Finite Group Databasebeta

← #27417651

Subgroups — #27417651

Order 13568.

1354 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C211yesmaximal, characteristic, complemented
2C22×#13133150C211yesmaximal, characteristic, complemented
3#20279023C211yesmaximal, characteristic, complemented
4#20276040C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275144C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic, complemented
8#20274962C212yesmaximal, complemented
9#20274903C212yesmaximal, complemented
10#20274801C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C2×#13133198—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15#16810934—24no—
16C2×#13133150—24no—
17C2×#13133150C2212yes—
18C2×#13133150C2212yes—
19C2×#13133150C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2212yes—
24#16810553—21no—
25#16810548—21no—
26#16810521C2212yes—
27#16810484—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C22×#41×C53—21no—
35C53×#547—24no—
36C53×#547—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322—24no—
49C22×#88322—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133198—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×#11719919—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61#13133150—24no—
62#13133150—24no—
63#13133150—24no—
64#13133138—24no—
65#13133138—24no—
66C22⋊#88322—24no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—21no—
70#13133124—22no—
71C2×#41×C53—28no—
72C2×#41×C53—44no—
73C2×#41×C53—44no—
74C22×Q8×C53—21no—
75C22×D4×C53—21no—
76C22×D4×C53—21no—
77C22×D4×C53D411yescharacteristic
78C22×D4×C53C2311yescharacteristic
79C23×C4×C53—21no—
80C23×C4×C53—21no—
81C23×C4×C53—21no—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C4⋊Q8×C53—22no—
85C4⋊Q8×C53—22no—
86#123×C53—28no—
87#123×C53—28no—
88C22⋊Q8×C53—28no—
89#121×C53—22no—
90#121×C53—22no—
91#121×C53—22no—
92#121×C53D414yes—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D414yes—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C24×C53—21no—
115C24×C53C2×D411yescharacteristic
116C24×C53C2×D411yescharacteristic
117#41×C53—432no—
118C2×Q8×C53—24no—
119C2×Q8×C53—42no—
120C2×Q8×C53—42no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—24no—
124C2×D4×C53—24no—
125C2×D4×C53—21no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—42no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53—42no—
132C2×D4×C53D814yes—
133C2×D4×C53C2×D412yes—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53—42no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D412yes—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C2×D412yes—
172C4⋊C4×C53C2×D412yes—
173C4⋊C4×C53C2×D412yes—
174C22⋊C4×C53—24no—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—24no—
178C22⋊C4×C53—22no—
179C42×C53—21no—
180C42×C53—21no—
181C22⋊Dic53—44no—
182C22⋊Dic53—42no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—42no—
188#11719919—44no—
189C2×#88322—28no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322#4114yescomplemented
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—21no—
199C23×C53—22no—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—42no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—42no—
207C23×C53#12012yes—
208C23×C53C2×D812yes—
209C23×C53C2×D812yes—
210C23×C53AGL(1,ℤ/8)12yes—
211C23×C53C22×D411yescharacteristic
212Q8×C53—48no—
213D4×C53—28no—
214D4×C53—24no—
215D4×C53—48no—
216D4×C53—48no—
217D4×C53—41no—
218D4×C53—41no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—48no—
233C2×C4×C53—48no—
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—48no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53#12012yes—
245C2×C4×C53C2×D812yes—
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C2×#4112yes—
249C2×C4×C532−1+412yes—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—84no—
257#88322—24no—
258#88322—42no—
259#24088—531nomaximal, Sylow
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—24no—
265C22×C53—22no—
266C22×C53—42no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—82no—
273C22×C53—82no—
274C22×C53#44714yes—
275C22×C53C2×#12011yescharacteristic
276C22×C53#54712yes—
277C22×C53C22×D811yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57812yes—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—41no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—41no—
287C4×C53#52511yescharacteristic
288C4×C53#53611yescharacteristic
289C4×C53#54712yes—
290Dic53—82no—
291Dic53—82no—
292C2×#547D5311yescharacteristic
293C2×#480—531no—
294C22×#107—531no—
295#2486—531no—
296#2057—532no—
297#2033—532no—
298#1893—531no—
299#1869—531no—
300#1862—532no—
301#1852—532no—
302#1823—531no—
303C2×C53—24no—
304C2×C53—42no—
305C2×C53—82no—
306C2×C53#240611yescharacteristic
307C2×C53C2×#44711yescharacteristic
308C2×C53#301111yescharacteristic
309C2×C53#333412yes—
310C2×C53#334312yes—
311C22×#41—21no—
312C23×C8—531no—
313#547—24no—
314#547—24no—
315C2×C4⋊Q8C2×D5311yescharacteristic
316C2×#123—21no—
317C2×#123—21no—
318C2×#123—1061no—
319C2×C22⋊Q8—21no—
320C2×#121C2×D5311yescharacteristic
321C2×#121C2×D5311yescharacteristic
322C2×C4×D4C2×D5311yescharacteristic
323C2×C4×D4C2×D5311yescharacteristic
324C22×C4⋊C4—531no—
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C4⋊C4C2×D5311yescharacteristic
327#480—1064no—
328C2×#107—532no—
329C2×#107—532no—
330C2×#107—532no—
331C2×#107—1064no—
332C2×#102—532no—
333C2×#102—532no—
334C2×C22⋊C8—531no—
335C2×C22⋊C8—532no—
336#398—1061no—
337#397—1061no—
338#389—532no—
339#381—1061no—
340#380—531no—
341#380—532no—
342#340—532no—
343#340—532no—
344#340—532no—
345#336—532no—
346C53#2408811yescharacteristic, complemented, Sylow
347C2×#41—28no—
348C2×#41—44no—
349C2×#41—44no—
350C22×Q8—21no—
351C22×D4—21no—
352C22×D4—21no—
353C22×D4C22⋊D5311yescharacteristic
354C22×D4C22×D5311yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360C23×C4C22×D5311yescharacteristic
361C22×C8—532no—
362C22×C8—532no—
363C22×C8—1064no—
364C22×C8—1062no—
365C4⋊Q8—22no—
366C4⋊Q8—22no—
367#123—28no—
368#123—28no—
369#123—2124no—
370C22⋊Q8—28no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121C22⋊D5314yes—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—532no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22⋊D5314yes—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4C22⋊D5311yescharacteristic
402C2×C22⋊C4C22⋊D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404#107—1064no—
405#107—1064no—
406#107—1064no—
407#102—1064no—
408#102—1064no—
409C22⋊C8—1064no—
410C22⋊C8—1062no—
411#95—1062no—
412#95—1061no—
413#95—1062no—
414C24—21no—
415C24C2×C22⋊D5311yescharacteristic
416C24C2×C22⋊D5311yescharacteristic
417#41—432no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4#1171992214yes—
433C2×D4C2×C22⋊D5312yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5312yes—
461C22×C4C2×C22⋊D5312yes—
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464C2×C8—534no—
465C2×C8—1068no—
466C2×C8—1062no—
467C2×C8—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C2×C8—1062no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—1062no—
482C4⋊C4—2122no—
483C4⋊C4—2124no—
484C4⋊C4C2×C22⋊D5312yes—
485C4⋊C4C2×C22⋊D5312yes—
486C4⋊C4C2×C22⋊D5312yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—2122no—
493C22⋊C4—2124no—
494C42—21no—
495C42—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23#1313318612yes—
508C23C2×#1171992212yes—
509C23C2×#1171992212yes—
510C23#1313324312yes—
511C23C22×C22⋊D5311yescharacteristic
512Q8—48no—
513D4—28no—
514D4—24no—
515D4—48no—
516D4—48no—
517D4—41no—
518D4—41no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—4244no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4#1313318612yes—
552C2×C4C2×#1171992212yes—
553C2×C4C22×C22⋊D5311yescharacteristic
554C2×C4C22×C22⋊D5311yescharacteristic
555C2×C4#41×D5312yes—
556C2×C4#1313326412yes—
557C8—1064no—
558C8—2122no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22#1681075714yes—
574C22C2×#1313318611yescharacteristic
575C22#1681127212yes—
576C22C22×#1171992211yescharacteristic
577C22C2×#1313324311yescharacteristic
578C22#1681143812yes—
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—4242no—
587C4—4242no—
588C4#1681113011yescharacteristic
589C4#1681119411yescharacteristic
590C4#1681127212yes—
591C2—24no—
592C2—42no—
593C2—82no—
594C2#2027804611yescharacteristic
595C2C2×#1681075711yescharacteristic
596C2#2028401011yescharacteristic
597C2#2028705812yes—
598C2#2028714912yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#24088
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810934C2semidirect
2. C22×#13133150C2semidirect
3. #20279023C2semidirect
4. #20276040C2semidirect
5. #20275894C2semidirect
6. #20275144C2semidirect
7. #20275008C2semidirect
8. #20274962C2semidirect
9. #20274903C2semidirect
10. #20274801C2semidirect
195. C2×#88322#41semidirect
346. C53#24088semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
C22×#131331502
#202790232
#202760402
#202758942
#202751442
#202750082
#202749622
#202749032
#202748012
C2×C53×#5472
#2408853