Finite Group Databasebeta

← #27387479

Subgroups — #27387479

Order 13568.

1058 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×#16810575C211yesmaximal, characteristic
4#20277196C211yesmaximal, characteristic
5#389×D53C211yesmaximal, characteristic
6#20275458C211yesmaximal, characteristic, complemented
7#20275455C211yesmaximal, characteristic, complemented
8#20275450C211yesmaximal, characteristic, complemented
9#20275449C211yesmaximal, characteristic, complemented
10#20275447C211yesmaximal, characteristic, complemented
11#20275442C211yesmaximal, characteristic, complemented
12#20275441C211yesmaximal, characteristic, complemented
13#20274433C211yesmaximal, characteristic
14#20274433C211yesmaximal, characteristic
15C53×#1943C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×#11719912C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C42×D53C2211yescharacteristic
24#16810664C418yescomplemented
25#16810662—24no—
26C2×#13133151C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133149C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32#16810575—24no—
33C2×#13133128C2211yescharacteristic
34C2×C8×Dic53C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810520—21no—
41#16810519C2211yescharacteristic
42#16810519C2211yescharacteristic
43#16810518—21no—
44#16810497—21no—
45#16810494—21no—
46C4⋊C4×Dic53—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×#97×C53C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#389C2211yescharacteristic
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×#11719912C2×C414yes—
70C2×#11719912D414yes—
71C2×#11719912Q814yes—
72C2×#11719912Q814yes—
73C22×#88322C2311yescharacteristic
74C22×#88322C2311yescharacteristic
75#13133175—22no—
76#13133175—22no—
77#13133175—22no—
78#13133175C2×C414yes—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C4⋊C4×D53—22no—
92C4⋊C4×D53—22no—
93C4⋊C4×D53—22no—
94C4⋊C4×D53—44no—
95C4⋊C4×D53C2×C414yes—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53D411yescharacteristic
101C2×C4×Dic53D411yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C42×D53—22no—
104C42×D53—22no—
105#13133151—22no—
106#13133150—22no—
107#13133150C2×C414yescomplemented
108#13133149C2×C414yescomplemented
109#13133148—22no—
110#13133148C2×C414yescomplemented
111#13133128—22no—
112C8×Dic53—22no—
113C8×Dic53—22no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C4⋊C4×C53—21no—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C42×C53C2311yescharacteristic
129#107×C53C2×C414yescomplemented
130#106×C53—22no—
131#97×C53—22no—
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—24no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—41no—
155C2×C4×D53—41no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
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×C53C22×C412yes—
170C2×C8×C53C2×D412yes—
171C2×C8×C53C2×Q812yes—
172C2×C8×C53C2×Q812yes—
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×C53—42no—
180C4⋊C4×C53—42no—
181C4⋊C4×C53C22×C412yes—
182C4⋊C4×C53C22×C412yes—
183C42×C53—21no—
184C42×C53—21no—
185#11719921—24no—
186#11719921—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53C22×C412yes—
196C4⋊Dic53C22×C412yes—
197#11719919—24no—
198#11719919—42no—
199#11719919—42no—
200#11719919—42no—
201#11719919—42no—
202#11719919—42no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—24no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—42no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322C22×C412yes—
212C2×#88322C2×D412yes—
213C2×#88322C2×Q812yes—
214C2×#88322C2×Q812yes—
215#11719912—28no—
216#11719912C4⋊C4116yescomplemented
217C23×C53C22×D411yescharacteristic
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53C4×D412yes—
223C22×D53#12112yes—
224C22×D53C4⋊Q812yes—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
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×C53—41no—
242C2×C4×C53—41no—
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—21no—
252C2×Dic53—21no—
253C2×Dic53—22no—
254C2×Dic53—22no—
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×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53C4×D412yes—
266C2×Dic53#12112yes—
267C2×Dic53C4⋊Q812yes—
268C2×Dic53AGL(1,ℤ/8)12yes—
269C2×Dic53#13712yes—
270C4×D53—44no—
271C4×D53—44no—
272C4×D53—48no—
273C4×D53—84no—
274C4×D53C2×C4⋊C414yes—
275C4×D53C4×Q814yes—
276C4×D53C22⋊Q814yes—
277C4×D53#12514yes—
278C8×C53—22no—
279C8×C53C2×C4⋊C414yescomplemented
280#88322—22no—
281#88322C2×C4⋊C414yescomplemented
282#21159—531nomaximal, Sylow
283C22×C53C2×C4×D411yescharacteristic
284C22×C53C2×#12111yescharacteristic
285C22×C53C2×C4⋊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—42no—
298C4×C53—42no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53C22×C4⋊C411yescharacteristic
304C4×C53C2×C4×Q811yescharacteristic
305C4×C53C2×C22⋊Q811yescharacteristic
306C4×C53C2×#12511yescharacteristic
307Dic53—22no—
308Dic53—22no—
309Dic53—42no—
310Dic53—42no—
311Dic53—81no—
312Dic53—81no—
313Dic53#38914yes—
314C2×#428—531no—
315C2×#428—531no—
316C2×C4×M16—531no—
317#2296—531no—
318C2×#389—531no—
319#1944—531no—
320#1944—531no—
321#1943—531no—
322#1943D5311yescharacteristic, complemented
323#1942—531no—
324#1942—531no—
325#1941—531no—
326#1941—531no—
327#1736—531no—
328#1736—531no—
329C2×C53C2×#38911yescharacteristic
330C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
331C2×C53C4×#13711yescharacteristic
332C2×C53#305511yescharacteristic
333C2×C53#305611yescharacteristic
334C2×C53#316511yescharacteristic
335C2×C53#316611yescharacteristic
336D53—24no—
337C22×M16—531no—
338C22×M16—531no—
339C2×#117—531no—
340C2×#117—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—1061no—
344C22×C42—531no—
345#428—538no—
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
355C4×M16—1064no—
356C2×#97—531no—
357C2×#97C2×D5311yescharacteristic
358C2×C4×C8—531no—
359C2×C4×C8—531no—
360#389—531no—
361#389—531no—
362#389—531no—
363#389—531no—
364#389—531no—
365#389—531no—
366#389—1061no—
367#389—1061no—
368#389C2×D5311yescharacteristic
369#389C2×D5311yescharacteristic
370#385—1061no—
371#385—1061no—
372C4×C4⋊C4—1061no—
373C4×C4⋊C4—1061no—
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#2115911yescharacteristic, 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—534no—
392C2×M16—534no—
393C2×M16—1062no—
394C2×M16—1062no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399#117—534no—
400#117—1062no—
401#117—1062no—
402#117—1062no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—534no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—2124no—
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—1062no—
432C2×C42—1062no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42C22×D5311yescharacteristic
436#107—534no—
437#107—1062no—
438#107—1062no—
439#107C4×D5314yescomplemented
440#106—22no—
441#106—534no—
442#106—534no—
443#106—1062no—
444#97—22no—
445#97—1062no—
446C4×C8—1062no—
447C4×C8—1062no—
448#95—21no—
449#95—21no—
450#95—1061no—
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—532no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1064no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—2121no—
480C22×C4—2121no—
481C22×C4—2122no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5311yescharacteristic
488C22×C4C23×D5311yescharacteristic
489M16—5316no—
490M16—1068no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—532no—
494C2×C8—532no—
495C2×C8—532no—
496C2×C8—532no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8C2×C4×D5312yes—
500C2×C8D4×D5312yes—
501C2×C8Q8×D5312yes—
502C2×C8Q8×D5312yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—532no—
512C4⋊C4—532no—
513C4⋊C4—1064no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4C2×C4×D5312yes—
528C4⋊C4C2×C4×D5312yes—
529C22⋊C4—1064no—
530C22⋊C4—2122no—
531C42—21no—
532C42—21no—
533C42—1062no—
534C42—1062no—
535C42—1064no—
536C42—1061no—
537C42—1061no—
538C42—2122no—
539C23—532no—
540C23—532no—
541C23—532no—
542C23—1061no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23C2×D4×D5311yescharacteristic
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
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—41no—
564C2×C4—41no—
565C2×C4—532no—
566C2×C4—532no—
567C2×C4—532no—
568C2×C4—534no—
569C2×C4—534no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—532no—
573C2×C4—532no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—4244no—
581C2×C4—2124no—
582C2×C4—2124no—
583C2×C4—2128no—
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×C4—2121no—
593C2×C4—2121no—
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—534no—
602C8—1062no—
603C8C4⋊C4×D5314yescomplemented
604C22—534no—
605C22—1062no—
606C22—1062no—
607C22—1062no—
608C22—1062no—
609C22—1062no—
610C22—1062no—
611C22C4×D4×D5311yescharacteristic
612C22#121×D5311yescharacteristic
613C22C4⋊Q8×D5311yescharacteristic
614C22AGL(1,ℤ/8)×D5311yescharacteristic
615C22#137×D5311yescharacteristic
616C22#1681141911yescharacteristic
617C22#1681142211yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—534no—
625C4—1062no—
626C4—1062no—
627C4—4241no—
628C4—4241no—
629C4—2122no—
630C4—2122no—
631C4C2×C4⋊C4×D5311yescharacteristic
632C4C4×Q8×D5311yescharacteristic
633C4C22⋊Q8×D5311yescharacteristic
634C4#125×D5311yescharacteristic
635C2—1064no—
636C2#389×D5311yescharacteristic
637C2#2028352611yescharacteristic
638C2#2028353111yescharacteristic
639C2#2028432911yescharacteristic
640C2#2028434011yescharacteristic
641C2#2028526011yescharacteristic
642C2#2028526611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1943C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1943D53
hypocenterC53#21159
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
6. #20275458C2semidirect
7. #20275455C2semidirect
8. #20275450C2semidirect
9. #20275449C2semidirect
10. #20275447C2semidirect
11. #20275442C2semidirect
12. #20275441C2semidirect
15. C53×#1943C2semidirect
24. #16810664C4semidirect
107. #13133150C2×C4semidirect
108. #13133149C2×C4semidirect
110. #13133148C2×C4semidirect
129. #107×C53C2×C4semidirect
216. #11719912C4⋊C4semidirect
279. C8×C53C2×C4⋊C4semidirect
281. #88322C2×C4⋊C4semidirect
322. #1943D53semidirect
383. C53#21159semidirect
439. #107C4×D53semidirect
603. C8C4⋊C4×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×#168105752
#202771962
#389×D532
#202754582
#202754552
#202754502
#202754492
#202754472
#202754422
#202754412
#202744332
C53×#19432
#2115953