Finite Group Databasebeta

← #27387560

Subgroups — #27387560

Order 13568.

1082 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810664C211yesmaximal, characteristic
2C2×#16810662C211yesmaximal, characteristic
3C2×#16810645C211yesmaximal, characteristic
4#20277121C211yesmaximal, characteristic
5C4×C4⋊C4×D53C211yesmaximal, characteristic
6#20275485C211yesmaximal, characteristic, complemented
7#20275483C211yesmaximal, characteristic, complemented
8#20275481C211yesmaximal, characteristic, complemented
9#20275477C211yesmaximal, characteristic, complemented
10#20275475C211yesmaximal, characteristic, complemented
11#20275472C211yesmaximal, characteristic, complemented
12#20275470C211yesmaximal, characteristic, complemented
13#20274433C211yesmaximal, characteristic
14#20274433C211yesmaximal, characteristic
15C53×#1949C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×#11719912C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C42×D53—21no—
23C2×C42×D53C2211yescharacteristic
24#16810664—24no—
25#16810662—24no—
26C2×#13133151C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133149C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32#16810645C418yescomplemented
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133146C2211yescharacteristic
35C2×#13133146C2211yescharacteristic
36#16810492C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810489C2211yescharacteristic
39#16810482—21no—
40#16810482—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919—21no—
44C4×C4⋊Dic53C2211yescharacteristic
45#16810474—21no—
46#16810474—21no—
47#16810474—21no—
48#95×D53—21no—
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C4⋊C8×C53C2211yescharacteristic
58C53×#382C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53D411yescharacteristic
65C22×C4×D53D411yescharacteristic
66C22×C4×D53C2311yescharacteristic
67C2×#11719912—22no—
68C2×#11719912—22no—
69C2×#11719912—22no—
70C2×#11719912—22no—
71C2×#11719912C2×C414yes—
72C2×#11719912C2×C414yes—
73C22×#88322C2311yescharacteristic
74C22×#88322C2311yescharacteristic
75#13133175—22no—
76#13133175—22no—
77#13133175—22no—
78#13133175—22no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—22no—
91C4⋊C4×D53D414yes—
92C4⋊C4×D53Q814yes—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53D411yescharacteristic
101C2×C4×Dic53D411yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C42×D53—44no—
104C42×D53C2×C414yes—
105C42×D53Q814yes—
106#13133151—22no—
107#13133150—22no—
108#13133150—22no—
109#13133149—22no—
110#13133148—22no—
111#13133148—22no—
112C4⋊#88322C2×C414yescomplemented
113#13133146C2×C414yescomplemented
114#13133146C2×C414yescomplemented
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C42×C53—21no—
127C2×C42×C53C2311yescharacteristic
128#107×C53—22no—
129#106×C53—22no—
130C4⋊C8×C53C2×C414yescomplemented
131#95×C53—21no—
132#95×C53—21no—
133#95×C53—21no—
134C23×D53C2×D411yescharacteristic
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53C4⋊C414yes—
160C2×C4×D53C4⋊C414yes—
161C2×C4×D53C22×C412yes—
162C2×C4×D53C2×D412yes—
163C2×C4×D53C2×Q812yes—
164C2×C4×D53C2×Q812yes—
165C2×C4×D53#4112yes—
166C2×C4×D53#4112yes—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53C22×C412yes—
172C2×C8×C53C22×C412yes—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53C2×D412yes—
180C4⋊C4×C53C2×Q812yes—
181C42×C53—42no—
182C42×C53C22×C412yes—
183C42×C53C2×Q812yes—
184#11719921—42no—
185#11719921—42no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53C2×D412yes—
193C4⋊Dic53C2×Q812yes—
194#11719919—24no—
195#11719919—24no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199#11719919—42no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C4×Dic53C4⋊C414yes—
206C4×Dic53C4⋊C414yes—
207C4×Dic53C22×C412yes—
208C4×Dic53C2×Q812yes—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322C22×C412yes—
214C2×#88322C22×C412yes—
215#11719912—28no—
216#11719912—28no—
217C23×C53C22×D411yescharacteristic
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53C2×C4⋊C412yes—
223C22×D53#12112yes—
224C22×D53C22⋊Q812yes—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53C2×C4⋊C412yes—
242C2×C4×C53C2×C4⋊C412yes—
243C2×C4×C53C23×C411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C22×Q811yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53C2×C4⋊C412yes—
264C2×Dic53C2×C4⋊C412yes—
265C2×Dic53C2×C4⋊C412yes—
266C2×Dic53#12112yes—
267C2×Dic53C22⋊Q812yes—
268C2×Dic53AGL(1,ℤ/8)12yes—
269C2×Dic53#13712yes—
270C4×D53—28no—
271C4×D53—44no—
272C4×D53—48no—
273C4×D53—48no—
274C4×D53C4×D414yes—
275C4×D53C4×Q814yes—
276C4×D53#12514yes—
277C4×D53C4⋊Q814yes—
278C8×C53—22no—
279C8×C53—22no—
280#88322—22no—
281#88322—22no—
282#21170—531nomaximal, Sylow
283C22×C53C22×C4⋊C411yescharacteristic
284C22×C53C2×#12111yescharacteristic
285C22×C53C2×C22⋊Q811yescharacteristic
286C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C53C2×#13711yescharacteristic
288C22×C53#57511yescharacteristic
289C22×C53#57511yescharacteristic
290C2×D53—22no—
291C2×D53—22no—
292C2×D53—22no—
293C2×D53—22no—
294C2×D53—22no—
295C2×D53—22no—
296C2×D53#38914yes—
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—42no—
300C4×C53—42no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53C2×C4×D411yescharacteristic
304C4×C53C2×C4×Q811yescharacteristic
305C4×C53C2×#12511yescharacteristic
306C4×C53C2×C4⋊Q811yescharacteristic
307Dic53—22no—
308Dic53—42no—
309Dic53—42no—
310Dic53—81no—
311Dic53—81no—
312Dic53#38914yes—
313Dic53#42814yescomplemented
314C2×#428—531no—
315C2×#428—531no—
316C2×#423—531no—
317#2285—531no—
318C2×C4×C4⋊C4—531no—
319#1949—531no—
320#1949—531no—
321#1949—531no—
322#1949D5311yescharacteristic, complemented
323#1948—531no—
324#1948—531no—
325#1947—531no—
326#1947—531no—
327#1736—531no—
328#1736—531no—
329C2×C53C2×#38911yescharacteristic
330C2×C53C2×#42811yescharacteristic
331C2×C53#291111yescharacteristic
332C2×C53#303611yescharacteristic
333C2×C53#303711yescharacteristic
334C2×C53#307711yescharacteristic
335C2×C53#307811yescharacteristic
336D53—24no—
337C22×M16—531no—
338C22×M16—531no—
339C2×#117—531no—
340C2×#117—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C42—531no—
344C22×C42—1061no—
345#428—1064no—
346#428—1064no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#107C2×D5311yescharacteristic
351C2×#106—531no—
352C2×#106—531no—
353C2×#106—531no—
354C2×#106C2×D5311yescharacteristic
355#423—538no—
356C2×C4⋊C8—531no—
357C2×C4⋊C8—531no—
358C2×C4⋊C8—531no—
359C2×C4⋊C8C2×D5311yescharacteristic
360#384—531no—
361#384—531no—
362#382—531no—
363#382C2×D5311yescharacteristic
364#380—1061no—
365#380—1061no—
366C4×C4⋊C4—531no—
367C4×C4⋊C4—531no—
368C4×C4⋊C4—531no—
369C4×C4⋊C4—1061no—
370C4×C4⋊C4—1061no—
371C4×C4⋊C4—1061no—
372C4×C4⋊C4—1061no—
373C4×C4⋊C4C2×D5311yescharacteristic
374C2×#95—1061no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340C2×D5311yescharacteristic
382#340C2×D5311yescharacteristic
383C53#2117011yescharacteristic, complemented, Sylow
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—1061no—
388C23×C4—1061no—
389C2×M16—534no—
390C2×M16—534no—
391C2×M16—1062no—
392C2×M16—1062no—
393C2×M16—1062no—
394C2×M16—1062no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399#117—1062no—
400#117—1062no—
401#117—1062no—
402#117—1062no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C42—21no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—534no—
428C2×C42—534no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42—2124no—
435C2×C42C22×D5311yescharacteristic
436#107—22no—
437#107—1062no—
438#107—1062no—
439#107—1062no—
440#106—22no—
441#106—1062no—
442#106—1062no—
443#106—1062no—
444C4⋊C8—534no—
445C4⋊C8—534no—
446C4⋊C8—534no—
447C4⋊C8C4×D5314yescomplemented
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456C24—531no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—531no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1064no—
475C22×C4—1064no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—2121no—
481C22×C4—2121no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5311yescharacteristic
488C22×C4C23×D5311yescharacteristic
489M16—1068no—
490M16—1068no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—532no—
496C2×C8—532no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8C2×C4×D5312yes—
502C2×C8C2×C4×D5312yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—1064no—
512C4⋊C4—1064no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4D4×D5312yes—
524C4⋊C4Q8×D5312yes—
525C22⋊C4—2122no—
526C22⋊C4—2122no—
527C42—42no—
528C42—534no—
529C42—534no—
530C42—532no—
531C42—532no—
532C42—2122no—
533C42—2122no—
534C42—2122no—
535C42—2122no—
536C42—2122no—
537C42C2×C4×D5312yes—
538C42Q8×D5312yes—
539C23—532no—
540C23—532no—
541C23—532no—
542C23—1061no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23C2×D4×D5311yescharacteristic
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—532no—
564C2×C4—532no—
565C2×C4—532no—
566C2×C4—532no—
567C2×C4—532no—
568C2×C4—532no—
569C2×C4—534no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—534no—
573C2×C4—532no—
574C2×C4—1068no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—2124no—
580C2×C4—2128no—
581C2×C4—2128no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4C4⋊C4×D5312yes—
593C2×C4C4⋊C4×D5312yes—
594C2×C4C22×C4×D5311yescharacteristic
595C2×C4C2×D4×D5311yescharacteristic
596C2×C4C2×Q8×D5311yescharacteristic
597C2×C4C2×Q8×D5311yescharacteristic
598C2×C4#41×D5311yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C8—22no—
601C8—22no—
602C8—1062no—
603C8—1062no—
604C22—534no—
605C22—1062no—
606C22—1062no—
607C22—1062no—
608C22—1062no—
609C22—1062no—
610C22—1062no—
611C22C2×C4⋊C4×D5311yescharacteristic
612C22#121×D5311yescharacteristic
613C22C22⋊Q8×D5311yescharacteristic
614C22AGL(1,ℤ/8)×D5311yescharacteristic
615C22#137×D5311yescharacteristic
616C22#1681142011yescharacteristic
617C22#1681142011yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—81no—
623C4—81no—
624C4—534no—
625C4—534no—
626C4—1062no—
627C4—4241no—
628C4—4241no—
629C4—2122no—
630C4—2122no—
631C4C4×D4×D5311yescharacteristic
632C4C4×Q8×D5311yescharacteristic
633C4#125×D5311yescharacteristic
634C4C4⋊Q8×D5311yescharacteristic
635C2—1064no—
636C2#389×D5311yescharacteristic
637C2#428×D5311yescharacteristic
638C2#2028330311yescharacteristic
639C2#2028417711yescharacteristic
640C2#2028418511yescharacteristic
641C2#2028451011yescharacteristic
642C2#2028451811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53#21170
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #20275485C2semidirect
7. #20275483C2semidirect
8. #20275481C2semidirect
9. #20275477C2semidirect
10. #20275475C2semidirect
11. #20275472C2semidirect
12. #20275470C2semidirect
15. C53×#1949C2semidirect
32. #16810645C4semidirect
112. C4⋊#88322C2×C4semidirect
113. #13133146C2×C4semidirect
114. #13133146C2×C4semidirect
130. C4⋊C8×C53C2×C4semidirect
313. Dic53#428semidirect
322. #1949D53semidirect
383. C53#21170semidirect
447. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106642
C2×#168106622
C2×#168106452
#202771212
C4×C4⋊C4×D532
#202754852
#202754832
#202754812
#202754772
#202754752
#202754722
#202754702
#202744332
C53×#19492
#2117053