C22×D411yescharacteristic
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53C2×C4⋊C412yes—
212C22×D53AGL(1,ℤ/8)12yes—
213C22×D53#13712yes—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53C2×C4⋊C412yes—
231C2×C4×C53C2×C4⋊C412yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C22×Q811yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53C2×C4⋊C412yes—
255C2×Dic53C2×C4⋊C412yes—
256C2×Dic53C2×C4⋊C412yes—
257C2×Dic53AGL(1,ℤ/8)12yes—
258C2×Dic53#13712yes—
259C4×D53—28no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—28no—
265C4×D53—48no—
266C4×D53—48no—
267C8×C53—22no—
268C8×C53—22no—
269#88322—22no—
270#88322—22no—
271#21365—531nomaximal, Sylow
272C22×C53C22×C4⋊C411yescharacteristic
273C22×C53#52511yescharacteristic
274C22×C53#52511yescharacteristic
275C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C53C2×#13711yescharacteristic
278C22×C53C2×#13711yescharacteristic
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53#42814yes—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53C4×#4111yescharacteristic
293C4×C53C4×#4111yescharacteristic
294C4×C53#53311yescharacteristic
295C4×C53#53311yescharacteristic
296Dic53—22no—
297Dic53—22no—
298Dic53—42no—
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302Dic53#42814yes—
303C2×#432—531no—
304#2285—531no—
305C2×C4×C4⋊C4—531no—
306#1948—531no—
307#1948—531no—
308#1947—531no—
309#1947—531no—
310#1853—531no—
311#1853—531no—
312#1852—531no—
313#1852—531no—
314#1776—531no—
315#1776—531no—
316#1776—531no—
317#1776D5311yescharacteristic, complemented
318C2×C53#229611yescharacteristic
319C2×C53C2×#42811yescharacteristic
320C2×C53C2×#42811yescharacteristic
321C2×C53#310911yescharacteristic
322C2×C53#311011yescharacteristic
323C2×C53#311211yescharacteristic
324C2×C53#311311yescharacteristic
325D53—24no—
326C2×#117—1061no—
327C22×C4⋊C4—531no—
328C22×C4⋊C4—531no—
329C22×C42—531no—
330C22×C42—1061no—
331#432—538no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107C2×D5311yescharacteristic
336C2×#106—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106C2×D5311yescharacteristic
340C2×C4⋊C8—531no—
341C2×C4⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—531no—
344C2×#97—531no—
345C2×#97C2×D5311yescharacteristic
346#384—531no—
347#384—531no—
348#382—1061no—
349#380—531no—
350#380—531no—
351#380—1061no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—1061no—
357C4×C4⋊C4—1061no—
358C4×C4⋊C4—1061no—
359C4×C4⋊C4C2×D5311yescharacteristic
360C4×C4⋊C4C2×D5311yescharacteristic
361C2×#95—1061no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#340C2×D5311yescharacteristic
370C53#2136511yescharacteristic, complemented, Sylow
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—1061no—
375C23×C4—1061no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380#117—2124no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—534no—
407C2×C42—534no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—2124no—
413C2×C42C22×D5311yescharacteristic
414#107—22no—
415#107—1062no—
416#107—1062no—
417#107—1062no—
418#106—22no—
419#106—1062no—
420#106—1062no—
421#106—1062no—
422C4⋊C8—534no—
423C4⋊C8—534no—
424C22⋊C8—534no—
425C22⋊C8—534no—
426#97—534no—
427#97C4×D5314yescomplemented
428#95—21no—
429#95—21no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—531no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—534no—
445C22×C4—534no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1064no—
453C22×C4—1064no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4C23×D5311yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—532no—
474C2×C8—532no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8C2×C4×D5312yes—
480C2×C8C2×C4×D5312yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—1064no—
490C4⋊C4—1064no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4#1171994512yes—
502C4⋊C4#1171994512yes—
503C22⋊C4—2122no—
504C22⋊C4—2122no—
505C42—42no—
506C42—42no—
507C42—534no—
508C42—534no—
509C42—532no—
510C42—532no—
511C42—2122no—
512C42—2122no—
513C42—2122no—
514C42—2122no—
515C42C2×C4×D5312yes—
516C42Q8×D5312yes—
517C23—532no—
518C23—532no—
519C23—532no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23C2×D4×D5311yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—1068no—
547C2×C4—1068no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—2128no—
559C2×C4—2128no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4C4⋊C4×D5312yes—
571C2×C4C4⋊C4×D5312yes—
572C2×C4C22×C4×D5311yescharacteristic
573C2×C4C2×Q8×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4C2×#1171994511yescharacteristic
577C2×C4C2×#1171994511yescharacteristic
578C8—22no—
579C8—22no—
580C8—1062no—
581C8—1062no—
582C22—534no—
583C22—1062no—
584C22—1062no—
585C22—1062no—
586C22—1062no—
587C22—1062no—
588C22—1062no—
589C22C2×C4⋊C4×D5311yescharacteristic
590C22#1681113211yescharacteristic
591C22#1681113211yescharacteristic
592C22AGL(1,ℤ/8)×D5311yescharacteristic
593C22AGL(1,ℤ/8)×D5311yescharacteristic
594C22#137×D5311yescharacteristic
595C22#137×D5311yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—534no—
603C4—1062no—
604C4—1062no—
605C4—4241no—
606C4—4241no—
607C4—2122no—
608C4—2122no—
609C4#1681109111yescharacteristic
610C4C4×#1171994511yescharacteristic
611C4#1681117711yescharacteristic
612C4#1681117711yescharacteristic
613C2—1064no—
614C2#2027719511yescharacteristic
615C2#428×D5311yescharacteristic
616C2#428×D5311yescharacteristic
617C2#2028480411yescharacteristic
618C2#2028481211yescharacteristic
619C2#2028483511yescharacteristic
620C2#2028484311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1776C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1776D53
hypocenterC53#21365
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #20275477C2semidirect
5. #20275475C2semidirect
6. #20275472C2semidirect
7. #20275470C2semidirect
12. #20274613C2semidirect
13. #20274613C2semidirect
14. #20274612C2semidirect
15. C53×#1776C2semidirect
21. #16810678C4semidirect
100. #13133146C2×C4semidirect
101. #13133146C2×C4semidirect
104. #13133128C2×C4semidirect
122. #97×C53C2×C4semidirect
317. #1776D53semidirect
370. C53#21365semidirect
427. #97C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106782
#202771192
C4×C4⋊C4×D532
#202754772
#202754752
#202754722
#202754702
#202749092
#202749062
#202749042
#202749012
#202746132
#202746122
C53×#17762
#2136553